Course Notes Math 275 Boise State University. Shari Ultman

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1 Course Notes Math 275 Boise State University Shari Ultman Fall 2017

2 Contents 1 Vectors Introduction to 3-Space & Vectors Working With Vectors Introduction to the Dot and Cross Products: Computation Geometry of the Dot and Cross Products Applications of the Dot Product: Projections and Work Applications of Vectors: Lines and Planes Introduction to Vector Functions: Curves in R 2 & R Derivatives of Vector Functions: Computation & Geometry Applications of Vector Derivatives: Motion Along Curves Partial Derivatives Introduction to Multivariate Functions Partial Derivatives Tangent Planes and Linear Approximations Applications of Differentiability: the Differential df The Gradient f and the Directional Derivative Dûf Chain Rules for Functions of Two or More Variables Optimization: Second Derivative Test Optimization: Lagrange Multipliers Multiple Integrals Introduction to Double Integrals (Cartesian Coordinates) Double Integrals in Polar Coordinates Triple Integrals in Cartesian Coordinates Triple Integrals in Cylindrical Coordinates Triple Integrals in Spherical Coordinates Vector Calculus Line Element and Scalar Line Integrals Introduction to Vector Fields Vector Line Integrals Conservative Vector Fields Green s Theorem i

3 CONTENTS ii 4.6 Surfaces and Surface Elements Scalar Surface Integrals Vector Surface Integrals (aka: Flux Integrals) Divergence & the Divergence Theorem Curl & Stokes Theorem

4 Chapter 1: Vectors Definitions SCALAR For this class, scalars are real numbers, c R. VECTOR Vectors are mathematical objects that encode two pieces of information: magnitude (length) and direction. For vectors in two and three dimensions, it can be useful to represent vectors as arrows. A unit vector is a vector whose magnitude (length) equals one unit. POINT A point is a location in space. Numerically, a point is specified by its coordinates. The number of coordinates needed to specify a point is the same as the dimension of the space; two coordinates for a point in the plane (R 2 ), three coordinates for a point in 3-space (R 3 ), etc. VECTOR ADDITION Two vectors of the same dimension (having the same number of components) can be added to produce a new vector. A vector can be multiplied by a scalar to pro- SCALAR MULTIPLICATION duce a new vector. DOT PRODUCT The dot product of two vectors of the same dimension is an operation that produces a scalar. The dot product can be used to find the angle between two vectors, and the magnitude of a vector. ORTHOGONAL VECTORS Two vectors are orthogonal if their dot product is equal to zero. If two vectors are orthogonal, then either they are perpendicular, or else one of the vectors was the zero vector. CROSS PRODUCT The cross product of two 3-dimensional vectors is an operation that produces a 3-dimensional vector. This vector is orthogonal to both input vectors, and its magnitude is the area of the parallelogram spanned by the input vectors. VECTOR FUNCTION A vector function is a function whose output is a vector. One application of vector functions is to describe the path traveled by an object as it moves through space. 1

5 2 Notation You will very likely see different notation used online or in other classes. SCALARS numbers or letters (usually lower-case) 1, 5, π, c, t, λ VECTORS symbolic: boldface or arrow over letter v, A, 0, or v, A, 0 component form in R 2 component form in R 3 a, b or a î + b ĵ a, b, c or a î + b ĵ + c ˆk POINTS symbolic: letters (usually upper-case) P, Q, A, O coordinate form in R 2 (a, b) coordinate form in R 3 (a, b, c) MAGNITUDE vertical bars around a vector v, v, or a, b, c UNIT VECTORS unit vectors wear a hat ˆv, û, î, ĵ, ˆk VECTOR ADDITION SCALAR MULTIPLICATION DOT PRODUCT v + w cv v w CROSS PRODUCT v w VECTOR FUNCTIONS in R 2 r(t) = x(t), y(t) in R 3 r(t) = x(t), y(t), z(t) Basis vector notation î, ĵ, ˆk may also be used instead of angle bracket notation: r(t) = x(t) î + y(t) ĵ or r(t) = x(t) î + y(t) ĵ + z(t) ˆk Important note for this class: Points are denoted by parentheses (...). Vectors are denoted by angle brackets... or basis vectors î, ĵ, ˆk.

6 1.1. INTRODUCTION TO 3-SPACE & VECTORS Introduction to 3-Space & Vectors Textbook Sections: 12.1, 12.2 The Big Picture Single variable calculus (Calc I and II) deals with one-dimensional problems. In multivariable and vector calculus (Calc III), we will learn how to work on problems in higher-dimensional space; primarily, the plane (R 2 ), which is 2- dimensional, and 3-space (R 3 ), which is 3-dimensional. In these problems, we often deal with physical systems in which involve both numerical values, and directions. For example, think of a force, which has both strength and direction in which it acts; or velocity, which accounts for both speed and direction of motion. Vectors are the mathematical tools that we use to represent and analyze these situations. From the Toolbox (what you need from previous classes): Cartesian Coordinates: Locating a point in the plane (R 2 ) based on its Cartesian coordinates. Checklist: Computational Methods & Important Concepts Computational Methods Locate a point in 3-space (R 3 ) using Cartesian coordinates. Recognize whether an object is a scalar, a point, or a vector. Compute the component form of a vector given the initial (starting) and terminal (ending) points of the vector. Important Concepts A scalar is a number. A point is a location. A vector is an arrow (directed line segment) from one point to another.

7 1.1. INTRODUCTION TO 3-SPACE & VECTORS 4 Scalars, points, and vectors are different objects, and are represented using different notation. Vectors encode two pieces of information: magnitude (length) and direction. Two vectors are equivalent if they have the same direction and magnitude (geometric properties), or, equivalently, if they have the same component form (algebraic representation). More Details R 2 = R R (the plane) is 2-dimensional space. Coordinates of points in R 2 are given by ordered pairs of numbers, (x, y). R 3 = R R R (3-space) is 3-dimensional space. Coordinates of points in R 3 are given by ordered triples of numbers, (x, y, z). R n is n-dimensional space. Coordinates of points in R n are given by ordered n-tuples of numbers, (x 1, x 2,..., x n ). In this class, we will focus on 2- and 3-dimensional space. Many of the methods we use, however, can be generalized to n-dimensions. Scalars, Point, and Vectors: Scalars are numbers. In this class, scalars are the real numbers (R). In future classes, you may encounter scalars taken from other number systems for example, complex scalars. Points are locations in space. Cartesian coordinates are used to give the location of a point in terms of distances from the origin along the coordinate axes, with sign indicating whether the coordinate lies along the positive or negative side of the axis. Vectors have a magnitude (length) and a direction. A vector can be represented by an arrow (directed line segment) that starts at one point (the initial point or tail), and ends at a second point (the terminal point or head).

8 1.1. INTRODUCTION TO 3-SPACE & VECTORS 5 Points and vectors are different, so they are represented using different notation. Vectors will be represented using boldface, arrows, angle brackets, or basis vectors î, ĵ, ˆk : Vector: v = a, b, c = a î + b ĵ + c ˆk or: v = a, b, c = a î + b ĵ + c ˆk. Points will be represented without boldface or arrows, and using parentheses: Point: P = (a, b, c) or P (a, b, c). The component form of the vector P Q from the initial point P = (x 1, y 1, z 1 ) to the terminal point Q = (x 2, y 2, z 2 ) is found by subtracting the coordinates of the initial point from those of the terminal point ( head minus tail ): P Q = x 2 x 1, y 2 y 1, z 2 z 1. There are two ways of describing vectors: Geometric: Magnitude and direction are geometric properties. They can be determined through physical measurement and by examining how a vector sits in space. This is useful if you want to draw diagrams using vectors to represent physical situations, for example: force diagrams or velocity fields. Algebraic: The component form is an algebraic representation. It assigns numbers to a vector, so you can apply mathematical methods like algebra and calculus to analyze situations involving vectors. This dual nature of vectors, as objects possessing both geometric properties and algebraic representations, appears in any situation in which vectors are used. For example: Two vectors v and w are equivalent ( v = w ) when: They have the same magnitude and the same direction (geometric properties), or:

9 1.1. INTRODUCTION TO 3-SPACE & VECTORS 6 They have the same component form: v = w = a, b, c (algebraic representation). There are infinitely many equivalent vectors. At every point, there will be a vector that is equivalent to exactly one other vector at every other point. Equivalence of vectors is useful when using vectors to represent physical objects. You can sketch a diagram using vectors, and then use equivalence to move them to more convenient locations as long as you don t change their magnitude or direction. The position vector of the point P is the vector OP with initial point at the origin and terminal point at P. The position vector is commonly represented as r or s. The zero vector 0 is the vector whose components are all equal to zero. In 2-dimensions, 0 = 0, 0. In 3-dimensions, 0 = 0, 0, 0. Moving a vector without changing its magnitude or direction is called translation. Computations Component Form of a Vector Given two points P = (x 1, y 1, z 1 ) and Q = (x 2, y 2, z 2 ), the component form of the vector P Q is: P Q = x 2 x 1, y 2 y 1, z 2 z 1.

10 1.2. WORKING WITH VECTORS Working With Vectors Textbook Sections: 12.2 The Big Picture Vectors encode two pieces of information: direction and magnitude (length). The magnitude of a vector is the distance between the endpoints of the vector. Magnitude is a scalar quantity. Two vectors can be added, and a vector can be multiplied by a scalar. A vector with magnitude equal to 1 ( v = 1) is called a unit vector. Dividing a non-zero vector by its magnitude results in a unit vector that has the same direction as the original vector: ˆv = v/ v. In addition to angle bracket notation v = a, b, c, vectors can also be represented using the basis vectors î, ĵ, ˆk, which are the unit vectors that point in the directions of the x-, y-, and z-coordinate axes: v = a, b = a î + b ĵ or v = a, b, c = a î + b ĵ + c ˆk From the Toolbox (what you need from previous classes): Trigonometry: Pythagorean theorem. Polar coordinates (x, y) = (r cos θ, r sin θ). Vectors: Know what a vector is. Be able to find the component form of a vector given its initial and terminal points. Checklist: Computational Methods & Important Concepts Computational Methods Compute the magnitude of a vector. Add two vectors (vector addition). Multiply a vector by a scalar (scalar multiplication).

11 1.2. WORKING WITH VECTORS 8 Find a unit vector in the direction of a given vector. Write a vector in component form using either angle bracket notation, or the basis vectors î, ĵ, ˆk. Important Concepts The magnitude of a vector is the distance between the endpoints of the vector. This is measured using the Pythagorean theorem. Vector addition results in a vector that is the diagonal of the parallelogram whose sides are the two vectors being added (this is called the parallelogram law). Scalar multiplication changes the magnitude of a vector. If the scalar is positive, the direction stays the same. If the scalar is negative, the direction reverses. v is a unit vector if v = 1. Unit vectors are often denoted by a hat : ˆv. If v is a non-zero vector, you can find a unit vector that has the same 1 direction by multiplying the vector by the scalar v (informally, you divide the vector by its magnitude): ˆv = ( 1/ v ) v = v/ v. The basis vectors î, ĵ, ˆk are the unit vectors in the directions of the x-, y-, and z- coordinate axes. A vector can be written using either angle-brackets or basis vectors: v = a, b = a î + b ĵ or v = a, b, c = a î + b ĵ + c ˆk More Details The magnitude of a vector is the distance between its endpoints, measured using the Pythagorean theorem. Later in this class, we will use an infinitesimal (very very small) form of the Pythagorean theorem to measure distance along curves. The only vector with magnitude equal to zero is the zero vector 0.

12 1.2. WORKING WITH VECTORS 9 Vector addition can be thought of in two ways: Algebraic: Add the components of two vectors to get a new vector. Geometric: Parallelogram law : Draw the parallelogram spanned by two vectors. The sum of the two vectors is the diagonal of this parallelogram. Scalar multiplication can be thought of in two ways: Algebraic: Multiply each component of a vector by a scalar. Geometric: Stretch or shrink the magnitude the vector by the absolute value of the scalar: cv = c v. If the scalar c is positive, then cv and v have the same direction. If c is negative, then cv and v have opposite directions. Scalar multiplication results in vectors that have either the same or opposite directions. So two (non-zero) vectors are parallel if and only if they are scalar multiples of each other. That is, v and w are parallel if and only if v = cw for some real number c). A linear combination of two vectors is a new vector that is a sum of scalar multiples of the two vectors: if u = av + bw, then u is a linear combination of v and w. Unit vectors are important when you want to focus on direction. For example: Basis Vectors: The set of vectors { î, ĵ, ˆk } 1 are unit vectors that are parallel to the x, y, and z-coordinate axes: In R 2 : î = 1, 0 In R 3 : î = 1, 0, 0 ĵ = 0, 1 ĵ = 0, 1, 0 ˆk = 0, 0, 1 These can be used to write a vector in terms of components parallel to the coordinate axes: v = a, b, c = a î + b ĵ + c ˆk. 1 Historical note: The vectors { î, ĵ, ˆk } have to do with a type of number called the quaternions These are related to complex numbers, with three imaginary axes.

13 1.2. WORKING WITH VECTORS 10 You may also see { î, ĵ, ˆk } written using other notation, for example: {ê 1, ê 2, ê 3 }, or {ˆx, ŷ, ẑ}. Direction/Magnitude Decomposition of a Vector: If v 0, the unit vector in the direction of v is ˆv = v/ v. It is sometimes useful to represent the vector v in terms of ˆv and the magnitude of v: v = v ˆv. More on basis vectors: The set of vectors { î, ĵ, ˆk }. is an example of a basis. A basis is a set of vectors that can be used to represent any n-dimensional vector of a given dimension as a linear combination of a set of n fixed vectors. There are many different sets of basis vectors. For example, two of the infinitely many bases in R 2 are the the Cartesian basis { î, ĵ } and the polar basis {ˆr, ˆθ }. There are infinitely many bases in R 3, including the Cartesian basis { î, ĵ, ˆk }, the cylindrical basis {ˆr, ˆθ, ẑ }, and the spherical basis { ˆρ, ˆφ, ˆθ }. Computations Magnitude of a Vector: In R 2 : v = v 1, v 2 In R 3 : v = v 1, v 2, v 3 In R n : v = v 1,..., v n v = v1 2 + v 2 2 v = v1 2 + v v 3 2 v = v v n 2 Vector Addition: In R 2 : v = v 1, v 2, w = w1, v w v + w = v 1 + w 1, v 2 + w 2 In R 3 : v = v 1, v 2, v 3, w = w1, w 2, w 3 v + w = v 1 + w 1, v 2 + w 2, v 3 + w 3 In R n : v = v 1,..., v n, w = w1,..., w n v + w = v1 + w 1,..., v n + w n

14 1.2. WORKING WITH VECTORS 11 Scalar Multiplication: In R 2 : c R, v = v 1, v 2 In R 3 : c R, v = v 1, v 2, v 3 In R n : c R, v = v 1,..., v n cv = cv 1, cv 2 cv = cv 1, cv 2, cv 3 cv = cv 1,..., cv n Unit Vectors: In R 2 : v = v 1, v 2 In R 3 : v = v 1, v 2, v 3 In R n : v = v 1,..., v n ˆv = v/ v = v 1 / v, v 2 / v ˆv = v/ v = v 1 / v, v 2 / v, v 3 / v ˆv = v/ v = v 1 / v,..., v n / v

15 1.3. INTRODUCTION TO THE DOT AND CROSS PRODUCTS: COMPUTATION Introduction to the Dot and Cross Products: Computation Textbook Sections: 12.3, 12.4 The Big Picture The dot and cross product are important vector operations that will be used extensively throughout this course and others. They are used to analyze geometric relationships between vectors. For example, the dot product can be used to measure the angle between two vectors, while the cross product of two vectors is a vector that is normal (perpendicular) to the plane containing the two vectors, and whose magnitude is the area of the parallelogram spanned by the two vectors. We will explore the geometric properties of the dot and cross product in the next section. For now, we will focus only on computational aspects and the definitions of the dot and cross products. The dot product of two vectors is a scalar. The cross product of two vectors is a vector. Both dot and cross products can be defined in two ways. The algebraic definitions depend only on the components of the two input vectors. The geometric definitions depend on the magnitudes of the two input vectors, and the angle between them. Computationally, either definition will result in the same answer. Practically, which one to use depends on the information you are given. From the Toolbox (what you need from previous classes): Trigonometry: Sine and cosine functions. Vectors: Know what a vector is. Be able to compute the magnitude of a vector.

16 1.3. INTRODUCTION TO THE DOT AND CROSS PRODUCTS: COMPUTATION 13 Checklist: Computational Methods & Important Concepts Computational Methods Compute the dot product of two vectors, using either the algebraic definition or the geometric definition. Compute the cross product of two vectors, using either the algebraic definition or the geometric definition. Important Concepts The dot product of two vectors can be defined in two ways: in terms of the components of the vectors (the algebraic definition), or in terms of the magnitudes of the vectors and the cosine of the angle between them (the geometric definition). The cross product of two vectors can be defined in two ways: in terms of the components of the vectors (the algebraic definition), or in terms of the magnitudes of the vectors and the sine of the angle between them (the geometric definition), and a unit vector giving the direction. The dot product of two vectors is a scalar. The cross product of two vectors is a vector. More Details The dot product is a vector operation that can be applied to any pair of vectors v and w, as long as they both have the same dimension (that is, the same number of components). The dot product of two vectors can be defined in two ways. Algebraic: Multiply the corresponding components of each vector and add the products. For example, if v = v 1, v 2 and w = w1, w 2, then: v w = v 1, v 2 w1, w 2 = v1 w 1 + v 2 w 2. Geometric: Multiply the magnitudes of the vectors by the cosine of

17 1.3. INTRODUCTION TO THE DOT AND CROSS PRODUCTS: COMPUTATION 14 the angle θ between them: v w = v w cos θ. The angle theta lies in the plane spanned by v and w, and is the smaller of the two angles between the vectors (0 θ π). These two definitions are equivalent. In theory, they will both give you the same answer. In practice, however, one of the definitions will usually work better for a given problem. In order to determine which definition you should use (algebraic or geometric) will depend on your analysis of the information included in the problem. The cross product is a vector operation that can be applied to pairs of vectors v = v 1, v 2, v 3 and w = w1, w 2, w 3. Note that the vectors must be 3-dimensional (have three components). The cross product can also be defined in two ways: Algebraic: v w = (v 2 w 3 v 3 w 2 ) î (v 1 w 3 v 3 w 1 ) ĵ + (v 1 w 2 v 2 w 1 ) ˆk. Geometric: v w = ( v w sin θ ) ˆn. Where θ is the same angle between v and w used in the dot product, v w sin θ = v w, and ˆn is the unit normal vector perpendicular to both v and w, with direction determined by the right-hand-rule. These two definitions are equivalent; they will both give you the same answer. As with the dot product, however, it is usually the case that one of the definitions will work better for a particular problem. Technical note: the geometric definitions of the dot and cross products given above do not work when v = 0 and/or w = 0, since you cannot determine the angle θ. Also, and when v and w are parallel, you cannot determine a normal vector ˆn for the geometric definition of the cross product. So, if v = 0 and/or w = 0, then v w = 0 and v w = 0, and if v and w are parallel, then v u = 0.

18 1.3. INTRODUCTION TO THE DOT AND CROSS PRODUCTS: COMPUTATION 15 The dot and cross product satisfy algebraic properties that are similar to the algebraic properties satisfied by the real numbers: Algebraic properties of the dot product: u v = v u u (v + w) = u v + u w (cu) v = u (cv) = c(u v) 0 v = 0 u u = u 2 commutativity distributive law for any scalar c Algebraic properties of the cross product: u v = v u u (v + w) = u v + u w (cu) v = u (cv) = c(u v) 0 v = v 0 = 0 v v = 0 anti-commutativity distributive law for any scalar c Computations Dot Product (Algebraic Definition) In R 2 : v = v 1, v 2, w = w1, v w In R 3 : v = v 1, v 2, v 3, w = w1, w 2, w 3 v w = v 1 w 1 + v 2 w 2 v w = v 1 w 1 + v 2 w 2 + v 3 w 3 In R n : v = v 1,..., v n, w = w1,..., w n v w = v1 w v n w n Dot Product (Geometric Definition) For two (non-zero) vectors v and w with the same dimension: v w = v w cos θ The angle theta lies in the plane spanned by v and w, and is the smaller of the two angles between the vectors (0 θ π).

19 1.3. INTRODUCTION TO THE DOT AND CROSS PRODUCTS: COMPUTATION 16 Cross Product (Algebraic Definition) Method 1: Determinant Method The determinant of a 2 2 matrix is: a b c d = ad bc. So, for a pair of 3-dimensional vectors v = v 1 î + v 2 ĵ + v 3 ˆk and w = w 1 î + w 2 ĵ + w 3 ˆk : î ĵ ˆk v w = v 1 v 2 v 3 w 1 w 2 w 3 = v 2 v 3 w 2 w 3 î v 1 v 3 w 1 w 3 ĵ + v 1 v 2 w 1 w 2 ˆk = (v 2 w 3 v 3 w 2 ) î (v 1 w 3 v 3 w 1 ) ĵ + (v 1 w 2 v 2 w 1 ) ˆk. Cross Product (Algebraic Definition) Method 2: Multiplication Method The basis vectors î, ĵ, ˆk can be multiplied individually: î ˆk î ĵ = ˆk ĵ î = ˆk ĵ ˆk = î ˆk ĵ = î ĵ ˆk î = ĵ î ˆk = ĵ î î = ĵ ĵ = ˆk ˆk = 0 Combine this with the algebraic properties of the cross product to compute the cross product of two vectors (this is similar to multiplying polynomials).

20 1.3. INTRODUCTION TO THE DOT AND CROSS PRODUCTS: COMPUTATION 17 Example: v = 2 î + 3 ĵ + ˆk, w = 4 î + 5 ˆk v w = ( 2 î + 3 ĵ + ˆk ) ( 4 î + 5 ˆk ) = ( 2 î 4 î ) + ( 2 î 5 ˆk ) + ( 3 ĵ 4 î ) + ( 3 ĵ 5 ˆk ) + ( ˆk 4 î ) + ( ˆk + 5 ˆk ) = 8 ( î î ) + 10 ( î ˆk ) + 12 ( ĵ î ) + 15 ( ĵ ˆk ) + 4 ( ˆk î ) + 5 ( ˆk ˆk ) = 8 ( 0 ) + 10 ( ĵ ) + 12 ( ˆk ) + 15 ( î ) + 4 ( ĵ ) + 5 ( 0 ) = 15 î 6 ĵ 12 ˆk Cross Product (Geometric Definition) For a pair of (non-zero) 3-dimensional vectors v and w, the magnitude of the cross product v w is: v w = v w sin θ The angle theta lies in the plane spanned by v and w, and is the smaller of the two angles between the vectors (0 θ π). The direction of the cross product v w (for non-zero, non-parallel vectors) is given by the unit vector ˆn, which is normal (perpendicular) to the plane spanned by v and w, and follows the right-hand rule.

21 1.4. GEOMETRY OF THE DOT AND CROSS PRODUCTS Geometry of the Dot and Cross Products Textbook Sections: 12.3, 12.4 The Big Picture The the dot and cross products can be used to tell us about the geometric relationship between vectors. The dot product tells us about angles. The cross product tells us about area, and can be used to produce a vector that is orthogonal (perpendicular) to two vectors. The geometric definitions of the dot and cross products the definitions involving the magnitudes and angle between the vectors reveal the geometric properties of these vector operations. When it come time to compute things, however, it is usually the algebraic definitions that are used. From the Toolbox (what you need from previous classes): Algebra & Trigonometry: be able to solve quadratic equations, know the cosine and sine, and inverse cosine functions. Vectors: angle-bracket and î, ĵ, ˆk notation; finding the components of a vector given initial and terminal points; computing vector magnitude; computing the dot and cross products of two vectors. Checklist: Computational Methods & Important Concepts Computational Methods Use the dot product to compute the angle between two vectors and determine when two vectors are orthogonal (perpendicular). Use the cross product to find a vector that is orthogonal to both u and v. Using the magnitude of the cross product to compute the area of parallelograms and triangles.

22 1.4. GEOMETRY OF THE DOT AND CROSS PRODUCTS 19 Important Concepts The dot product can be used to find the angle between two vectors. If v w = 0, we say that v and w are orthogonal. This happens when either: v and w are perpendicular, or: Either v or w (or both) equal 0 (the zero vector). Since the cross product v w is a vector, it has both direction and magnitude. Both have important geometric properties: Direction: v w is orthogonal to both v and w. When v and w are both non-zero, and are not parallel, this means v w is normal to the plane spanned by v and w. Magnitude: v w is the area of the parallelogram having sides v and w. More Details Recall the geometric definition of the dot product: v w = v w cos θ. The magnitudes v and w cannot be negative, so the sign of the dot product v w depends only on cos θ. Since cos θ is positive for 0 θ < π/2 and cos θ is negative for π/2 < θ π, the sign of the dot product v w tells you whether the angle between two vectors is acute or obtuse. Two vectors v and w are orthogonal when v w = 0. This happens when two vectors are perpendicular, since cos(π/2) = 0. But the dot product will also equal zero if one (or both) of the vectors is the zero vector, since 0 = 0. So, for most pairs of vectors, being orthogonal is the same as being perpendicular, but orthogonality also includes the extra case of one (or both) of the vectors being 0.

23 1.4. GEOMETRY OF THE DOT AND CROSS PRODUCTS 20 Recall the geometric definition of the cross product: v w = ( v w sin θ ) ˆn. Magnitude: You may recall from earlier math classes that the area of a parallelogram with sides of length a and b, separated by the angle θ is ab sin θ. So the magnitude of the cross product v w = v w sin θ is the area of the parallelogram spanned by v and w. Direction: The cross product v w is orthogonal to both v and w. This means, if v and w are non-zero, non-parallel vectors, then v w is normal to the plane spanned by v and w. (Normal to a plane means orthogonal to every vector in the plane. So a non-zero normal vector of a plane is perpendicular to the plane.) If v w = 0, then either: v and w are parallel (θ = 0 or θ = π), or either v or w (or both) is 0 (the zero vector). Computations Angle Between Vectors (Dot Product) If v and w ( have ) the same dimension, the angle θ between v and w is θ = arccos. v w v w Test for Orthogonality (Dot Product) If v and w have the same dimension, they are orthogonal if and only if v w = 0. Area of Parallelogram (Cross Product Magnitude) If v and w are 3-dimensional vectors, the area of the parallelogram with sides v and w is Area = v w. Producing Orthogonal Vectors (Cross Product) If v and w are non-zero, non-parallel vectors, a vector orthogonal to both v and w is v w. This vector is also normal to the plane spanned by v and w.

24 1.5. APPLICATIONS OF THE DOT PRODUCT: PROJECTIONS AND WORK Applications of the Dot Product: Projections and Work Textbook Section: 12.3 The Big Picture A vector projection proj u v can be thought of as the shadow cast by a vector v along the line spanned by a second vector u. The magnitude of the scalar projection scal u v is the length of the shadow. The sign of the scalar projection indicates whether the direction of the vector projection is the same as or opposite to the vector u. The orthogonal decomposition of the vector v with respect to a non-zero vector u is a way of writing v as the sum of two orthogonal (perpendicular) vectors, one of which is parallel to u. From the Toolbox (what you need from previous classes): Trigonometry: Polar coordinates (x, y) = (r cos θ, r sin θ). Vectors: Dot product (computation, and algebraic and geometric definitions. Computing the magnitude of a vector. Unit vectors ˆv = v/ v. Checklist: Computational Methods & Important Concepts Computational Methods Compute scalar and vector projections, Compute the work performed by a constant force as it displaces an object over a straight path. Important Concepts Projections are the shadow of one vector along the line spanned by a second vector.

25 1.5. APPLICATIONS OF THE DOT PRODUCT: PROJECTIONS AND WORK 22 The vector projection of v onto u (proj u v) is a vector that is parallel to u. The scalar projection of v onto u (comp u v) is a scalar whose absolute value is the magnitude of the vector projection, and whose sign reflects whether the vector projection of v onto u points in the same direction as u, or the opposite direction as u. The work performed by a constant force F acting along a straight line displacement D can be computed using the dot product: W = F D. The work performed against (or in the presence of ) a field has the opposite sign: W = F D. More Details The (vector) projection proj u v of a vector v along a (non-zero) vector u is the vector parallel to u that follows the shadow cast by v onto the line spanned by u. The scalar component comp u v of the projection is a scalar value that is related to the projection of v along u. The absolute value of comp u v gives the length of the shadow of v along u (the magnitude proj u v). The sign of comp u v indicates whether the projection of v points in the same direction as u, or the opposite direction: if comp u v > 0, the projection and u point in the same direction; if comp u v < 0, the projection and u have opposite directions.) The formula for the scalar component of the projection comes from trigonometry. If θ is the angle from u to v, then the shadow of v along u is: comp u v = v cos θ = v û cos θ (since û = 1) = v û (geometric definition of dot product) = v u/ u (û = u/ u ) = v u u

26 1.5. APPLICATIONS OF THE DOT PRODUCT: PROJECTIONS AND WORK 23 The formula for the vector projection comes from writing the vector projection as the scalar projection times the unit vector û: proj u v = ( comp u v ) û ( ) v u = û u ( ) v u u = u u = v u u u 2 If the scalar projection comp u v is positive, then u and proj u v have the same direction. If comp u v is negative, then u and proj u v have opposite directions. Work: Suppose F is a constant force acting on an object, displacing it along a straight path. Let D be the displacement vector (the force moves the object along D), with magnitude D = D (the distance the object is moved by the force). The component of force that acts on the object is the part of the force that is parallel to the displacement that is, the scalar projection comp D F. Since work = force distance: W = ( comp D F )( D ) ( ) F D ( D ) = D = F D The work performed against (or in the presence of ) a field F has the opposite sign as the work performed by the field, or: W = F D. (Optional) The orthogonal decomposition of a vector v with respect to a (non-zero) vector u writes the vector v as the sum of its projection onto u, which is parallel to u, and the normal component v, which is a vector orthogonal to u. So: v = proj u v + v. The orthogonal decomposition is used to break down a vector into two orthogonal components, one of which is parallel to another vector. For

27 1.5. APPLICATIONS OF THE DOT PRODUCT: PROJECTIONS AND WORK 24 example, decomposing a force F into its parallel and normal components, relative to a given direction. Or, examining the tangential (parallel) and normal components of acceleration relative to velocity. Computations (Vector) Projection Projection of the vector v along u, use the formula: proj u v = v u u 2 u. Scalar Component of (Vector) Projection Scalar component of the projection of v along u: comp u v = v u u. Work Work performed by a force F over a displacement D, use the formula: W = F D. Work performed against (or in the presence of ) a force F over a displacement D: W = F D. Normal Component of Orthogonal Decomposition (Optional:) To compute the normal component v in an orthogonal decomposition v = proj u v + v : v = v proj u v.

28 1.6. APPLICATIONS OF VECTORS: LINES AND PLANES Applications of Vectors: Lines and Planes Textbook Section: 12.5 The Big Picture Lines: The familiar equation y = mx + b represents a line in the plane with slope m and y-intercept b. We will now learn a new way of representing lines using a vector equation: r(t) = tv + r 0 v is a vector that gives the direction of the line; similar to the slope m). r 0 is the position vector of a point on the line; similar to the y- intercept b. t, called the parameter, is the independent variable. Planes: A plane ax + by + cy = d in R 3 can also be represented using a vector equation: n P 0 P = 0 n = a, b, c is normal to the plane. P 0 = (x 0, y 0, z 0 ) is a point on the plane. P 0 P = x x 0, y y 0, z z 0 is the vector from the fixed point P 0 to an arbitrary point P = (x, y, z). Lines and Planes: There are a few similarities between the vector equations of lines and planes. Both equations require a point and a vector. The point is a point on the line or the plane. The vector is used to describe the orientation of the line or plane that is, how it is positioned in space. For a line, this directional vector is parallel to the line. For a plane, the directional vector is normal to the plane.

29 1.6. APPLICATIONS OF VECTORS: LINES AND PLANES 26 Parallel lines have parallel directional vectors. Parallel planes have parallel normal vectors. From the Toolbox (what you need from previous classes): Points and Vectors: Plotting points, sketching vectors, finding the component form a vector given two points. Vector operations: Vector addition, scalar multiplication, dot product, cross product. Using the dot product to test for orthogonality of vectors. Using the cross product to produce a normal vector. Checklist: Computational Methods & Important Concepts Computational Methods Lines: Find a vector equation r(t) = tv + r 0 and parametric equations for a line, given a fixed direction vector v and a fixed point P 0 on the line. Find a direction vector v for a line given a vector equation or parametric equations for the line. In two dimensions, find a direction vector using the graph of a line. Use the direction vectors of two lines to determine whether or not the lines are parallel. Planes: Find a vector equation n P 0 P = 0 of a plane, given a normal vector n and a point P 0 on the. Use the vector equation to find the algebraic equation ax + by + cz = d. Find a vector and/or algebraic equation of a plane given three points in the plane, using the cross product to produce a normal vector. Use the algebraic equation ax +by +cz = d of a plane to find a normal vector to the plane.

30 1.6. APPLICATIONS OF VECTORS: LINES AND PLANES 27 Use the normal vectors of two planes to determine whether or not the planes are parallel. Important Concepts Lines: To find the vector equation of a line, r(t) = tv + r 0, you need two pieces of information: a vector v giving the direction of the line, and the coordinates of a point P 0 on the line. The vector r 0 is the position vector of the point P 0 (that is, r 0 = OP is the vector from the origin to the point P on the line). A vector equation specifies location along a line in terms of a single variable, called the parameter. The components of a vector-valued function are functions of a single variable (the parameter). These are called component functions or parametric equations. A line can be described by infinitely many vector equations. Parallel lines have parallel direction vectors. Planes: To find the vector equation of a plane, n P 0 P = 0, you need two pieces of information: a vector n normal to the plane, and the coordinates of a point P 0 on the plane. The vector 0 P is the vector from the point P 0 on the plane, to an arbitrary vector P = x, y, z. If ax +by +cz = d is an algebraic equation of a plane, then n = a, b, c is a normal vector to a plane. If n is normal to a plane, then all vectors parallel to n are also normal to the plane. Algebraically, this means all scalar multiples cn of a normal vector n are also normal to the plane. Parallel planes have the same normal vectors. The equation of a plane can be found beginning with three non-collinear points on the plane, by using the three points to make two vectors, then using the cross product to produce a normal vector.

31 1.6. APPLICATIONS OF VECTORS: LINES AND PLANES 28 More Details Lines: Why is using vector equations to represent lines is useful? Consider the following application. Suppose an object is traveling along a straight path through space with velocity v. It passes through the point P 0 at time t = 0. Then the position of the object at any time t is given by the vector equation r(t) = tv + r 0. Different objects can travel along the same path with different velocities, and may pass through different points at t = 0 (think about cars driving along a road). The slope-intercept equation of a line y = mx + b only works for lines in the xy-plane (R 2 ). It does not work for lines in higher dimensions, like 3-space (R 3 ). Vector equations for lines can be used in any dimension. A line can be represented by infinitely many vector equations, depending on the choice of directional vector v and fixed point P 0. A similarity between the slope-intercept equation of a line y = mx+b and the vector equation r(t) = tv + r 0, is that both require two pieces of information: a point on the line, and the orientation of the line in space. Point on the line: In the slope-intercept equation, the point on the line is the y-intercept, (x, y) = (0, b). In the vector equation, the point P on the line is indicated by its position vector, r 0 = OP. Orientation of the line: In the slope-intercept equation, the orientation of the line is indicated by the slope m. In the vector equation, the orientation is indicated by the direction vector v, which is parallel to the line. To parameterize a line means to find a vector equation r(t) = tv + r 0 for the line. The parameter is the variable in this case, t. In applications, the parameter often represents time, and the vector-function r(t) represents the position of an object at time t. The coordinate functions (or parametric equations) describing a line are

32 1.6. APPLICATIONS OF VECTORS: LINES AND PLANES 29 the functions that appear in the components of the vector equation of the line. If the parameter is t, these will all be functions of the variable t. For example, for a line in R 2 with v = v 1, v 2 and r 0 = x 0, y 0 : The parametric equations are: r(t) = tv + r 0 = tv 1, tv 2 + x0, y 0 = tv 1 + x 0, tv 2 + y 0 x(t) = tv 1 + x 0, y(t) = tv 2 + y 0 If the vector equation represents the location at time t of an object traveling along a line, then the parametric equations give the x, y, and z coordinates of the object at time t. Planes in R 3 : A vector is normal to a plane if it is orthogonal to every vector in the plane. This leads to the vector equation of a plane. Suppose P 0 is a fixed point in the plane, and n is a normal vector to the plane. A second point P will also be in the plane if and only if the vector P 0 P is parallel to the plane, which in turn means P 0 P is normal to n, so n P 0 P = 0. There are infinitely many vectors that are normal to a plane. If n 0 is normal to a plane P, then every scalar multiple cn is also normal to the plane. The line parameterized by r(t) = tn is called a normal line of P. A normal line of a plane is perpendicular to the plane. Parallel planes have the same normal vectors, so they share a normal line. Another way of writing the vector equation of the plane: If P = (x 0, y 0, z 0 ) is a fixed point on the plane with position vector r 0 = x 0, y 0, z 0, and P = (x, y, z) is an arbitrary point with position vector r = x, y, z,

33 1.6. APPLICATIONS OF VECTORS: LINES AND PLANES 30 then P 0 P = x x 0, y y 0, z z 0 = r r 0, so: n P 0 P = 0 n (r r 0 ) = 0 n r n r 0 = 0 n r = n r 0 To see the relationship between a vector equation of a plane and an algebraic equation of the plane, let n = a, b, c for some constants a, b, and c (not all equal to zero), and let P 0 = (x 0, y 0, z 0 ) be a fixed point on a plane (so x 0, y 0, and z 0 are constants). Then: n P 0 P = 0 n r = n r 0 a, b, c x, y, z = a, b, c x0, y 0, z 0 ax + by + cz = d where: d = a, b, c x 0, y 0, z 0 = ax0 + by 0 + cz 0 From the above, you can see that starting with an algebraic equation ax + by + cz = d, a normal vector to the plane is n = a, b, c. In other words, the x, y, and z coefficients in the algebraic equation of a plane give the î, ĵ, and ˆk components of a vector normal to the plane. Suppose the vectors u and v determine a plane. Since u v is orthogonal to both u and v, then u v is normal to the plane spanned by u and v. This can be used to find the equation of a plane given three non-collinear points on the plane, by first using the three points to construct two vectors, then using the cross product of these vectors as a normal vector.

34 1.6. APPLICATIONS OF VECTORS: LINES AND PLANES 31 Computations Vector Equation of Line General Equation: r(t) = tv + r 0 In R 2 : r(t) = t v 1, v 2 + x0, y 0 = tv1 + x 0, tv 2 + y 0 In R 3 : r(t) = t v 1, v 2, v 3 + x0, y 0, z 0 = tv1 + x 0, tv 2 + y 0, tv 3 + z 0 Parametric Equations of Line x(t) = tv 1 + x 0, y(t) = tv 2 + y 0, z(t) = tv 3 + z 0 Vector Equation of Plane With normal vector n = a, b, c and fixed point on plane P 0 = (x 0, y 0, z 0 ): n P 0 P = 0 a, b, c x x0, y y 0, z z 0 = 0 Normal Vector to Plane Beginning with an algebraic equation ax + by + cz = d: n = a, b, c Beginning with three (non-collinear) points P, Q, R on the plane: n = P Q P R

35 1.7. INTRODUCTION TO VECTOR FUNCTIONS: CURVES IN R 2 & R Introduction to Vector Functions: Curves in the Plane (R 2 ) & 3-space (R 3 ) Textbook Sections: 13.1 The Big Picture Scalar-valued functions of a single variable, for example, y = f (t), are very limited when it comes to describing paths. For example, if t is a time variable, and y is a space variable, then y = f (t) is position of an object on the y-axis at time t. So we can only describe motion along a coordinate axis. To describe more complicated paths, you need to specify position relative to all coordinate axes. This can be done using a vector function r(t) = x(t) î +y(t) ĵ +z(t) ˆk, or the coordinate functions x(t), y(t), z(t). You have already seen an example of this: vector equations of lines (Section 1-6). The vector equation r(t) = tv + r 0 describes a line. An application of vector functions is to represent the path followed by a moving object. In this case, the parameter t represents time, and the coordinate functions represent spacial locations relative to the coordinate axes. Notation alert: the textbook uses parametric equations x = f (t), y = g(t), z = h(t) instead of coordinate functions x(t), y(t), z(t). From the Toolbox (what you need from previous classes): Cartesian Coordinates: Coordinates of points in general, P = (x, y) or P = (x, y, z); cooridinates (x, f (x)) of points on the graph of a function. Trigonometry: Polar coordinates (x, y) = (r cos θ, r sin θ). Vectors: Sketching a vector using its component form. Vector equations of lines.

36 1.7. INTRODUCTION TO VECTOR FUNCTIONS: CURVES IN R 2 & R 3 33 Checklist: Computational Methods & Important Concepts Computational Methods Evaluate and sketch curves parameterized by vector functions. Use this to determine position at a given parameter value, and direction of travel along a curve as the parameter increases. Match curves to vector functions that parameterize them. Use vector functions to find locations on a curve based on parameter values, and to find parameter values corresponding to particular locations. Use a CAS or graphing app to graph parametric curves. Important Concepts A vector function is a function whose input is a scalar (called the parameter), and whose output is a vector. The components of a vector function are scalar-valued functions of a single variable (the kind you worked with in Calc I and Calc II). They are called parametric equations or coordinate functions. A vector function r(t) is a moving position vector. As the parameter varies, the terminal points of the vector function are points on a curve. We say the vector function parameterizes the curve. Application of vector functions: If the parameter t represents time, and the x, y, z coordinates represent space, then the vector function r(t) = x(t) î + y(t) ĵ + z(t) ˆk determines location at time t. This can be used to describe the path of a moving object. More Details A vector function r(t) is a function with a scalar input and a vector output:

37 1.7. INTRODUCTION TO VECTOR FUNCTIONS: CURVES IN R 2 & R 3 34 Input: A scalar, called the parameter (in this case, t). Only one parameter is needed, since curves are 1-dimensional. The parameter is often t (for time) or θ (for an angle). Output: A vector r(t) that gives position based on the parameter value. It points to a location in space, determined by the parameter. The general position vector r = x, y, z starts at the origin and points to a location P = (x, y, z) in space. A vector function r(t) is a moving position vector, indicating position by pointing to different points based on values of the parameter. The coordinate functions (or parametric equations) of a vector function r(t) = x(t) î + y(t) ĵ + z(t) ˆk are the scalar-valued functions x(t), y(t), z(t) that make up the components of the vector function. These are single-variable functions that you are already familiar with from precalculus, Calc I, and Calc II. A parameterization of a curve C is a vector function r(t), where each point on the curve C is the terminal point of r(t) for some value of t. To parameterize a path in the plane (R 2 ), you need two coordinate functions x(t) and y(t): r(t) = x(t) î + y(t) ĵ. For each value of t, r(t) = x(t) î + y(t) ĵ gives a position in the plane; the vector r(t) starts at the origin and ends at the point P (t) = ( x(t), y(t) ). To parameterize a path in 3-space (R 3 ) requires three coordinate functions, x(t), y(t), and z(t): r(t) = x(t) î + y(t) ĵ + z(t) ˆk. For each value of t, r(t) = x(t) î + y(t) ĵ + z(t) ˆk gives a position in 3-space; the vector r(t) starts at the origin and ends at the point P (t) = ( x(t), y(t), z(t) ). From pre-calculus, you know that adding a constant c to a function y = f (x) translates the graph of the function up and down relative to the

38 1.7. INTRODUCTION TO VECTOR FUNCTIONS: CURVES IN R 2 & R 3 35 x-axis. Similarly, adding a constant vector C to a vector function r(t) translates the curve relative to the co-ordinate axes. For example, a circle of radius r = 2 centered at the origin in the xy-plane can be parameterized as: r(t) = 2 cos t, 2 sin t, 0. To move this circle so that it is centered at the point C = (1, 2, 3): r(t) = 2 cos t, 2 sin t, 0 + C = 2 cos t, 2 sin t, 0 + 1, 2, 3 = 2 cos t + 1, 2 sin t + 2, 3 Recall from Pre-calculus & Calc I: The domain of a function y = f (t) is the set of all independent variables for which the function is defined. Similarly, the domain of a vector function is the set of all parameters for which the coordinate functions x(t), y(t), z(t) are defined. Recall from Calc I: If a function is continuous its graph has no holes or jumps, and is not ± at any point of its domain, so you can draw the graph without picking up the pen and moving it to a different location. The same idea applies to vector functions: the curve it parameterizes has no holes or jumps, and is not ±. One way to think about this: If r(t) represents the path of an object moving through space, there are no points at which the object vanishes from one location, and spontaneously appears at another. Continuity of a vector function r(t) = x(t), y(t), z(t) depends only on continuity of the coordinate functions x(t), y(t), and z(t) are continuous. So you can determine continuity of a vector function by looking at the coordinate functions, which are functions you are already familiar with from Calc I. Computations Link to Notes: Common Parameterizations for Some Important Curves (You may use these during exams.) shariultman/275/paramcurves.pdf

39 1.8. DERIVATIVES OF VECTOR FUNCTIONS: COMPUTATION & GEOMETRY Derivatives of Vector Functions: Computation & Geometry Textbook Sections: 13.2, 13.3 The Big Picture There are many similarities between the derivative of a scalar-valued function of a single variable, and derivatives of vector functions of a single parameter. Both are defined as limits of difference quotients. Both are used to describe the direction of tangent lines. Both represent instantaneous rates of change. Derivatives (and integrals) of vector functions are computed by differentiating (or integrating) the coordinate functions. This is good news, since you learned how to do this in Calc I/II. As long as the vector derivative r (t) is non-zero, it is tangent to the curve parameterized by the vector function r(t), and points in the direction along the curve in which the parameter t is increasing. In addition to r (t), there are two other tangent vectors we will be using in this class. The vector differential (also called the vector line element) dr is an infinitesimal (very very small) tangent vector, measuring an infinitesimal displacement along a curve. Its magnitude ds = dr (called the scalar line element) is an infinitesimal version of the Pythagorean theorem, and is used to measure distance along curves. The unit tangent vector ˆT is a unit vector tangent to a curve. From the Toolbox (what you need from previous classes): Calc I: Computing derivatives of single-variable functions y = f (t). Geometrically: these derivatives represent slopes of tangent lines. In real world applications, these derivatives represent instantaneous rates of change. Vectors: Vector operations (vector addition, scalar multiplication, the dot and cross products); computing the magnitude of a vector; know

40 1.8. DERIVATIVES OF VECTOR FUNCTIONS: COMPUTATION & GEOMETRY 37 what a vector function is, and be able to evaluate vector functions at specific parameter (input) values. Checklist: Computational Methods & Important Concepts Computational Methods Compute the vector derivative r (t), and sketch it relative to the curve parameterized by r(t) for specified values of t. Compute the vector differential dr, and its magnitude ds. Compute the unit tangent vector ˆT. Find equations of tangent lines to curves parameterized by vector functions. Apply any previous knowledge about vectors to vector functions (vector addition, scalar multiplication, magnitude, dot and cross products, finding angles, projections, etc). Important Concepts The derivative r (t) of a vector function r(t) is computed by differentiating the coordinate functions. If r (t) 0, it spans a line tangent to the curve parameterized by r(t). For this reason, r (f ) is often called the tangent vector. The vector differential (or vector line element) dr is an infinitesimal (very very small) tangent vector, measuring an infinitesimal displacement along a curve. Its magnitude ds = dr (the scalar line element) is an infinitesimal version of the Pythagorean theorem, and is used to measure distance along curves. The unit tangent vector ˆT is a unit vector tangent to a curve. Because r(t) and its derivatives are vectors, one can perform any vector operation on them.

41 1.8. DERIVATIVES OF VECTOR FUNCTIONS: COMPUTATION & GEOMETRY 38 More Details Derivatives of vector functions are computed by differentiating the component functions using methods learned in Calc I. Integrals of vector functions are computed by integrating the component functions using methods learned in Calc I/II. So, if r(t) = x(t) î + y(t) ĵ + z(t) ˆk : r (t) = x (t) î + y (t) ĵ + z (t) ˆk or dr dt = dx dt î + dy dt ĵ + dz dt ˆk ˆ ˆ b a (ˆ r(t) dt = (ˆ b r(t) dt = a ) (ˆ x(t) dt î + ) (ˆ y(t) dt ĵ + ) z(t) dt ˆk ) (ˆ b ) (ˆ b ) x(t) dt î + y(t) dt ĵ + z(t) dt ˆk a a Some similarities between the Calc I derivative f (t) and the vector derivative r (t): Both f (t) and r(t) are functions of a single variable (their input is a scalar), so their derivatives are computed with respect to a single variable t. Tangent lines: The derivative f (t) is the slope of the tangent line to the graph of the function f (t). The derivative r (t) is the direction vector of the tangent line to the curve r(t) (provided r (t) 0). Rates of change: Both f (t) and r (t) are defined as the limits of difference quotients: f f (t) = lim t 0 t and r r (t) = lim t 0 t These difference quotients f / t and r/ t represent the average rate of change of the functions f (t) and r(t) over an increment t, so the limits f (t) and r (t) represent the instantaneous rate of change of f (t) and r(t) with respect to t. r (t) is a vector, so this change can occur in either the magnitude or the direction of r(t) (or both).

42 1.8. DERIVATIVES OF VECTOR FUNCTIONS: COMPUTATION & GEOMETRY 39 Velocity, speed, and acceleration: If f (t) represents the position of an object on a coordinate axis at time t, then f (t) is velocity, f (t) is speed, and f (t) is acceleration. If r(t) represents the position of an object in the plane or in 3-space at time t, then r (t) is velocity, r (t) is speed, and r (t) is acceleration. A major difference between f (t) and r (t): f (t) is a scalar valued function (output is a scalar); r (t) is a vector valued function (output is a vector). Geometry of the vector derivative: If the derivative r (t) exists and is non-zero (r(t) 0), then: If r (t) 0, it points in the direction along the curve in which t increases. If r (t) 0, it spans the tangent line to the curve parameterized by r(t). For this reason, r (t) is sometimes called the tangent vector. Tangent lines to curves: If r (t 0 ) 0, it can be used as the directional vector for the tangent line to the curve r(t) when t = t 0. Since r(t 0 ) gives the position of a point both on the curve r(t) and on the curve s tangent line R(t), a vector equation for this tangent line is: R(s) = sr (t 0 ) + r(t 0 ) Notation alert: Since r(t) is being used for the original curve, we need different names for the vector and the parameter of the tangent line, so we use R(s). A unit tangent vector ˆT is a vector of length 1 ( ˆT = 1) that is tangent to a curve. If r (t) 0, then ˆT (t) = r (t)/ r (t). The vector line element (or vector differential) dr is another useful vector that is tangent to a curve. dr represents an infinitesimal (very very small) change in position between a point (x, y, z) and a second point

43 1.8. DERIVATIVES OF VECTOR FUNCTIONS: COMPUTATION & GEOMETRY 40 (x + dx, y + dy, z + dz). So: dr = x + dx, y + dy, z + dz x, y, z = dx, dy, dz = dx î + dy ĵ + dz ˆk The scalar line element ds is the magnitude of dr (ds = dr ): ds = dr = dx 2 + dy 2 + dz 2. ds is an infinitesimal version of the Pythagorean theorem. So the magnitude of the vector differential gives a way of measuring distance along a curve. To compute the line elements of the vector function r(t) = x(t) î + y(t) ĵ + z(t) ˆk, use the method of computing differentials that you learned for u substitution in Calc II: d[x(t)] = x (t)dt, d[y(t)] = y (t)dt, d[z(t)] = z (t)dt So: and: dr = d[x (t)] î + d[y (t)] ĵ + d[z (t)] ˆk = x (t) dt î + y (t) dt ĵ + z (t) dt ˆk ( ) = x (t) î + y (t) ĵ + z (t) ˆk dt = r (t) dt ds = r (t) dt If r(t) represents the position of an object at time t, then r (t) is the object s velocity, and the vector differential dr = r (t)dt represents (the object s velocity) (an infinitesimal time increment dt). Computations Vector Derivatives: For a vector function r(t) = x(t) î + y(t) ĵ + z(t) ˆk : r (t) = x (t) î + y (t) ĵ + z (t) ˆk or dr dt = dx dt î + dy dt ĵ + dz dt ˆk

44 1.8. DERIVATIVES OF VECTOR FUNCTIONS: COMPUTATION & GEOMETRY 41 Vector Line Element: dr = r (t)dt = ( x (t) î + y (t) ĵ + z (t) ˆk ) dt Scalar Line Element: ds = dr = r (t) dt ( [x = (t) ] 2 [ + y (t) ] 2 [ + z (t) ] ) 2 1/2 dt Unit Tangent Vector: ˆT (t) = r (t)/ r (t) (for r (t) 0) Equation of Tangent Line: For a curve is parameterized by r(t), if the tangent vector r (t 0 ) 0, a vector equation of the tangent line to the curve r(t) at t = t 0 is: R(s) = sr (t 0 ) + r(t 0 ) Differentiation Rules for Vector Functions: Constant Rule If C is a constant vector (all components are constants), then: d [ ] C = 0 dt Linearity If a and b are scalars and r(t) and s(t) are vector functions: d [ ] a r(t) + b s(t) = a r (t) + b s (t) dt Product Rules There are three product rules: Multiplication of vector function by a scalar-valued function: d [ ] f (t) r(t) = f (t) r(t) + f (t) r (t) dt

45 1.8. DERIVATIVES OF VECTOR FUNCTIONS: COMPUTATION & GEOMETRY 42 Dot product of two vector functions: d [ ] [ ] [ ] r(t) s(t) = r (t) s(t) + r(t) s (t) dt Cross product of two vector functions: d [ ] [ ] [ ] r(t) s(t) = r (t) s(t) + r(t) s (t) dt

46 1.9. APPLICATIONS OF VECTOR DERIVATIVES: MOTION ALONG CURVES Applications of Vector Derivatives: Motion Along Curves Textbook Sections: 13.3, 13.4 The Big Picture Derivatives of vector functions have many applications. One of the most immediate is using vector functions to represent the motion of an object traveling through space. If a vector function r(t) represents the position of an object at time t, then the first derivative r (t) represents velocity and the second derivative r (t) represents acceleration. Often velocity is denoted by v(t) and acceleration by a(t). The speed v(t) of the object is the magnitude of the velocity: v(t) = r (t). Another application of vector derivatives is measuring the length of a curve (or arc length). Recall that the scalar line element ds represents the length of an infinitesimally small piece of a curve. Adding up (or, integrating) these infinitesimal lengths gives the total length L of a curve between points P and Q on the curve: L = ˆ Q A related application is distance traveled along a curve as a function of time. If r(t) represents the position of an object at time t, then the total distance traveled by the object as a function of time, beginning at time t 0, is: s(t) = P ds ˆ r(t) (Optional) Other applications of vector derivatives include: arc length parameterizations, moving frames, curvature, and the decomposition of acceleration into tangential and normal components. r(t 0 ) ds. From the Toolbox (what you need from previous classes):

47 1.9. APPLICATIONS OF VECTOR DERIVATIVES: MOTION ALONG CURVES 44 Calc I: Computing derivatives of single-variable functions y = f (t). Geometrically: these derivatives represent slopes of tangent lines. In real world applications, these derivatives represent instantaneous rates of change. If f (t) is position, then f (t) is velocity and f (t) is acceleration. Vectors: For a vector function r(t), be able to compute and understand the geometry behind the derivative r (t), the line elements dr and ds, and the unit tangent vector ˆT (t). Checklist: Computational Methods & Important Concepts Computational Methods Given a vector function representing position along a curve at time t, compute and answer questions about position, velocity, speed, and acceleration. Find the length of a curve between two points, and the total distance traveled along a curve as a function of time. Important Concepts As with scalar-valued functions from Calc I, if r(t) represents position at time t, then the first derivative r (t) represents velocity (often denoted v(t)), and the second derivative r (t) represents acceleration (often denoted a(t)). The magnitude of velocity is speed (v(t) = v(t) ). The length L of a curve C between two points P and Q = r(b) can be measured by chopping up the curve into infinitesimally small pieces of length ds, and adding up (integrating) the lengths: L = Q P ds. If r(t) represents the position of an object at time t, then the total distance traveled by the object as a function of time, beginning at time t 0, is s(t) = r(t) r(t 0 ) ds.

48 1.9. APPLICATIONS OF VECTOR DERIVATIVES: MOTION ALONG CURVES 45 More Details Velocity, speed, and acceleration. As in Calc I, the derivatives of a position function r(t) give velocity and acceleration: Velocity: Often denoted v(t), velocity is the first derivative of position. Speed: Often denoted v(t), speed is the magnitude of velocity. Acceleration: Often denoted a(t), acceleration is the derivative of velocity, or the second derivative of position. v(t) = r (t) v(t) = r (t) = v(t) a(t) = r (t) = v (t) Length L of a curve between two points. To approximate the length of a curve C between two points P and Q, you could chose finitely many points on the curve between P and Q, and measure the straight-line distance s i between adjacent points. Then, the approximate length is: L n s i. i=1 Using more points that get closer and closer together gives a better approximation. The exact length of the curve between P and Q is defined to be the limit (if it exists) of the approximations: L = ˆ Q P ds = lim n n s i. Another way of thinking about length, using the differential d s: The differential ds = dx 2 + dy 2 + dz 2 is an infinitesimal version of the Pythagorean theorem. It can be used to measure an infinitesimally small length along the curve. To add up these lengths from the starting point P to the ending point Q, we integrate, so: L = ˆ Q P ds. i=1

49 1.9. APPLICATIONS OF VECTOR DERIVATIVES: MOTION ALONG CURVES 46 Computationally, for a curve parameterized by r(t), ds = r (t) dt, so if P = r(a) and Q = r(b), and r (t) is continuous: L = ˆ Q P ds = ˆ b a r (t) dt. Distance s(t) traveled along a curve at time t (or, the arc length function). If you travel along a curve r(t), starting at a fixed point P 0 = r(t 0 ) and ending at a second point P = r(t), then the distance you have traveled can be viewed as a function of t: s(t) = ˆ P P 0 ds = ˆ t t 0 r (t) dt. (Optional) Arc length parameterizations and unit speed curves. This is a topic that is very important for the mathematical theory behind curves and their applications, but is usually difficult or impossible to carry out computationally. The general idea is, one can use the arc length function to re-parameterize a curve, replacing the parameter t with the parameter s. This is called an arc length parameterization. The advantage of an arc length parameterization r(s) is that the derivative r (s) = ˆT (s), and r (s) = 1. In other words, an object traveling along a curve according to an arc length parameterization is traveling with unit speed: one unit of distance is covered in one unit of time s. (Optional) The { ˆT, ˆN, ˆB} moving frame. Along with the unit tangent vector ˆT, there are two additional unit vector associated with curves: The unit normal vector ˆN is a unit vector that is orthogonal to ˆT. ˆN points inside the curve. For example, on a circle, ˆN points towards the center of the circle. Computationally, ˆN(t) = ˆT (t)/ ˆT (t). The unit binormal vector ˆB is a unit vector that is orthogonal to both ˆT and ˆN. Computationally, ˆB = ˆT ˆN, so its direction relative to the ˆT ˆN-plane is determined by the right-hand rule. The three vectors ˆT, ˆN, ˆB form a moving frame, and are used to describe vectors in a way that is natural with respect to motion along the curve. For example, acceleration is often decomposed into its tangential and normal components: a = aˆt ˆT + a ˆN ˆN.

50 1.9. APPLICATIONS OF VECTOR DERIVATIVES: MOTION ALONG CURVES 47 (Optional) Curvature κ of a curve. Curvature of a curve can be defined in two ways: In terms of the radius ρ of the osculating circle. The osculating circle of a curve at a point is the circle that best approximates the curve at that point (compare this to the tangent line at a point, which is the line that best approximates the curve at that point). This circle lies in the ˆT ˆN-plane (also called the osculating plane). If the radius of the osculating circle is ρ, the curvature is defined to be κ = 1/ρ. So, the larger the radius, the smaller the curvature. This makes sense, if you think about circles: a circle with a smaller radius is more curvy than a circle with a larger radius. (The word osculating comes from the Latin osculum to kiss.) In terms of the rate of change of the unit tangent vector. Suppose ˆT (s) is the unit vector for a curve that is being traveled at unit speed (covering one unit of distance per unit time). Then the curvature is also defined as κ = d ˆT /ds. It is inconvenient to find unit-speed parameterizations; fortunately, curvature can be computed in terms of an arbitrary parameterization r(t). If ˆT (t) is the unit vector of this parameterization, then κ = ˆT (t) / r (t). This computation can be simplified to: κ(t) = r (t) r (t) r (t) 3 Somewhat surprisingly, both of these definitions result in the same value for curvature.

51 1.9. APPLICATIONS OF VECTOR DERIVATIVES: MOTION ALONG CURVES 48 Computations Velocity, Speed, and Acceleration: is given by r(t), then: Velocity: Speed: Acceleration: If the position of an object at time t v(t) = r (t) v(t) = r (t) = v(t) a(t) = r (t) = v (t) Length s of a curve: If r (t) is continuous, and P = r(a) and Q = r(b), the length L of the curve C parameterized by r(t) between the points P and Q is: L = ˆ Q P ds = ˆ b a r (t) dt Distance s(t) traveled along a curve at time t (or, the arc length function): If r (t) is continuous, and P 0 = r(t 0 ) and P = r(t), the distance s(t) traveled along the curve C at time t is: s(t) = ˆ t t 0 r (t) dt

52 Chapter 2: Partial Derivatives 2.1 Introduction to Multivariate Functions (Functions of Two or More Variables) Textbook Section: 14.1 The Big Picture Real-world processes often depend on more than one parameter. Examples: the ideal gas law, P V = nrt ; the Pythagorean theorem, c 2 = a 2 + b 2 ; Newton s law of universal gravitation F = Gm 1 m 2 /r 2. For this reason, we study functions with two or more independent variables (called multivariate functions). We will primarily focus on functions with two or three independent variables in this class; many of the techniques and results we encounter also hold for functions with an arbitrary number of independent variables. As with functions of a single variable y = f (x), functions of more than one variable have a domain (set of allowable inputs) and a range (set of possible outputs). The range will always be subset of the real numbers R. The domain, however, depends on the number of independent variables. The domain of a function y = f (x) is a subset of R. The domain of a function z = f (x, y) is a subset R 2. The domain of a function w = f (x, y, z) is a subset of R 3. In general, the domain of a function of n variables is a subset of R n. For functions of two variables z = f (x, y), there are several types of visual representations, including graphs, traces, and level curves/contour diagrams. Visual representations of functions of three variables w = f (x, y, z) are limited. From the Toolbox (what you need from previous classes): Precalculus: Find the domain and range for functions of a single variable, y = f (x). Evaluate and graph functions of a single variable. 49

53 2.1. INTRODUCTION TO MULTIVARIATE FUNCTIONS 50 Checklist: Computational Methods & Important Concepts Computational Methods Evaluate scalar valued functions of two and three variables. Find the domain and range of scalar valued functions of two and three variables. Use an app to generate a graph and level curves/contour maps for a function z = f (x, y). Match the graph of a function z = f (x, y) with its contour map. Sketch vertical and horizontal traces (by hand) on the graph of a function z = f (x, y). Find equations for level curves of a function z = f (x, y). Sketch the level curves (by hand), and use the level curves to construct a contour map. For a function z = f (x, y), use a contour map to determine the function s value, the change in the function s value between two points in the domain, and the average rate of change of a function s value with respect to distance in the domain. Important Concepts The domain of a multivariate function is the set of all allowable inputs. An input for a function z = f (x, y) is a point P = (x, y); for a function w = f (x, y, z), a point P = (x, y, z). The dimension of the domain is the same as the number input variables. So, the domain of a function z = f (x, y) is a subset of R 2 (2-dimensional), and the domain of a function w = f (x, y, z) is a subset of R 3 (3-dimensional). The range of a multivariate function is set of all possible outputs. Since these outputs are scalars, the range is a subset of R. The graph of a function z = f (x, y) is the set of all points (x, y, f (x, y)) where (x, y) is in the domain of f. For the functions we study in this class, this graph is a surface (two-dimensional object) in R 3.

54 2.1. INTRODUCTION TO MULTIVARIATE FUNCTIONS 51 Traces of a function z = f (x, y) are curves in the graph of the function that are generated by holding one of the variables x, y, or z constant. Vertical traces are generated by holding either x or y constant, producing curves on the graph that are parallel to the xz- or yzcoordinate planes. These are the curves of intersection of the graph z = f (x, y) with the vertical planes x = c or y = c (c is a constant). Horizontal traces are generated by holding z constant, producing curves on the graph that are parallel to the xy-coordinate plane. These are the curves of intersection of the graph z = f (x, y) with the horizontal planes z = c (c is a constant). Level curves of a function z = f (x, y) are curves in the xy-plane defined by the equations c = f (x, y). They are the projections of horizontal traces into the domain of the function in the xy-plane. A collection of level curves with a constant change in the value c between adjacent curves is called a contour map or contour diagram. The difference of the c-values between adjacent curves is called the contour interval. This corresponds to projecting evenly-spaced horizontal traces from the graph of the function into the xy-plane. More Details Domain: The domain of a function is the set of allowable inputs for that function. Example, the domain of the function f (x, y) = (x 2 + y 2 ) 1/2 is all of R 2 except for the origin (in set-notation, {(x, y) R 2 : (x, y) (0, 0)}), since you cannot divide by zero. Range: The range of a function is the set of all possible outputs of a function. Example, the range of the function f (x, y) = x 2 + y 2 is all non-negative real numbers (in set-notation, {z R : z 0}), since the real-valued square root function cannot be negative. Visual representations of functions of two variables z = f (x, y) : Graphs: The graph of a function z = f (x, y) is a subset in R 3.

55 2.1. INTRODUCTION TO MULTIVARIATE FUNCTIONS 52 These graphs can be sketched hand, or using a computer. If the function is continuous, its graph is a 2-dimensional surface (called the surface generated by the graph). Traces: A trace is a curve in the graph of the function generated by holding all but one of the independent variables constant. We will use traces when we define partial derivatives, and again later to define area elements for surfaces. For functions z = f (x, y), traces are generated by intersecting the graph of the function with planes parallel to the coordinate planes. Vertical traces of a function z = f (x, y) are the curves of intersection of the graph of f with planes parallel to the xz- or yzcoordinate planes. These planes have equations y = c (parallel to the xz-plane) or x = c (parallel to the yz-plane), so the vertical traces are defined by the equations z = f (x, c), y = c or z = f (c, y), x = c. (Preview: Partial derivatives are slopes of tangent lines to vertical traces.) Horizontal traces of a function z = f (x, y) are the curves of intersection of the graph of f and planes parallel to the xycoordinate plane. These planes have equations z = c, so the horizontal traces are defined by the equations c = f (x, y), z = c. Level Curves A level set is a subset of a function s domain over which the function s value remains constant. For functions of two variables z = f (x, y), these are curves in the xy-plane, called level curves. Level curves are defined by the equations f (x, y) = c, z = 0, or simply f (x, y) = c. They are projections of the horizontal traces into the xy-plane. The value of a function over a level curve is constant. Example: if z = f (x, y) represents the height of a surface relative to the xy-plane, then an ant walking along a level curve would look up (or down) at the surface, and the distance from the ant to the surface would not change. Example: If z = T (x, y) represents the temperature of a flat plate at the location (x, y), then an ant

56 2.1. INTRODUCTION TO MULTIVARIATE FUNCTIONS 53 walking along a level curve would experience constant temperature (for temperature functions, level curves are called isotherms ). Contour Maps Contour maps (or, contour diagrams) are collections of level curves, where the change in the function value on adjacent level curves, called the contour interval, is constant. So the steepness of the graph of the function is reflected by how closely spaced curves are: the closer the spacing, the steeper the graph. Contour maps can be used to determine the change in a function s value between two points in the domain, and the average rate of change of a function between two points in the domain. Real-world examples of contour maps are topographic maps, used to indicate terrain, and maps of air pressure used in meteorology. Visual representations of functions of three variables w = f (x, y, z) : Methods for visualizing functions w = f (x, y, z) are limited. The graph of a function w = f (x, y, z) is a three-dimensional hypersurface in R 4. Since we cannot visualize objects in four dimensions, these graphs do not give us useful visual information. The same is true for functions of more than three variables: their graphs are n- dimensional hyper-surfaces in R n+1, and do not provide useful visual information. Level sets of a function of three variables are surfaces in R 3, which can be graphed. Coloring these level sets to correspond to function s values can help organize the information. However, level sets of a function of three variables does not provide the same level of visual clarity that a contour map does for a function of two variables, since these level surfaces often intersect and/or obscure each other, and there is no guiding intuition about R 4 to help intrepret the information they present.

57 2.2. PARTIAL DERIVATIVES Partial Derivatives Textbook Section: 14.3 The Big Picture Jedi Mind Tricks. Computationally: To compute the partial derivative with respect to one variable, treat all other variables as constants. Geometrically: Look at the curve of intersection of the surface with a plane parallel to a coordinate plane (the vertical traces from Topic 2.1). Since all but one variable is constant, these curves look like graphs of functions of a single variable. The geometric interpretation of the derivative as the slope of the tangent line the limit of the slopes of secant lines to these curves, is the same as in Calc I. From the Toolbox (what you need from previous classes): Algebra/Pre-calculus: Find the slope of a line in the plane. Calc I: Know the definition of a derivative as the slope of a tangent line (the limit of the slopes of secant lines). Compute the derivative of a single-variable function y = f (x). Checklist: Computational Methods & Important Concepts Computational Methods Explore the geometric meaning of a partial derivative, see how that is similar to the geometric meaning of derivatives from Calc I, and use it to justify using computational techniques from Calc I to compute partial derivatives. Estimate partial derivatives using graphical data (graphs or contour maps).

58 2.2. PARTIAL DERIVATIVES 55 Compute partial derivatives, and evaluate them at a given point. Use partial derivatives to answer questions about rates of change with respect to a specified input parameter. Important Concepts A partial derivative of a multivariate function is analogous to derivatives of functions of a single variable. For partial derivatives, all variables except one are held constant, and the derivative is computed using the methods developed in Calc I. Geometrically, a partial derivative is the slope of a line that is tangent to a curve on the surface that is generated by holding all but one of the independent variables constant. More Details Geometrically, a partial derivative has the same interpretation as an ordinary derivative: it is the slope of a tangent line to a curve in the graph of the function. This is defined to be the limit of the slopes of secant lines as the secant lines approach the tangent line. For the partial derivative f / x of f (x, y) with respect to x, y is held constant, and the points defining the secant lines lie along a vertical trace z = f (x, c) in the graph of f, parallel to the xzplane. The partial derivative with respect to x is the slope of the tangent line to the vertical trace z = f (x, c). For the partial derivative f / y of f (x, y) with respect to y, x is held constant, and the points defining the secant lines lie along a vertical trace z = f (c, y) in the graph of f, parallel to the yzplane. The partial derivative with respect to y is the slope of the tangent line to the vertical trace z = f (c, y). Derivatives of functions of more three or more variables have the same interpretation, but the traces are curves in a higher-dimensional graph.

59 2.2. PARTIAL DERIVATIVES 56 To compute partial derivatives with respect to a given variable, use Jedi mind tricks: treat that variable like the only variable, treat the other variables like constants, and compute derivatives using the rules learned in Calc I. Higher-order derivatives are derivatives of derivatives. Second-order mixed partial derivatives are second partial derivatives taken with respect to two different variables, eg: f xy (derivative taken first with respect to x, then with respect to y), or f yx (derivative taken first with respect to y, then with respect to x). Clairaut s Theorem states that, if the second-order mixed partial derivatives exist and are continuous, then the order of differentiation does not matter, that is: f xy = f yx.

60 2.2. PARTIAL DERIVATIVES 57 Computations Partial Derivatives: Function of two variables, f (x, y): Partial wrt x: f x (x, y) or f / x Treat x as variable, y as constant. Partial wrt y: f y (x, y) or f / y Treat y as variable, x as constant. Function of three variables, f (x, y, z): Partial wrt x: f x (x, y, z) or f / x Treat x as variable, y and z as constants. Partial wrt y: f y (x, y, z) or f / y Treat y as variable, x and z as constants. Partial wrt z: f z (x, y, z) or f / z Treat z as variable, x and y as constants. Functions of more than three variables follow the same pattern. Second-Order Partial Derivatives: Function of two variables, f (x, y): Mixed partial derivatives: f xy (x, y) or 2 f / y x Take the x-derivative first, then take the y-derivative of the new function. f yx (x, y) or 2 f / x y Take the y-derivative first, then take the x-derivative of the new function. Second partial wrt x only: f xx (x, y) or 2 f / x 2 Take the x-derivative both times. Second partial wrt y only: f yy (x, y) or 2 f / y 2 Take the y-derivative both times. By Clairaut s theorem, it will often be the case that the mixed partial derivatives are equal, so order of differentiation does not matter. Functions of three or more variables follow the same pattern.

61 2.3. TANGENT PLANES AND LINEAR APPROXIMATIONS Tangent Planes and Linear Approximations Textbook Section: 14.4 The Big Picture Geometrically, the derivative of a function of a single variable is the slope of the tangent line to the graph of the function. For a function of more than one variable, there are potentially infinitely many lines tangent to its graph. This leads to the question: what does it mean for a function of more than one variable to be differentiable? For a function of two variables, differentiability at a point is related to the existence of a well-defined tangent plane (R 2 ), which is made up of all of the tangent lines to all of the curves passing through that point in the graph of the function. For functions of three variables, the collection of all tangent lines is a 3-dimensional tangent space (R 3 ). The (local) linearization L gives a way of approximating the value of a function at a point near a fixed point. Because the tangent space is the graph of the linearization L, they have the same equation. For example, if you have a function z = f (x, y) of two variables, then at the point (a, b): z = f x (a, b) ( x a ) + f y (a, b) ( y b ) + f (a, b) f (x, y) L(x, y) = f x (a, b) ( x a ) + f y (a, b) ( y b ) + f (a, b) equation of tangent plane equation of linearization From the Toolbox (what you need from previous classes): Calc III: Equations of planes. Partial derivatives (be able to compute, and know they are the slope of tangent lines to the graph of a function).

62 2.3. TANGENT PLANES AND LINEAR APPROXIMATIONS 59 Checklist: Computational Methods & Important Concepts Computational Methods Find the equation of the tangent plane at a point on the graph of a function of two variables, and the equation of the 3-dimensional tangent space at a point on the graph of a function of three variables. Compute the linearization L (aka linear approximation) for a function of two or three variables. Apply the linearization L to approximate the value of a function of two or three variables. Important Concepts The tangent plane of a differentiable function of two variables, or the 3- dimensional tangent space of a function of three variables, is analogous to the tangent line of a differentiable function of a single variable. If f is differentiable at a point P, the linearization L is the best linear approximation of f at P. The graph of L is the tangent space to the graph of f at P. The linearization L can be used to approximate the values of f for points near P. More Details If a function is differentiable, the tangent plane (or space) at a point has the property that, as you zoom in on the point of tangency, the tangent plane looks more and more like the graph of the function. The linearization L approximates the value of a function at points near a fixed point (f L). Linear functions are are computationally tractable. You can write algorithms that will run on a computer in relatively short time.

63 2.3. TANGENT PLANES AND LINEAR APPROXIMATIONS 60 One thing to consider when using an approximation is how good it is. What is the difference between the approximated value and the actual value? What degree of precision does the problem at hand require? There are methods of determining error, but we will leave those for other courses. Equations of tangent planes: If f (x, y) is a differentiable function of two variables, the equation of its tangent plane (and also the linearization, since they have the same equation) at a point P = (a, b) can be derived using what we know about partial derivatives and equations of planes: We know two vectors in the tangent plane. Since the slope of the line tangent to the curve generated by holding y constant is m = f x (a, b), then a vector tangent to this curve is v x = 1, 0, f x (a, b) (you can see that this has the correct direction by using the x and z components to find the slope m). Similarly, v y = 0, 1, f y (a, b) is tangent to the curve generated by holding x constant. This means we can find a normal vector to the tangent plane: n = v x v y The point of tangency ( a, b, f (a, b) ) lies in the tangent plane. Since we have a point and a normal vector, the equation of the tangent plane is given by the vector equation: ( v x v y ) x a, y b, z f (a, b) = 0 which leads to: z = f x (a, b) ( x a ) + f y (a, b) ( y b ) + f (a, b)

64 2.4. APPLICATIONS OF DIFFERENTIABILITY: THE DIFFERENTIAL DF Applications of Differentiability: the Differential df. Textbook Section: 14.4 The Big Picture The linearization L, which we studied last class, gives a way of approximating the value of a function at a point near a fixed point. The differential df is related to the linearization L, but instead of approximating the value of the function, it approximates the change in the function s value between a fixed point (the center) and nearby points. So df answers the question, What is the (approximate) change in f if there is a small change in x (denoted dx) and/or a small change in y (denoted dy). The linearization and the differential are linear functions that can be used to analyze the behavior of a function, but they do different things. The linearization L is used to approximate the values of the function. The differential df relates changes in the values of the function to changes in the input variables. Neither the linearization L nor the differential df is a derivative. L approximates the value of a function. df measures the change in a function. A derivative is a rate of change.. From the Toolbox (what you need from previous classes): Calc III: Concept: differentiability of multivariate functions is related to the existence of tangent spaces. Computation: Tangent spaces and the linearization of multivariate functions. Checklist: Computational Methods & Important Concepts Computational Methods Find the differential df of a function f of two or three variables.

65 2.4. APPLICATIONS OF DIFFERENTIABILITY: THE DIFFERENTIAL DF. 62 Apply the differential df to estimate the change in the value of a function between nearby points, or the error in a computation based on errors in input and output. Determine whether a problem calls for using the linearization L or the differential df. Important Concepts Suppose f is differentiable (has a well-defined tangent space) at a fixed point P. The differential df can be used to approximate the difference or change in values of f between P and nearby points. d means a (small) change, so dx means a small change in x, dy means a small change in y, dz means a small change in z), and df means a small change in f. Note that neither the linearization nor the differential is a derivative! A derivative is a rate of change of a function. A differential is a (small) change in the value of a function. A linearization is an approximate value of a function. More Details We are beginning to build a toolbox for analyzing the behavior of multivariate functions. Knowing which tool to use for a particular problem depends on the context of the problem. Partial derivatives measure (instantaneous) rates of change with respect to a single variable. Key words/phrases include: rate of change. The linearization L approximates the value of a function. Key words/phrases include: estimate the value ; find the approximate the value. The differential df measures changes in the values of a function relative to changes in the input variables. Key words/phrases include: estimate the change in (or increase or decrease) of

66 2.4. APPLICATIONS OF DIFFERENTIABILITY: THE DIFFERENTIAL DF. 63 the value ; find the change in (something) if (something else) changes. Equation of the differential df for functions of two and three variables: df = f x (a, b) dx + f y (a, b) dy df = f x (a, b, c) dx + f y (a, b, c) dy + f z (a, b, c) dz The differential df is related to the linearization L. For example, suppose L is the linearization of the function f (x, y), centered at the point (a, b). Then the equation of the linearization L(x, y) is: where: f x (a, b)(x a) + f y (a, b)(y b) + f (a, b) f (a, b) is the exact value of the function f at the center point (a, b). f x (a, b)(x a)+f y (a, b)(y b) is the approximate change in f between the point (a, b) and the nearby the point (x, y). So: L(x, y) = f x (a, b)(x a) + f y (a, b)(y b) + f (a, b) = estimated change in f from (a, b) to (x, y) + exact value of f at (a, b) = df + f (a, b) Notation: d vs.. In engineering and the sciences, the notation d and are often used interchangeable, to denote a (usually small) change in a value, or error. For example, dx and x are both used to denote a small input value, or measurement error; df or f are both used to denote the resulting change in output value, or total error. Warning to Math Majors: In mathematics, the notation and d are used differently.

67 2.5. THE GRADIENT F AND THE DIRECTIONAL DERIVATIVE DÛF The Gradient f and the Directional Derivative Dûf Textbook Section: 14.6 The Big Picture A differentiable multivariate function has derivatives in every direction. If f (x, y) is a differentiable function of two variables, these correspond to the slope of the tangent plane in any given direction. Directional derivatives are computed by taking the dot product of an object called the gradient with a unit vector in the indicated direction. The gradient is the vector composed of the partial derivatives of the function. Not only does the gradient encode information about all possible derivatives; it is also an interesting object in its own right. From the Toolbox (what you need from previous classes): Calc III Dot product; unit vectors; partial derivatives. Checklist: Computational Methods & Important Concepts Computational Methods Compute the gradient of a multivariate function. Use the gradient to compute the directional derivative (slope of the tangent plane) in any direction from a given point in space. Use the geometric properties of the gradient to answer questions about level curves, and about the direction and value of the greatest rate of increase (directional derivative) of a function. Important Concepts The gradient of a function f denoted f is the vector field made up of the partial derivatives of f.

68 2.5. THE GRADIENT F AND THE DIRECTIONAL DERIVATIVE DÛF 65 Geometric properties of the gradient: it gives the direction and value of the maximum directional derivative, and is orthogonal to level curves. Directional derivatives give the rate of change of a multivariate function in the direction of a given unit vector in the domain of the function. For functions of two variables, this can be thought of as the slope of the tangent plane in any given direction. Directional derivatives are computed by taking the dot product of the gradient vector with the unit vector in the indicated direction. In this sense, the gradient encodes information about all other derivatives. Partial derivatives are the directional derivatives in the directions of the coordinate axes in the domain of the function. More Details The gradient is an example of a vector field. This is a function whose input is a point, and whose output is a vector. f can be thought of as a function that puts a vector at every point in the domain of the function f. The gradient has three important geometric properties: i. f is orthogonal to the level sets of f. For example, if f (x, y) is a function of two variables, f is orthogonal to the level curves of f. ii. The direction of f is the direction in which the rate of increase of f is the greatest (that is, the direction where D u f is greatest). iii. f is the value of the greatest rate of change of f in any direction at a point (that is, it has the same value as the greatest value taken on by the directional derivatives at a point). Note that (i) and (ii) are related to the direction of the gradient, while (iii) is related to the magnitude of the gradient. A directional derivative Dûf of a function f is the slope of the tangent space in a direction given by the unit vector û in the domain of f. To compute a directional derivative, two pieces of information are needed:

69 2.5. THE GRADIENT F AND THE DIRECTIONAL DERIVATIVE DÛF 66 the gradient f (which is entirely determined by the function f ), and a direction û (which depends only on a direction in the domain of f ). To see why the directional derivative maximizes the directional derivative, use the geometric definition of the dot product: Dûf = f û = f û cos θ = f (1) cos θ = f cos θ This will be maximized when cos θ = 1, which occurs when f and u point in the same direction. When this happens, Dûf = f cos 0 = f, and û = f / f. A similar argument shows that Dûf is minimized in the direction opposite of f, and that in this direction, Dûf = f. To see why the gradient is orthogonal to level curves, suppose ˆT is a unit vector tangent to a level curve of f. The DˆT f = f ˆT. Since the function remains constant (ie: does not change) on level curves, then DˆT f = f ˆT = 0.

70 2.6. CHAIN RULES FOR FUNCTIONS OF TWO OR MORE VARIABLES Chain Rules for Functions of Two or More Variables Textbook Sections: 14.5 The Big Picture As with functions of a single variable, the chain rules for multivariate functions gives a way of computing the derivative of composite functions. From the Toolbox (what you need from previous classes): Calc I: Chain rule for single-variable functions. Calc III Dot product; partial derivatives; the gradient; directional derivatives. Checklist: Computational Methods & Important Concepts Computational Methods Compute derivatives of multivariate functions using the multivariate chain rules. Apply the multivariate chain rules to answer applied questions about rates of change. Important Concepts The chain rules for multivariate functions are similar to the chain rule for functions of a single variable, in that they give a way to compute the derivative of a composite function.

71 2.6. CHAIN RULES FOR FUNCTIONS OF TWO OR MORE VARIABLES 68 More Details The multivariate chain rule in a single parameter gives a way to compute the ordinary derivative df /dt of the composition f ( x(t), y(t) ), f ( x(t), y(t), z(t) ), etc. (This is true for a function f of any number of variables, as long as all the variables are functions of the same parameters.) The multivariate chain rule in two parameters gives a way to compute the partial derivatives f / s and f / t of a composition f ( x(s, t), y(s, t) ), f ( x(s, t), y(s, t), z(s, t) ), etc. (This is true for a function f of any number of variables, as long as all the variables are functions of the same parameters. Similarly, there can be more than two parameters, as long as they are all the same.) Using Leibniz notation, the similarities of the different versions of the chain rule are apparent: single variable (Calc I): multivariate, single parameter: multivariate, general: df dt = df dx dx dt df dt = f dx x dt + f dy y dt + f dz z dt f t = f x x t + f y y t + f z z t The chain rule & directional derivatives: There is a connection between the multivariate chain rule for a function with a single parameter, and directional derivatives. Suppose an object is traveling along a path with position at time t given by r(t) = x(t), y(t), z(t), and f (x, y, z) is a function of position. Then df /dt is the change the object experiences in the function f with respect to time, as it travels along its path.

72 2.6. CHAIN RULES FOR FUNCTIONS OF TWO OR MORE VARIABLES 69 Suppose r (t) is the velocity of the object. Then: df dt = f x dx dt + f y = f r (t) = f ( r (t) ˆT ) = r (t) ( f ˆT ) = r (t) Dˆv f dy dt + f dz z dt So the derivative of f with respect to t is the directional derivative of f in the direction of velocity, times speed. When working with units, the final units of the derivative are the units of the function values over the units of the parameter eg: if the function gives the temperature in degrees celsius and the units of the parameter is time measured in seconds, then the units of the chain rule will be C/s).

73 2.6. CHAIN RULES FOR FUNCTIONS OF TWO OR MORE VARIABLES 70 Tree Diagrams: f ( x(t), y(t) ) f One Parameter t f ( x(t), y(t), z(t) ) f f x f y f x f y f z x y x y z dx dt dy dt dx dt dy dt dz dt t t Two Parameters s & t f ( x(s, t), y(s, t) ) f ( x(s, t), y(s, t), z(s, t) ) f f f x f y f x f y f z x y x y z x s y s x t y t x s y s z s x t y t z t s t s t

74 2.7. OPTIMIZATION: SECOND DERIVATIVE TEST Critical Points, Local Extrema, and the Second Derivative Test for Functions of Two Variables Textbook Section: 14.7 The Big Picture The theory and application of critical points and local extrema for functions of two variables are directly analogous to those for functions of a single variable. For functions of a single variable, critical points occur when the derivative equals zero, or does not exist. For functions of two variables, critical points occur when either both partial derivatives equal zero (which is the same as saying f = 0) or when one or both of the partial derivatives do not exist. For both single-variable and bivariate functions, local maximum and minimum values (extrema) can only occur at critical points. Also in both cases, a second derivative test can be used to determine whether critical points are local maxima, local minima, or inflection/saddle points. From the Toolbox (what you need from previous classes): Algebra: Solving systems of two equations in two variables. Calc I: Critical points, local extrema (maxima and minima) and the second derivative test for functions of a single variable (from Calc I). Calc III: First and second partial derivatives, and gradients of functions of two variables. Familiarity with level curves and contour maps.

75 2.7. OPTIMIZATION: SECOND DERIVATIVE TEST 72 Checklist: Computational Methods & Important Concepts Computational Methods Find critical points of functions of two variables. Use contour maps to determine whether a point is a local maximum, local minimum, or saddle point. Use the second derivative test to determine whether critical points are local minima, local maxima, or saddle points. Important Concepts Critical points occur when f (x, y) = 0 (a horizontal tangent plane), or when one or both of the partial derivatives does not exist (no tangent plane). Local maxima and minima occur only at critical points. Not every critical point corresponds to a local maximum or minimum. For example: saddle points can occur at critical point. (Saddle points are two-dimensional versions of inflection points.) For a critical point (a, b) where f (a, b) = 0 and D(a, b) 0, the second derivative test will determine whether there is a local maximum, local minimum, or saddle point at (a, b). More Details Definitions of local minimum, local maximum, and saddle points for a function f (x, y): f (a, b) is a local maximum if: f (a, b) f (x, y) for all points (x, y) near the point (a, b).

76 2.7. OPTIMIZATION: SECOND DERIVATIVE TEST 73 f (a, b) is a local minimum if: f (a, b) f (x, y) for all points (x, y) near the point (a, b). (a, b) is a saddle point if: f (a, b) = 0 f (a, b) is neither a local maximum nor a local minimum. (No matter how close you get to (a, b), there are some points (x, y) such that f (a, b) > f (x, y), and others such that f (a, b) < f (x, y).) The second derivative test for functions of two variables may look more complicated than the second derivative test for a function of a single variable, but it really measures the same thing: The Calc I second derivative test tells you whether the graph of the function y = f (x) lies entirely on one side of the tangent line. The Calc III second derivative test tells you whether the graph of the function z = f (x, z) lies entirely on one side of the tangent plane. When the test fails: There are two instances in which the second derivative test does not work, and you need to try something else to determine whether there is a local max or min at a critical point (a, b). These two instances are: The critical point occurs because one or both of the partial derivatives does not exist. In this case, the function D can t be used, since you need partial derivatives for D. If D(a, b) = 0, the test fails. Recall from Calc I: These are the same two conditions that will cause the second derivative test to fail.

77 2.7. OPTIMIZATION: SECOND DERIVATIVE TEST 74 Using the Second Derivative Test The second derivative test can be used when two conditions are met: i. f (a, b) = 0. ii. The second partial derivatives of f (x, y) exist and are continuous near (a, b). The function used to classify a critical point (a, b) is: D(a, b) = f xx (a, b)f yy (a, b) [ ] 2 f xy (a, b) or, in Leibnitz notation: [ ] D(a, b) = 2 f 2 f 2 2 x 2 y f 2 y x The test works as follows: If D(a, b) > 0 and f xx (a, b) < 0, then f (a, b) is a local maximum. (The graph of the function is concave up at the critical point.) If D(a, b) > 0 and f xx (a, b) > 0, then f (a, b) is a local minimum. (The graph of the function is concave down at the critical point.) If D(a, b) < 0, then f has a saddle point at (a, b). Note: If D(a, b) = 0, the test is inconclusive. You need to find another way to determine whether f (a, b) is a local maximum, minimum, or saddle point.

78 2.8. OPTIMIZATION: LAGRANGE MULTIPLIERS Optimization Subject to Constraint: the Method of Lagrange Multipliers Textbook Section: 14.8 The Big Picture The method of Lagrange multipliers makes use of the fact that, at the critical points of a function f subject to a constraint g = 0, the gradients f and g are parallel. Since parallel vectors are scalar multiples of each other, this results in the vector equation: f = λ g where λ is an unknown constant called the Lagrange multiplier. This vector equation, together with the equation g = 0, gives a system of n equations in n unknowns. The solutions to the system are the critical points of f subject to the constraint. Evaluating these critical points allows one to determine the maximum and minimum values of the function f, subject to the constraint g = 0. From the Toolbox (what you need from previous classes): Algebra: Solving systems of equations. Calc III: Computing gradients, and understanding the relationship between the gradient field and the level curves of a function f (x, y). Checklist: Computational Methods & Important Concepts Computational Methods Recognize a constraint on a function. Use the method of Lagrange multipliers to find the critical points of a function subject to a constraint.

79 2.8. OPTIMIZATION: LAGRANGE MULTIPLIERS 76 Evaluate the function at these critical points to determine which of these critical points are maxima and which are minima, and find the corresponding maximum and minimum values of the subject to the constraint. Important Concepts The method of Lagrange multipliers uses gradients to identify critical points. Critical points of a function f subject to a constraint g = 0 occur when the gradients f and g are parallel. In general, maxima and minima of function subject to a constraint are not the same as the local maxima and minima of the function over its entire domain. More Details The method of Lagrange multipliers is used to detect extrema of a multivariate function, subject to some constraint of the function s domain. For example, a function f (x, y) may be restricted to a curve in the xyplane. Or, one may want to find the minimum surface area of a box (the function) of minimum volume (constraint). The method of Lagrange multipliers assumes that the constraint on a differentiable function f can be expressed as the level curve g = 0 of a differentiable function g, so that: The level set g = 0 lies in the domain of f, and: g 0 on the level set g = 0. (This requirement is in place because, if f 0 by g = 0, there is no λ that makes the equation f = λ g = λ 0 true!) If this is the case, critical points of f subject to the constraint occur when the gradient of f is parallel to the gradient of g along the level set g = 0. This leads to an vector equation: f = λ g.

80 2.8. OPTIMIZATION: LAGRANGE MULTIPLIERS 77 Setting the coordinate functions of f equal to those of λ g gives a system of equations. Solving this system, together with the equation given by the constraint curve g = 0, returns the set of critical points of f subject to the constraint g = 0. To determine which of these critical points correspond to maxima or minima, evaluate f on these points and choose those for which the function takes on the largest and smallest values. Any points on the set g = 0 where g = 0 need to be evaluated separately, and the results compared with the maxima and minima produced using the method of Lagrange multipliers. Summary of Method: Identify f, the function being optimized. Identify the constraint, and express it as the level set g = 0 of a function g. Note that g and f have the same number of variables. Set f = λ g, where λ is an unknown constant (called the Lagrange multiplier). Find the critical points of f subject to the constraint, by solving the system of equations: f = λ g and g = 0. Evaluate f at the critical points to determine the maximum and minimum values among the critical points.

81 Chapter 3: Multiple Integrals 3.1 Introduction to Double Integrals (Cartesian Coordinates) Textbook Sections: 15.1, 15.2, 15.4 The Big Picture A double integral is an integral where the region of integration is a 2- dimensional region in the plane R 2, and the integrand is a function of two variables. There are many similarities between evaluating double integrals, and evaluating single integrals. In fact, double integrals are generally evaluated by evaluating two single integrals, using the techniques from in Calc I/II. When evaluating the first of the two single integrals, the second variable is treated as a constant (more Jedi mind tricks, analogous to those used to compute partial derivatives). One of the major differences in evaluating double integrals lies in finding the limits of integration. In Calc I and Calc II, the domain of integration was an interval in R, so the limits of integration were just constants the endpoints of the interval. For a double integral, however, the region of integration is a two-dimensional region in the plane R 2, and the limits of integration are given by the equations of the curves bounding the region. So in fact, the new thing to be learned for double integrals is finding limits of integration. From the Toolbox (what you need from previous classes): Algebra: Find points of intersection of the graphs of two functions; solve an equation in x and y in terms of either variable: y = g(x) or x = h(y). Calc I/II: Evaluate definite integrals of functions of a single variable. 78

82 3.1. INTRODUCTION TO DOUBLE INTEGRALS (CARTESIAN COORDINATES) 79 Checklist: Computational Methods & Important Concepts Computational Methods Set up and evaluate double integrals. This includes finding limits of integration, and determining whether a given region of integration requires more than one double integral. Use the limits of integration for a given double integral to sketch the region of integration, and change the order of integration. Use double integrals to compute area, volume, and mass. Important Concepts: The region of integration for a double integral is a two-dimensional region in the plane. Double integrals are evaluated by evaluating two single integrals, using the techniques from Calc I/II, while treating the second variable as a constant (more Jedi mind tricks, analogous to those used to compute partial derivatives). The area element da is the area of an infinitesimal (very very small) rectangle in the domain of integration. In Cartesian coordinates, da = dx dy = dy dx. The way da is written indicates the order of integration. Applications of the double integral include: Measuring the area A D of a planar region: A D = D da. Measuring the volume V of a region between a planar region D and the graph of a non-negative function f (x, y): V = D f (x, y) da, where f (x, y) 0 over D. Computing the mass m D of a 2-dimensional flat object: m D = D σ(x, y) da, where σ(x, y) 0 is a function giving the density per unit area at each point in D (a surface density).

83 3.1. INTRODUCTION TO DOUBLE INTEGRALS (CARTESIAN COORDINATES) 80 How to Read a Double Integral Double integrals over planar regions, in Cartesian coordinates: f (x, y) da D D is the region (or domain) of integration. D is a planar region: D R 2. f (x, y) is called the integrand. da is the area element. da represents the area of an infinitesimal (very very small) rectangle in R 2. In Cartesian coordinates: da = dx dy = dy dx Evaluating Double Integrals A double integral is evaluated by evaluating two single integrals, using the methods already learned in Calc I and II. The boundary curves of the region D determine the limits of integration. ˆ b ˆ g2 (x) ˆ [ b ˆ ] g2 (x) f (x, y) da = f (x, y) dy dx = f (x, y) dy dx D D f (x, y) da = a g 1 (x) ˆ d ˆ h2 (y) c h 1 (y) or: f (x, y) dx dy = a ˆ d c g 1 (x) [ ˆ ] h2 (y) f (x, y) dx dy h 1 (y) Evaluate the inside integral first, with respect to the variable indicated by the differential. Then evaluate the outside integral, with respect to the other variable. Jedi Mind Tricks: If the differential is dx, integrate with respect to x and treat y as a constant. If the differential is dy, integrate with respect to y and treat x as a constant.

84 3.1. INTRODUCTION TO DOUBLE INTEGRALS (CARTESIAN COORDINATES) 81 Note: the limits of integration of the outside integral are always constant. More Details If f (x, y) is continuous, and the boundary curves of D are continuous, the double integral D f (x, y) da can be evaluated as two single integrals (sometimes called iterated integrals): If the order of integration is da = dy dx, and the boundary curves are y = g 1 (x) and y = g 2 (x) with endpoints a x b, then: ˆ b ˆ g2 (x) ˆ [ b ˆ ] g2 (x) f (x, y) da = f (x, y) dy dx = f (x, y) dy dx D a g 1 (x) a g 1 (x) If the order of integration is da = dx dy, and the boundary curves are x = h 1 (y) and x = h 2 (y) with endpoints c y d, then: ˆ d ˆ h2 (y) ˆ [ d ˆ ] h2 (y) f (x, y) da = f (x, y) dx dy = f (x, y) dx dy D c h 1 (y) c h 1 (y) If f (x, y) is continuous, and the boundary curves of D are continuous, the order of integration does not matter: ˆ b ˆ g2 (x) ˆ d ˆ h2 (y) f (x, y) da = f (x, y) dy dx = f (x, y) dx dy D a g 1 (x) c h 1 (y) Some Applications of Double Integrals All applications of integrals begin with the idea that an integral works by chopping up and adding. For double integrals in particular, the region of integration is chopped up into infinitely many infinitesimal (very very small) rectangles of area da. The adding up is accomplished by integrating over the region D. Double Integrals and Area Chop up the region D into infinitesimal rectangles, each having area da. To find the total area of the

85 3.1. INTRODUCTION TO DOUBLE INTEGRALS (CARTESIAN COORDINATES) 82 region D, add up the areas of all of the rectangles by integrating da over D: Area D = da Double Integrals and Volume Suppose f (x, y) 0 over D. Chop up the region D into infinitesimal rectangles, each having area da. Then f (x, y) da is the volume of a box with height f (x, y) and (an infinitesimal) base of area da. To find the total volume between the surface generated by the graph of f and the xy-plane over the region D, add up the volume of all of the boxes by integrating f (x, y) da over D: Volume = f (x, y) da. Double Integrals and Mass Chop up the region D into infinitesimal rectangles, each having area da. If σ(x, y) 0 represents the surface density (density per unit area) at each point of the planar region D, then σ(x, y) da = density (infinitesimal) area = the mass of an infinitesimal rectangle with area da. To find the total mass m D of D, add up the masses of all the rectangles by integrating σ(x, y) da over D: m D = σ(x, y) da. D D (This application can be extended to probability. If σ(x, y) 0 over D and σ(x, y) da = 1, then σ is a probability density D function, and can be used to measure bivariate probabilities.) D Technical Details: Double Integrals as Limits of Riemann Sums The definition of a double integral is analogous to that of the definite integral of a function of a single variable from Calc I.

86 3.1. INTRODUCTION TO DOUBLE INTEGRALS (CARTESIAN COORDINATES) 83 Recall: Calc I In the single-variable case, the definite integral b a defined as follows: f (x) dx was The region of integration is an interval [a, b] on the real line R. This interval is is partitioned ( chopped up ) into n sub-intervals of length x. In each sub-interval, the integrand f (x) is evaluated at a point x i, and multiplied by x (the length of the sub-interval). These products f (x 1 ) x are added up (a Riemann sum) to approximate the integral: n f (x i ) x i=1 ˆ b a f (x) dx The smaller the subintervals, the better the approximation. If the integral exists, then the exact value of the definite integral is the limit of the Riemann sums as n ) (which forces the length x 0): lim n n f (x i ) x = i=1 ˆ b a f (x) dx Double Integrals The double integral f (x, y) da is defined as follows: D The region of integration is a two-dimensional region D in the plane R 2. This region is partitioned ( chopped up ) into small rectangles of area A = x y = y x. In each sub-rectangle, the integrand f (x, y) is evaluated at a point (x i, y j ), and multiplied by A (the area of the small rectangle): f (x i, y j ) A These products f (x i, y j ) A are added up (a Riemann sum) to approximate the double integral: M j=1 N f (x i, y j ) A i=1 D f (x, y) da

87 3.1. INTRODUCTION TO DOUBLE INTEGRALS (CARTESIAN COORDINATES) 84 If the integral exists, then the smaller the rectangles in the partition of D, the better the approximation, and the exact value of the double integral is the limit of the Riemann sums as M, N (in such a way that x, y 0): lim M,N M j=1 N f (x i, y j ) A = i=1 D f (x, y) da

88 3.2. DOUBLE INTEGRALS IN POLAR COORDINATES Double Integrals in Polar Coordinates Textbook Section: 15.3 The Big Picture Polar coordinates are used to express the location of a point in the plane in terms of its distance from the origin r, and the angle θ measured counterclockwise from the x-axis to the line through the point and the origin. They are useful when dealing with planar problems that possess rotational symmetry about the origin. When expressing a double integral in polar coordinates, it is important to remember that not only the limits and the integrand need to be expressed in polar coordinates; the area element da must be the polar area element, da = r dr dθ = r dθ dr. So: f (r, θ) da = f (r, θ) r dr dθ = f (r, θ) r dθ dr D From the Toolbox (what you need from previous classes): D Trig/Calc II: Convert equations in x and y into r and θ, using the change of variable functions x = r cos θ, y = r sin θ. Calc III: Set up and evaluate double integrals in Cartesian coordinates (this includes finding limits of integration); use the limits of integration for a given double integral to sketch the region of integration; be familiar with applications of the double integral. In particular, know how double integrals are used to compute area, volume, and mass. D Checklist: Computational Methods & Important Concepts Computational Methods

89 3.2. DOUBLE INTEGRALS IN POLAR COORDINATES 86 Set up and evaluate double integrals in polar coordinates. This includes finding limits of integration, converting the integrand from Cartesian to polar coordinates, and using the polar area element. Use the limits of integration for a given double integral in Cartesian coordinates to sketch the region of integration, and then change the integral to polar coordinates. Use double integrals in polar coordinates to compute area, volume, and mass. Important Concepts: Polar coordinates are used to express the location of a point in the plane in terms of its distance from the origin r, and the angle θ measured counterclockwise from the x-axis to the line through the point and the origin. Expressions in Cartesian coordinates x and y are converted to polar coordinates r and θ using the change of variable functions (also called coordinate functions) x(r, θ) = r cos θ and y(r, θ) = r sin θ. The polar area element is da = r dr dθ = r dθ dr. Any application of double integrals in Cartesian coordinates also exists for double integrals in polar coordinates. More Details In order to maintain well-defined coordinates, the values of the polar coordinates are restricted to radial values r 0 and angular values in a range of 2π that is, ω θ < ω + 2π (often, ω = 0, but there may be times where other values are used, for example: π θ < π). Polar Grid Curves The polar grid is defined by two families of curves:

90 3.2. DOUBLE INTEGRALS IN POLAR COORDINATES 87 Circles of radius r centered at the origin. These grid curves arise from holding the radius r constant: r = c. Rays (half-lines) from the origin. These grid curves arise from holding the angle θ constant: θ = c. Together, these two families of grid curves define the polar grid:

91 3.2. DOUBLE INTEGRALS IN POLAR COORDINATES 88 Polar Area Element da A polar rectangle is a region bounded on four sides by grid curves. The polar area element da is the area of an infinitesimal polar rectangle. The length of the sides of an infinitesimal polar rectangle parallel to the radial grid curves is dr. The length of the sides of an infinitesimal polar rectangle parallel to the circular grid curves is r dθ. So the polar area element da is: da = r dθ dr = r dr dθ. For comparison, the Cartesian area element is the area of an infinitesimal rectangle with sides parallel to the x- and y-coordinate axes. The lengths of the sides of this infinitesimal rectangle are dx and dy: da = dx dy = dy dx.

92 3.3. TRIPLE INTEGRALS IN CARTESIAN COORDINATES Triple Integrals in Cartesian Coordinates Textbook Section: 15.6 The Big Picture Computationally, triple integrals behave exactly like double integrals, with one additional integral to evaluate. To find limits of integration over a 3-dimensional region W requires first examining the bounding surfaces, then projecting the 3-dimensional region onto the coordinate plane perpendicular to the direction of the first (innermost) integral. The projection D is a planar region, whose boundary curves are the curves of intersection of the surfaces bounding W. The remainder of finding limits proceeds as it did with double integrals. From the Toolbox (what you need from previous classes): Calc I/II: Evaluate integrals of a single variable. Calc III: Find limits of integration for and evaluate double integrals over a planar region; sketch basic surfaces (planes, spheres, cones, paraboloids) and three-dimensional regions bounded by these surfaces. Checklist: Computational Methods & Important Concepts Computational Methods Set up and evaluate triple integrals. Identify the regions over which a triple integral is being evaluated. Use triple integrals to compute volume and mass. Important Concepts: The region of integration for a triple integral is a 3-dimensional region in 3-space.

93 3.3. TRIPLE INTEGRALS IN CARTESIAN COORDINATES 90 As with double integrals, triple integrals are evaluated by computing three single integrals, using the techniques from Calc I/II, and using the Jedi mind trick of treating the other variable(s) as constant(s). The volume element dv is the volume of an infinitesimal (very very small) box in the domain of integration. In Cartesian coordinates, dv = dx dy dz = dx dz dy =.... There are a total of six ways of writing dv, depending on the order of the differentials dx, dy, dz. Each way of writing dv indicates an order of integration. Applications of the triple integral include: Measuring the volume of a 3-dimensional region: V W = W dv. Computing the mass of a 3-dimensional object: Mass W = W ρ(x, y, z) dv, where ρ(x, y, z) 0 is a function giving the density per unit volume at each point in W (a volume density). How to Read a Triple Integral Triple integrals over 3-d regions: W f (x, y, z) dv W is the region (or domain) of integration. W is a 3-dimensional region: W R 3. f (x, y, z) is called the integrand. dv is the volume element. dv represents the volume of an infinitesimal (very very small) rectangular box in R 3. In Cartesian coordinates: dv = dx dy dz = dx dz dy = dy dx dz = dy dz dx = dz dx dy = dz dy dx More Details The definition of triple integrals is analogous to that of double integrals, with one additional dimension. Computationally, this means a triple integral can be evaluated as a series of three iterated (single) integrals.

94 3.3. TRIPLE INTEGRALS IN CARTESIAN COORDINATES 91 Some Applications of Triple Integrals All applications of integrals begin with the idea that an integral works by chopping up and adding. For triple integrals, the region of integration W is sliced by planes parallel to to the coordinate planes, chopping the region into infinitesimal boxes of volume dv. The adding up is accomplished by integrating over the region W. Triple Integrals and Volume Chop up the region W into infinitesimal boxes, each of volume dv. To find the total volume of the region W, add up the volumes of all of the boxes by integrating dv over W : dv W Triple Integrals and Mass Chop up the region W into infinitesimal boxes, each having volume dv. If ρ(x, y, z) 0 represents the volume density (that is, the density per unit volume at each point of 3-dimensional region W ), then ρ(x, y, z) dv = density (infinitesimal) volume = the mass of an infinitesimal box with volume dv. Then add up the masses of all the boxes by integrating ρ(x, y, z) dv over W : ρ(x, y, z) dv W Technical Details: Sums Triple Integrals as Limits of Riemann The definition of a triple integral is analogous to that of a double integral, with one additional dimension. For triple integrals, the integrand is a function of three variables f (x, y, z), and the domain W is a 3-dimensional region in R 3. Partition (chop up) the 3-dimensional region W using planes parallel to each of the coordinate planes, to create small boxes with side-lengths x, y, and z, and volume V = x y z (you can multiply in any order).

95 3.3. TRIPLE INTEGRALS IN CARTESIAN COORDINATES 92 In each sub-box, the integrand f (x, y, z) is evaluated at a point (x i, y j, z k ), and multiplied by V (the volume of a small box): f (x i, y j, z k ) V These products f (x i, y j, z k ) V are added up (a Riemann sum) to approximate the triple integral: L M k=1 j=1 i=1 N f (x i, y j, z k ) x y z W f (x, y, z) dx dy dz If the integral exists, then the smaller the boxes in the partition of W, the better the approximation, and the exact value of the double integral is the limit of the Riemann sums as L, M, N (in such a way that x, y, z 0): lim L,M,N L M k=1 j=1 i=1 N f (x i y j, z k ) V = W f (x, y, z) dv

96 3.4. TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES Triple Integrals in Cylindrical Coordinates Textbook Section: 15.7 The Big Picture Cylindrical coordinates (r, θ, z) are a combination of the polar coordinates (r, θ) in the xy-plane, and the Cartesian coordinate z along the z-axis. Cylindrical coordinates express the location of a point in 3-space in terms of its distance r from the z-axis, the polar angle θ measured counter-clockwise from the xz-plane, and the Cartesian z-coordinate. Cylindrical coordinates are useful when dealing with 3-dimensional problems that possess rotational symmetry about the z-axis. When expressing a triple integral in cylindrical coordinates, it is important to remember that not only do the limits of integration and the integrand need to be expressed in cylindrical coordinates, but also that dv must be the cylindrical volume element: f (x, y, z) dv = f ( x(r, θ, z), y(r, θ, z), z(r, θ, z) ) r dr dθ dz W W From the Toolbox (what you need from previous classes): Calc III: Convert functions from Cartesian to polar coordinates using the change of coordinate functions; know the area element da in polar coordinates, and be able to find limits of integration for double integrals in polar coordinates; know the volume element dv and be able to find limits of integration for triple integrals in Cartesian coordinates. Checklist: Computational Methods & Important Concepts Computational Methods Use the cylindrical change of coordinate functions to convert expressions in Cartesian coordinates to equations in cylindrical coordinates.

97 3.4. TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES 94 Set up and evaluate triple integrals in cylindrical coordinates. This includes finding limits of integration, converting the integrand from Cartesian to cylindrical coordinates, and using the cylindrical volume element. Identify the regions over which a triple integral is being evaluated. Use triple integrals to compute volume and mass. Important Concepts: Cylindrical coordinates (r, θ, z) are a combination of polar coordinates and Cartesian coordinates. They are used to express the location of a point in 3-space in terms of its distance r from the z-axis, the polar angle θ, and its z-coordinate from Cartesian coordinates. This gives the location of the point on a cylinder of radius r centered about the z-axis. Expressions in Cartesian and cylindrical coordinates are related by the change of variable functions (also called coordinate functions): x = r cos θ, y = r sin θ, z = z The cylindrical volume element is the product of the polar area element (base) times dz (height): dv = da polar dz = r dr dθ dz The differentials dr, dθ, and dz may be written in any order, specifying a total of six different orders of integration. More Details In order to have well-defined coordinates, the values of the cylindrical radius and angle are restricted to positive radial values, and angular values in a range of 2π: r 0

98 3.4. TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES 95 ω θ < ω + 2π, for example: 0 θ < 2π or π θ < π The cylindrical grid is defined by three families of surfaces generated by holding one of the parameters constant: Constant Radius: r = c Surfaces are cylinders centered about the z- axis. Constant Angle: θ = c Surfaces are half-planes perpendicular to the xy-plane. Constant Height: z = c Surfaces are planes parallel to the xy-plane.

99 3.4. TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES 96 The volume element dv in cylindrical coordinates is the volume of an infinitesimal box, with sides determined by the cylindrical grid. The area of the sides of this box that are parallel to the xy-plane is the polar area element da = r dr dθ. The height of this box is dz. So the cylindrical volume element is: dv = da polar dz = r dr dθ dz Equations of Some Common Surfaces Surface: \\ Coordinate System: Cartesian Cylindrical Cone z = k x 2 + y 2 r = z/k Sphere x 2 + y 2 + z 2 = c 2 r 2 + z 2 = c 2 Cylinder x 2 + y 2 = c 2 r = c Plane parallel to xy-plane z = c z = c Plane parallel to yz-plane x = c r cos θ = c Plane parallel to xz-plane y = c r sin θ = c (k and c are constants)

100 3.5. TRIPLE INTEGRALS IN SPHERICAL COORDINATES Triple Integrals in Spherical Coordinates Textbook Section: 15.8 The Big Picture Spherical coordinates (ρ, ϕ, θ) are used to express the location of a point in 3-space in terms of its distance ρ from the origin, the angle of declination ϕ measured from the positive z-axis (like latitude, but measured from North instead of the equator), and the polar angle θ measured counterclockwise from the xz-plane (this is the same angle from polar and cylindrical coordinates). In spherical coordinates, points are thought of a being located on spheres centered at the origin. Spherical coordinates are useful when dealing with 3-dimensional problems that possess symmetry about the origin. When expressing a triple integral in spherical coordinates, it is important to remember that not only do the limits of integration and the integrand need to be expressed in spherical coordinates, but also that dv must be the spherical volume element: f (x, y, z) dv = f (ρ sin ϕ cos θ, ρ sin ϕ sin θ, ρ cos ϕ) ρ 2 sin ϕ dρ dϕ dθ W W From the Toolbox (what you need from previous classes): Know what the volume element dv represents; find limits of integration for triple integrals in Cartesian and cylindrical coordinates. Checklist: Computational Methods & Important Concepts Computational Methods Use the spherical change of coordinate functions to convert expressions in Cartesian coordinates to equations in spherical coordinates.

101 3.5. TRIPLE INTEGRALS IN SPHERICAL COORDINATES 98 Set up and evaluate triple integrals in spherical coordinates. This includes finding limits of integration, converting the integrand from Cartesian to spherical coordinates, and using the spherical volume element. Use triple integrals to compute volume and mass. Important Concepts: Spherical coordinates (ρ, ϕ, θ) 1 are used to express the location of a point in 3-space in terms of its distance ρ from the origin, the angle of declination ϕ measured from the positive z-axis (like latitude, but measured from the North pole instead of the equator), and the polar angle θ measured counterclockwise from the xz-plane (this is the same angle from polar and cylindrical coordinates). Expressions in Cartesian coordinates are converted to expressions in spherical coordinates using the coordinate functions: x = ρ sin ϕ cos θ, y = ρ sin ϕ sin θ, z = ρ cos ϕ The spherical volume element is: dv = ρ 2 sin ϕ dρ dϕ dθ You can integrate in any order; however, it usually works best if your integrate with respect to ρ first. More Details The spherical radius ρ and the cylindrical radius r are not the same. ρ is the distance from the origin; r is the distance from the z-axis. The cylindrical radius r is the projection of the spherical radius ρ into the xy-plane, so: r = ρ sin ϕ 1 Note: Our textbook uses the order (ρ, θ, ϕ) instead of (ρ, ϕ, θ). The problem with using the order (ρ, θ, ϕ) is that it is not a right-handed coordinate system. In this class, as in most applications in science and engineering, we use the order (ρ, ϕ, θ) which is a right-handed coordinate system.

102 3.5. TRIPLE INTEGRALS IN SPHERICAL COORDINATES 99 In order to have well-defined coordinates, the values of the spherical radius and angles are restricted to: ρ 0 0 ϕ π (in other contexts, you may see π/2 ϕ π/2) ω θ < ω + 2π (for example, 0 θ < 2π or π θ < π). The spherical grid is defined by three families of surfaces generated by holding one of the parameters constant: Constant Radius: ρ = c Surfaces are spheres centered about the origin. (These spheres are cut open so you can see how they are nested inside each other.) Constant Angle of Declination: ϕ = c Surfaces are cones about the z-axis. Constant Polar Angle: θ = c Surfaces are half-planes planes perpendicular to the xy-plane. The volume element dv in spherical coordinates is the volume of an infinitesimal box, with sides determined by the spherical grid. The lengths of the sides of this box are: Sides that lie on the curves where ϕ and θ are constant (only ρ is changing) have length dρ.

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