MULTIVARIABLE CALCULUS MATH SUMMER 2013 (COHEN) LECTURE NOTES

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1 MULTIVARIABLE CALCULUS MATH SUMMER 2013 (COHEN) LECTURE NOTES These lecture notes are intended as an outline for both student and instructor; for much more detailed exposition on the topics contained herein, the student should consult Chapters of our standard textbook Calculus by Briggs and Cochran (NOT Early Transcendentals!). For the student s convenience these notes have the same section titles as Briggs/Cochran, arranged in the same order. 1 Preliminaries Definition 1.1. The symbol R denotes the set of all real numbers. Geometrically we think of R as the set of all points on the number line, or as 1-dimensional space. 2 Vectors in the Plane Definition 2.1. The plane, or Cartesian plane, or xy-plane, is the full set of all ordered pairs of the form (x, y) where x and y are both real numbers. We denote the plane by R 2. We think of R 2 as 2-dimensional space. The following definition is non-rigorous but well captures our intuitive notion of what a vector should be, and will be helpful when we consider applications of calculus to physical problems. We will give a more clear, i.e. purely mathematical, definition shortly. (Students who have taken a previous course in linear algebra may be familiar with a certain very general definition of a vector; the definitions we will end up working with here in fact a special case of the definition they have already learned.) Definition 2.2 (Naive Definition of a Vector). A vector is an object which consists of both a (strictly positive) length (or magnitude) and a direction. We typically denote vectors as lower-case letters with an arrow hat, e.g. u, v, w, a, b, etc. We consider two vectors u and v to be equal if they have both the same length and the same direction, and we write u = v. (For equality, note that we do not require u and v to have the same location!) We also allow the existence of a unique zero vector, denoted 0, which we consider to have 0 length and no direction. We may represent a vector v pictorially by drawing an arrow. We refer to pointy part at the end of the arrow as the head, and the base of the arrow as the tail. If P and Q are two points in (two-dimensional or three-dimensional) space, then we denote by P Q the vector which has its tail at P and its head at Q. (Note that unless P = Q, we always have P Q QP.) A scalar is just a magnitude, with no direction. In other words, a scalar is just a real number (a member of the set R). Example 2.3. Using the intuitive definition above, classify the following physical quantities as either a vector or a scalar. (a) A wind blowing southwest at 22 miles per hour. (b) The mass of an apple. (c) The force exerted by gravity on the apple. (d) The temperature in Denton, TX at 2pm today. (e) The velocity of a parked car at rest. Definition 2.4 (Naive Definition of Scalar Multiplication). Let v be a vector and let c be a scalar. The scalar product of c and v, denoted c v, is defined as follows: (1) if c > 0 then c v is the vector which points in the same direction as v, and whose length is c times the length of v. (2) if c < 0 then c v is the vector which points in the opposite direction from v, and whose length is c times the length of v. (3) if c = 0 then c v = 0. 1

2 2 MULTIVARIABLE CALCULUS MATH SUMMER 2013 (COHEN) LECTURE NOTES We call the vector c v a scalar multiple of v. Example 2.5. Draw any non-zero vector v. Then draw the vectors 3 v, 1 2 v, v = ( 1) v, 5 2 v, 0 v, and π v. Definition 2.6. Two vectors are called parallel if one is a scalar multiple of the other. Example 2.7. Section 12.1 Example 1. Definition 2.8 (Naive Definition of Vector Addition and Subtraction). Let u and v be vectors. If the tail of v is placed at the head of u, then the vector sum of u and v, denoted u + v, is the vector whose tail coincides with the tail of u and whose head coincides with the head of v. (Picture helps here.) The vector difference of u and v, denoted u v, is defined to be the vector sum u + ( 1) v. Example 2.9. Section 12.1 Example 2. Next we will give a more rigorous definition of vectors and vector operations by coordinatizing them in R 2. Definition A vector in the plane is an ordered pair v = (v 1, v 2 ) in R 2. (We think of the tail of v as being the origin (0, 0) and the head as the point (v 1, v 2 ).) Two vectors u = (u 1, u 2 ) and v = (v 1, v 2 ) are considered equal if both u 1 = v 1 and u 2 = v 2, in which case we write v = u. The number v 1 is called the x-component of v and v 2 is called the y-component of v. The magnitude of a vector v = (v 1, v 2 ), denoted by v, is the number v = v v2 2. Of course by the Pythagorean theorem, the magnitude of v is exactly its genuine geometric length. If c is a scalar, we define the scalar product of c and v to be the vector c v = (cv 1, cv 2 ). We define the vector sum of u and v to be the vector u + v = (u 1 + v 1, u 2 + v 2 ), and the vector difference to be the vector u v = (u 1 v 1, u 2 v 2 ). Example Let P = (x 1, y 1 ) and Q = (x 2, y 2 ) be two points in R 2. Compute the magnitude of the vector P Q. Example Let v = (v 1, v 2 ) be any vector and let c be any scalar. between the magnitudes v and c v. Fact c v = c v for any vector v and any scalar c. Compute the relationship Definition Let u = (u 1, u 2 ) and v = (v 1, v 2 ) be vectors in R 2 and let c be a scalar. Define the vector sum u + v, the vector difference u v, and the scalar product c v as follows: u + v = (u 1 + v 1, u 2 + v 2 ); u v = (u 1 v 1, u 2 v 2 ); c v = (cv 1, cv 2 ). Example Let u = ( 1, 2) and v = (2, 3). (a) Evaluate u + v. (b) Write 2 u 3 v as a single vector. (c) Find two distinct vectors half as long as u and parallel to u. Definition A unit vector is a vector of length 1. In particular we single out the coordinate unit vectors, which we permanently denote by i = (1, 0) and j = (0, 1). Note that if v = (v 1, v 2 ) is any vector, then we may write v = (v 1, v 2 ) = (v 1, 0) + (0, v 2 ) = v 1 (1, 0) + v 2 (0, 1) = v 1 i + v 2 j. So in general v = (v 1, v 2 ) = v 1 i + v 2 j; this gives us another notation for writing vectors. Fact If v is a non-zero vector, then the vector v v = 1 v v has the same direction as v, and magnitude 1. In other words v v is the unique unit vector which points in the same direction as v. Example Let P = (1, 2) and Q = (6, 10) be two points in the plane.

3 MULTIVARIABLE CALCULUS MATH SUMMER 2013 (COHEN) LECTURE NOTES 3 (a) Find P Q and two distinct unit vectors parallel to P Q. (b) Find two distinct vectors of length 3 parallel to P Q. Example Assume the water in a river moves southwest at 4 mi/hr. If a motorboat is traveling due east at 15 mi/hr relative to the shore, determine the speed of the boat and its heading relative to the moving water. Example A child pulls a wagon with a force of F = 20 lb at an angle of θ = 40 to the horizontal. Find the force vector F. Example A 400-lb engine is suspended from two chains that form 60 angles with a horizontal ceiling. How much weight must each chain withstand? 3 Vectors in Three Dimensions Definition 3.1. The set of all ordered triples (x, y, z) where x, y, and z are all real numbers is called xyz-space or three-dimensional space, and denoted permanently by R 3. Example 3.2. (1) Plot the points P = (3, 4, 5) and ( 2, 3, 4) in xyz-space. (2) Compute the distance between P and Q. Definition 3.3. A sphere is the set of all points in three-dimensional space which are exactly a fixed distance r from a single point (a, b, c) in R 3. (The point (a, b, c) is called the center of the sphere and the distance r is called the radius.) A ball is the set of all points which are strictly less than a fixed distance r from a single point (a, b, c). In other words a ball is the interior of a sphere. Example 3.4. (a) Find an equation in three variables whose graph is the sphere centered at the origin of radius 4. (b) Find an inequality in three variables whose graph is the ball centered at the origin of radius 4. (c) Find an equation for the sphere centered at (1, 2, 5) and containing the point (3, 4, 6). (d) Describe the set of points that satisfy the equation x 2 + y 2 + z 2 2x + 6y 8z = 1. Definition 3.5. A vector in three dimensions is an ordered triple in R 3. The magnitude of a vector v = (v 1, v 2, v 3 ) in three dimensions is the quantity v = v v2 2 + v2 3. The real numbers v 1, v 2, and v 3 are called the x-component, y-component, and z-component of v respectively. The notions of being parallel and of vector addition, vector subtraction, and scalar multiplication are analogously to the two-dimensional case. Definition 3.6. When working in R 3, the unit coordinate vectors are the following three distinguished vectors: i = (1, 0, 0); j = (0, 1, 0); k = (0, 0, 1). Note that for any vector v = (v 1, v 2, v 3 ) we have v = v 1 i + v 2 j + v 3 k. Example 3.7. Let u = (2, 4, 1) and v = (3, 0, 1). (a) Find 4 u + 2 v. (b) Find u v. (c) Find the unique unit vector with the same direction as u v. (d) Write the unit vector from part (c) as a sum of scalar multiples of i, j, and k. HOMEWORK (Due 7/9/13): Section 12.1 #18, 20, 26, 28, 30, 32, 34, 38, 40, 45, 51; Section 12.2 #36, 38, 39, 40

4 4 MULTIVARIABLE CALCULUS MATH SUMMER 2013 (COHEN) LECTURE NOTES 4 Dot Products Definition 4.1. Given two vectors u = (u 1, u 2, u 3 ) and v = (v 1, v 2, v 3 ), we define the dot product, denoted u v, to be the scalar given by u v = u 1 v 1 + u 2 v 2 + u 3 v 3. Example 4.2. (a) Let u = ( 3, 1, 0) and v = (1, 3, 0). Compute u v. (b) Let u = (2, 1, 2 3) and v = (2, 2, 3). Compute u v. The usefulness of the dot product emerges in the upcoming Theorem 4.4; in order to prove Theorem 4.4, we need to recall the following fact about triangles. Fact 4.3 (Law of Cosines). If a triangle has angle measures A, B, and C and corresponding opposite side lengths a, b, and c, then the following equalities hold: c 2 = a 2 + b 2 2ab cos C; b 2 = a 2 + c 2 2ac cos B; a 2 = b 2 + c 2 2bc cos A. The following theorem says that we can use dot products to compute the angles between given vectors. Theorem 4.4. Let u = (u 1, u 2, u 3 ) and v = (v 1, v 2, v 3 ) be non-zero vectors, and let θ be the angle between u and v with 0 θ π. Then u v = u v cos θ. Proof. Consider the triangle which has u and v for two of its sides. The third side is equal as a vector to u v, and hence the Law of Cosines (previous fact) implies that u v 2 = u 2 + v 2 2 u v cos θ. Now solving for u v cos θ in the above, we get: u v cos θ = 1 2 ( u v 2 u 2 v 2 ). Now by the definition of magnitude in R 3, we have u 2 = ( u u2 2 + u2 3 )2 = u u u 2 3. Similarly we have v 2 = v v v 2 3 and u v 2 = (u 1 v 1 ) 2 + (u 2 v 2 ) 2 + (u 3 v 3 ) 2. In that case, expanding out our equality from earlier and simplifying, we get: u v cos θ = 1 2 ( u v 2 u 2 v 2 ) = 1 2 ((u 1 v 1 ) 2 + (u 2 v 2 ) 2 + (u 3 v 2 ) 2 u 2 1 u 2 2 u 2 3 v1 2 v2 2 v3) 2 = 1 2 (2u 1v 1 + 2u 2 v 2 + 2u 3 v 3 ) = u 1 v 1 + u 2 v 2 + u 3 v 3 = u v. This proves the equality in the theorem. Example 4.5. (a) Let u = ( 3, 1, 0) and v = (1, 3, 0). Compute the angle θ between u and v. (b) Let u = (2, 1, 2 3) and v = (2, 2, 3). Compute the angle θ between u and v. Corollary 4.6. If the angle θ between two vectors u and v is π 2 (90 degrees), then u v = 0. Definition 4.7. Two vectors u and v are called orthogonal if u v = 0. (Orthogonal and perpendicular are synonyms in two dimensions.)

5 MULTIVARIABLE CALCULUS MATH SUMMER 2013 (COHEN) LECTURE NOTES 5 Definition 4.8. Given two vectors u and v, define the othogonal projection of u onto v, denoted proj v u, to be the vector component of u which lies in the direction of v. Define the scalar component of u in the direction of v, denoted scal v u, to be the length of the vector proj v u. We have already observed that v v is the unit vector which points in the same direction as v. Elementary geometric considerations show that scal v u = u cos θ, where θ is the angle formed by u and v. So we immediately have proj v u = u cos θ( v v. Theorem 4.9. For any two vectors u and v, ( u v proj v u = v v scal v u = ) v, and u v v. Example Find proj v u and scal v u for the following vectors. (a) u = (4, 1), v = (3, 4). (b) u = ( 4, 3), v = (1, 1). HOMEWORK (Due 7/10/13): Section 12.2 #17, 18, 19, 23, 24; Section 12.3 #11, 12, 14, 15, 17, 19, 20, 23, 24, 25, 26, 27, 28 Definition Let u and v be vectors in R 3. Define the cross product u v to be the vector with magnitude given by u v = u v sin θ, where θ is the angle between u and v (0 θ π), and with direction given by the right-hand rule: when you put the vectors tail to tail and let the fingers of your right hand curl from u to v, the direction of u v is the direction of your thumb, orthogonal to both u and v. (Note that the right-hand rule only makes sense if u and v are not parallel; if they are parallel and θ = 0 or θ = π, check that the given magnitude is 0 and hence u v = 0.) Example Find the direction and magnitude of u v, where u = (1, 1, 0) and v = (1, 1, 2). Example (a) In general is u v = v u? (b) In general is u v = v u? Fact 4.14 (Properties of the Cross Product). Let u, v, and w be any vectors in R 3 and let a and b be any scalars. The following properties all hold. (a) u v = ( v u) (b) (a u) (b v) = ab( u v) (c) u ( v + w) = ( u v) + ( u w) (d) ( u + v) w = ( u w) + ( v w) Example Evaluate all possible cross products of the coordinate vectors i, j, and k. Example Let u = (u 1, u 2, u 3 ) and v = (v 1, v 2, v 3 ) be any two vectors in R 3. Find a closed formula for the coordinates of u v. Theorem 4.17 (The Cross Product as a Determinant). Let u = (u 1, u 2, u 3 ) and v = (v 1, v 2, v 3 ). Then i j k [ ] [ ] [ ] u v = det u 1 u 2 u 3 u2 u = det 3 u1 u i det 3 u1 u j + det 2 k. v 2 v 3 v 1 v 3 v 1 v 2 v 1 v 2 v 3 Example Find a vector orthogonal to u = i + 6 k and v = 2 i 5 j 3 k.

6 6 MULTIVARIABLE CALCULUS MATH SUMMER 2013 (COHEN) LECTURE NOTES 5 Lines and Curves in Space Definition 5.1. A vector-valued function is a function which takes a real number for input, and outputs a vector (typically in R 2 or R 3 ). We will typically denote vector-valued functions with the arrow hat notation, e.g. r(t), f(t), etc., where t is regarded as the input variable. If r(t) outputs vectors in R 3, we may write r(t) = (x(t), y(t), z(t)), where x, y, and z are all real-valued functions. In this way we regard r(t) as an ordered triple of real-valued parametric functions. The graph of a vector-valued function r is the set of all possible outputs r(t). (Note that this definition differs somewhat from our usual definition of a graph.) Fact 5.2. An equation of the line in three-dimensional space passing through the point (x 0, y 0, z 0 ) in the direction of the vector v = (a, b, c) is r(t) = r 0 + t v, or (x, y, z) = (x 0, y 0, z 0 ) + t(a, b, c), for < t <. Equivalently, the parametric equations of the line are x(t) = x 0 + at; y(t) = y 0 + bt; z(t) = z 0 + ct. Example 5.3. (a) Find an equation of the line passing through (1, 2, 4) in the direction of v = (5, 3, 1). (b) Find an equation of the line passing through ( 3, 5, 8) and (4, 2, 1). Example 5.4 (Helix Curve). Graph the curve described by the equation r(t) = 4 cos t i + sin t j + t 2π k. Example 5.5 (Roller Coaster Curve). Graph the curve r(t) = cos t i + sin t j sin 2t k. Definition 5.6. Let a be a real number. A vector-valued function r(t) approaches the limit L as t approaches a, written lim = L, t a provided lim r(t) L = 0. t a The function r(t) is continuous at a provided r(a) exists, lim r(t) exists, and lim r(t) = r(a). The t a t a function r(t) is simply called continuous if it is continuous at every point in its domain. Fact 5.7. Let r(t) = (x(t), y(t), z(t)) be a vector-valued function. If lim x(t) = L 1, lim y(t) = L 2, and t a t a lim z(t) = L 3, then t a lim r(t) = (L 1, L 2, L 3 ). t a Also, r(t) is continuous at a point a provided x(t), y(t), and z(t) are all continuous at a. Example 5.8. Consider the function r(t) = cos πt i + sin πt j + e t k for t 0. (a) Evaluate lim r(t). t 2 (b) Evaluate lim r(t). t (c) At what points is r continuous? HOMEWORK (Due 7/11/13): Section 12.4 #10, 11, 15, 17, 23, 24, 26, 30; Section 12.5 #13, 15, 21, 23, 27, 34, 35

7 MULTIVARIABLE CALCULUS MATH SUMMER 2013 (COHEN) LECTURE NOTES 7 6 Calculus of Vector-Valued Functions Definition 6.1. Let r(t) = (x(t), y(t), z(t)) be a vector-valued function. We define the derivative of r(t), denoted r (t), to be r r(t + h) r(t) (t) = lim, h 0 h if the limit exists. If the derivative exists at a point t then we say r is differentiable at t. We also denote the derivative by d r(t). Note that the derivative r is a vector-valued function just like r. For any given t, the vector r (t) points in the same direction as the curve given by r(t); for this reason r (t) is called a tangent vector at r(t). We regard r (t) as the rate of change of the function r(t); for example, if r(t) is a position function in three-dimensional space, then r (t) is the associated velocity function. Fact 6.2. Let r(t) = x(t) i + y(t) j + z(t) k, where x, y, and z are all differentiable functions of t. Then r (t) = x (t) i + y (t) j + z (t) k. Example 6.3. Compute the derivative of the following functions. (a) r(t) = (t 3, 3t 2, t3 6 ) (b) r(t) = e t i + 10 t j + 2 cos(3t) k Example 6.4. Observe the behavior of r = (0, t 2, t 3 ) and r at t = 0 in part (a) of the example above. Definition 6.5. A vector-valued function r(t) is called smooth on an interval if it is differentiable on that interval, and also r (t) 0 on that interval. Let r(t) be a smooth vector-valued function on some interval [a, b]. The unit tangent vector for r(t) on [a, b] is T (t) = r (t) r (t). Example 6.6. Find the unit tangent vectors for the following functions. (a) r(t) = (t 2, 4t, ln t) for t > 0. (b) r(t) = (10, 3 cos t, 3 sin t) for 0 t 2π. Fact 6.7 (Derivative Rules). Let u(t) and v(t) be differentiable vector-valued functions and let f(t) be a differentiable scalar-valued function. Let a be a constant vector. (1) d a = 0 (2) d ( u(t) + v(t)) = u (t) + v (t) (3) d (f(t) u(t)) = f (t) u(t) + f(t) u (t) (Product Rule) (4) d u(f(t)) = f (t) u (f(t)) (Chain Rule) (5) d ( u(t) v(t)) = u (t) v(t) + u(t) v (t) (Dot Product Rule) (6) d ( u(t) v(t)) = u (t) v(t) + u(t) v (t) (Cross Product Rule) Example 6.8. Compute the following derivatives, where u(t) = t i+t 2 j t 3 k and v(t) = sin t i+2 cos t j + cos t k. (a) d v(t2 ) (b) (c) d [t2 v(t)] d [ u(t) v(t)]

8 8 MULTIVARIABLE CALCULUS MATH SUMMER 2013 (COHEN) LECTURE NOTES Example 6.9. Compute the first, second, and third derivatives of r(t) = (t 2, 8 ln t, 3e 2t ). Definition An antiderivative of a vector-valued function r is a function R for which R = r. If r(t) = x(t) i + y(t) j + z(t) k, then R(t) = X(t) i + Y (t) j + Z(t) k, where X, Y, and Z are antiderivatives of x, y, and z respectively. The collection of all antiderivatives R of r is called the indefinite integral of r, and denoted r(t). As is the case with real-valued functions, any two antiderivatives of r differ only by some constant vector C. So if R is any antiderivative of r, we may write r(t) = R(t) + C, where C is an arbitrary constant. Example Evaluate [ ] t + e t2 + 2 i 3t j + (sin 4t + 1) k. Example Find r(t) such that r (t) = (e 2, sin t, t) and r(0) = j. Definition Let r(t) = f(t) i + g(t) j + h(t) k be a vector-valued function, where f, g, and h are integrable on the interval [a, b]. We define the definite integral of r(t) across [a, b] to be the vector b a r(t) = [ b a f(t)] i + [ b a g(t)] j[ b a h(t)] k. Example Compute π 0 [ i + 3 cos( t 2 ) j 4t k]. HOMEWORK (Due 7/15/13): Section 12.6 #7, 8, 11, 13, 14, 19, 20, 21, 22, 26, 28, 29, 30, 31, 32, 36, 42, 43 7 Motion in Space Example 7.1. Consider the two-dimensional motion given by the position vector r(t) = (x(t), y(t)) = (3 cos t, 3 sin t) for 0 t 2π. (a) Sketch the trajectory of the object. (b) Find the velocity of the object. (c) Find the speed of the object. (d) Find the acceleration of the object. (e) Sketch the position, velocity, and acceleration vectors for t = 0, t = π 2, t = π, and t = 3π 2. Example 7.2. Consider the vector functions given by f(t) = (3t, t+2, t+3) and g(t) = 3t 2, t 2 +2, t 2 +3). (a) Sketch graphs of both functions. How do the graphs compare? (b) Compute the velocity functions f and g. How do they compare? (c) Graph both speed functions f and g. Definition 7.3. We say that a function r(t) models a uniform (constant velocity) straight-line motion if r is of the form r(t) = (x 0 + at, y 0 + bt, z 0 + ct) for some constants (x 0, y 0, z 0 ) and (a, b, c) in R 3. In this case the velocity is r (t) = (a, b, c) and the acceleration is r (t) = (0, 0, 0). We say that a function r(t) models a circular motion, or moves on a sphere, if r(t) = r for some fixed radius r 0, for every t in the domain of r. Example 7.4. An object moves on a trajectory described by r(t) = (3 cos t, 5 sin t, 4 cos t), for 0 t 2π. (a) Show that the object moves on a sphere, and find the radius of the sphere. (b) Find the velocity and speed of the object. Theorem 7.5. Suppose r(t) is differentiable and moves on a sphere, i.e. r(t) = r for some fixed r, for all t in the domain of r. Then the position vector r(t) and the velocity vector r (t) are orthogonal at every t in the domain of r.

9 MULTIVARIABLE CALCULUS MATH SUMMER 2013 (COHEN) LECTURE NOTES 9 Proof. First notice that since r(t) 2 = r(t) r(t) = r 2 is a constant function, we have d r(t) r(t) = 0. So, applying the dot product rule for derivatives, we have: 0 = d r(t) r(t) = r (t) r(t) + r(t) r (t) = 2 r (t) r(t). So we must have r (t) r(t) = 0, i.e. r and r are orthogonal. Fact 7.6. Gravity accelerates objects downward at a rate of approximately 32 ft/s 2, or 9.8 m/s 2. Example 7.7. A baseball is hit from 3 ft above home plate with an initial velocity in ft/s of v(0) = (80, 80). (a) Find functions v(t) and r(t) which model the position and velocity of the ball between the time it is hit and the time when it first hits the ground. (Neglect all forces acting on the ball except gravity.) (b) Show that the trajectory of the ball is a segment of a parabola. (c) Assuming a flat playing field, how far does the ball travel horizontally? Plot the trajectory of the ball. (Answer: t = 5.04s, x = 403ft.) (d) What is the maximum height of the ball? (Answer: y = 103ft.) (e) Does the ball clear a 20-ft fence that is 380 ft from home plate (directly under the path of the ball)? (t = 4.75s, y 22ft.) Example 7.8. Suppose an object is launched from the origin (in two dimensions) at some acute angle α (0 α π 2 ) with an initial speed of v 0 (where v 0 represents the initial velocity). Further suppose gravity is the only force acting on the object. (a) If v 0 = (u 0, v 0 ), use trigonometry to find expressions for u 0 and v 0 in terms of the direction α and the speed v 0. (b) Find general expressions for the velocity function v(t) and the position function r(t). (c) Find a closed formula for the total flight time of the object. (d) Find a closed formula for the total horizontal distance traveled by the object. (Which angle α maximizes the distance? Minimizes the distance?) (e) Find a closed formula for the maximum height of the object. Example 7.9. A golf ball is driven down a horizontal fairway with an initial speed of 55m/s at an initial angle of 25 (from a tee with negligible height). Neglect all forces except gravity and assume the ball s trajectory lies in a plane. (a) How far does the ball travel horizontally and when does it land? (236m, 4.7s.) (b) What is the maximum height of the ball? (27.6m.) (c) At what angles should the ball be hit to reach a green that is 300 m from the tee? (sin 2α =.972, α = 38.2 or α = 51.8.) Example A small projectile is fired over horizontal ground in an easterly direction with an initial speed of v 0 = 300 m/s at an angle of α = 30 above the horizontal. A crosswind blows from south to north producing an acceleration of the projectile of 0.36 m/s 2 to the north. (a) Where does the projectile land? (t = 30.6 s, (x, y, z) = (7953, 169, 0) m.) (b) In order to correct for the crosswind and make the projectile land due east of the launch site, at what angle from due east must the projectile be fired? ( 1.21.) HOMEWORK (Due 7/16/13): Section 12.7 #8, 9, 10, 19, 25, 26, 30, 32, 34, 36, 37, 38, 39 8 Length of Curves Fact 8.1. Let r(t) = (f(t), g(t), h(t)) be a parametrized curve, where f, g, and h are continuous, and the curve is traversed once for a t b. The arc length of the curve between (f(a), g(a), h(a)) and (f(b), g(b), h(b)) is L = b a r (t) = b a f (t) 2 + g (t) 2 + h (t) 2.

10 10 MULTIVARIABLE CALCULUS MATH SUMMER 2013 (COHEN) LECTURE NOTES We will omit the construction of the above formula here, but please see the intro to Section 12.8 in Briggs/Cochran for a plausibility argument. Example 8.2. Compute the circumference of a circle of radius a. Example 8.3. Find the length of the hypocycloid parametrized by r(t) = (cos 3 t, sin 3 t), where 0 t 2π. Example 8.4. An eagle rises at a rate of 100 vertical ft/min on a helical path given by r(t) = (250 cos t, 250 sin t, 100t), where r is measured in feet and t is measured in minutes. How far does it travel in 10 minutes? ( ) 9 Curvature and Normal Vectors Definition 9.1. Let r(t) be a smooth vector-valued function for all t a and let v(t) = r (t). If t = 1 for all t a, then we say that r is parametrized by arc length. Keep in mind for the following fact, we usually define r in such a way that a = 0. Fact 9.2. If r(t) is parametrized by arc length, and s(t) denotes the length of the curve parametrized by r from a to t, then s(t) = t a for every t a. Proof. By our arc length formula, s(t) = t a r (t) = t a v(t). Since v(t) = 1 for all t a, we have s(t) = t 1 = t a. a Example 9.3. Consider the helix parametrized by r(t) = 2 cos t, 2 sin t, 4t). Find the arc length function s(t). Is r parametrized by arc length? Definition 9.4. Recall from Definition 6.5 that the unit tangent vector associated to a smooth vectorvalued function r(t) is the function T (t) = r (t) r (t). The curvature of a smooth parametrized curve, denoted by κ, is the magnitude of the rate of change of T with respect to arc length s. In other words, if s denotes arc length, κ(s) = d T ds. We wish to have a method of computing curvature which works for any parameter t, and not just for the arc length parameter s as above. The following theorem gives us such a method. Theorem 9.5. Let r(t) be a smooth parametrized curve, where t is any parameter. If v = r is the velocity and T is the unit tangent vector, then the curvature is κ(t) = 1 v d T = T (t) r (t). Proof. Let s denote arc length of r. Note that the relationship between the derivatives of T with respect to s and to t is given by the chain rule: d T = d T ds ds. Now by the Fundamental Theorem of Calculus, we have ds = v, and hence, solving for d T ds, we get dt ds = 1 v d T. So by our original definition of curvature, we have

11 as claimed. MULTIVARIABLE CALCULUS MATH SUMMER 2013 (COHEN) LECTURE NOTES 11 κ(t) = d T ds = 1 v d T = T (t) r Example 9.6. Which has greater curvature: A circle with a large radius or a circle with a small radius? Theorem 9.7. Let r be the position of an object moving on a smooth curve. Let v = r be the velocity and a = v the acceleration of the object. Then the curvature at a point on the curve is a v κ = v 3. Proof. Since T = v v, write v = v T. Differentiate both sides above using the product rule. Next compute the cross product a v: a = ( d v ) T + v d T. a v = [( d v ) T + v d T ] [ v T ] = ( d v T v T + v d T v T = 0 + v 2 ( d T T ) = v 2 ( d T T ). Now notice that T and d T between T and d T are orthogonal since T moves in a sphere! (See Theorem 7.5. So the angle is π 2, and hence the cross product d T T has magnitude Putting everything together, we have d T T = d T T sin π 2 = d T 1 1 = d T. a v = v 2 d T T = v 2 d T = v 3 = v 3 κ. 1 v d T So κ = a v v 3 as claimed. Example 9.8. Find the curvature of the parabola r(t) = (t, at 2 ). Example 9.9. Find the curvature of the helix r(t) = a cos t, a sin t, bt), where a, b > 0 are real numbers.

12 12 MULTIVARIABLE CALCULUS MATH SUMMER 2013 (COHEN) LECTURE NOTES Definition Let r be a smooth parametrized curve. The principal unit normal vector associated to r is N = d T /ds dt /ds = 1 κ d T ds. We think of the principal unit normal vector as the (unit vector) direction in which the graph is curving. In practice it suffices to use the equivalent formula: N = d T / d T /. Example Find the principal unit normal vector for the helix r(t) = (a cos t, a sin t, bt), where a, b > 0 are real numbers. Fact Let r be a smooth parametrized curve with unit tangent vector T and principal unit normal vector N. Then T and N are orthogonal. Proof. Since T moves in a sphere, it is orthogonal to its own derivative d T by Theorem 7.5. Since N points in the same direction as d T by definition, T and N are orthogonal. Theorem Let r be a smooth parametrized curve describing the motion of an object through space. Then the acceleration of the function a = r has a unique representation as the sum of its tangential component a T and its normal component a N : a = a N N + a T T, where a N = κ v 2 = a v v and a T = d2 s 2. Proof. We begin with the fact that T = v v and hence v = T v = T ds. Differentiating both sides and using the product rule and chain rule respectively, we get Since d T ds = κ N and ds = v, we get a = d v = d ( T ds ) = d T = d T ds ds + T d2 s 2 ds ds + T d2 s 2. a = κ v 2 N + d 2 s 2 T and we are done. Example The driver of a car follows the parabolic trajectory r(t) = (t, t 2 ) for 2 t 2 through a sharp bend in the road. Find the tangential and normal components of the acceleration of the car. HOMEWORK (Due 7/17/13): Section 12.8 #7, 10, 13, 14; Section 12.9 #9, 10, 13, 15, 17, 18, 24, 26 Test 1 7/18/13. No HW due! HOMEWORK: (Due 7/22/13): Section 12.9 #24, 26, 28, 29, 30, 32, 37, 38, 40

13 MULTIVARIABLE CALCULUS MATH SUMMER 2013 (COHEN) LECTURE NOTES Planes and Surfaces In this section we begin to study functions which depend on multiple real inputs (or vector inputs), most typically of the form f(x, y), and their graphs. We also study equations in three variables and their graphs. We start with perhaps the simplest possible graphs in R 3 : Definition Given a fixed point P 0 = (x 0, y 0, z 0 ) in R 3 and a non-zero vector n = (a, b, c) in R 2, the set of all points P = (x, y, z) in R 3 for which P P 0 is orthogonal to n is called a plane. (The vector n is called the normal vector to the plane.) Example Given a point P 0 and a normal vector n as in the above definition, find an equation in three variables whose graph is the plane determined by P 0 and n. Fact The plane passing through P 0 = (x 0, y 0, z 0 ) with normal vector n = (a, b, c) is described by the equation a(x x 0 ) + b(y y 0 ) + c(z z 0 ) = 0. Example Find an equation of the plane passing through (2, 3, 4) and with normal vector n = ( 1, 2, 3). Example Find an equation of the plane that passes through the points (2, 1, 3), (1, 4, 0), and (0, 1, 5). Definition Two distinct planes are called parallel if their respective normal vectors are parallel. Two planes are orthogonal if their respective normal vectors are orthogonal. Example Which of the following distinct planes are parallel and which are orthogonal? Q: 2x 3y + 6z = 12 R: x + 3 2y 3z = 14 S: 6x + 8y + 2z = 1 T : 9x 12y 3z = 7 Example Find a parametrized vector-valued function whose graph is the line of intersection of the planes Q: x + 2y + z = 5 and R: 2x + y z = 7. Definition Given a curve C in a plane P and a line l not in P, a cylinder is the surface consisting of all lines parallel to l that pass through C. (Note that this is a rare occasion where our definition probably does not coincide with the student s intuition from a previous math course.) Definition A trace of a surface is the set of points at which the surface intersects a plane which is parallel to one of the coordinate planes. (Informally, a trace is a cross-section of a surface.) The traces in the coordinate planes are called the xy-trace, the xz-trace, and the yz-trace. Example Use traces to sketch graphs of the following cylinders in R 3. (a) x 2 + 4y 2 = 16 (b) z sin x = 0 See Section 13.1, Table 13.1 (p. 774) in Briggs/Cochran for a full summary of the next few examples, which we will probably have limited in-class time for. Example (Ellipsoid). An ellipsoid is the graph of an equation of the form x2 a 2 Graph the ellipsoid with a = 3, b = 4, and c = 5. + y2 b 2 + z2 c 2 = 1. Example (Elliptic Paraboloid). An elliptic paraboloid is the graph of an equation of the form z = x2 a + y2 2 b. Graph the elliptic paraboloid with a = 4 and b = 2. 2 Example (Hyperboloid of One Sheet). Graph the equation x2 4 + y2 9 z2 = 1. Example (Hyperbolic Paraboloid). Graph the equation z = x 2 y2 4. Example (Elliptic Cone). Graph y2 4 + z2 = 4x 2. Example (Hyperboloid of Two Sheets). Graph 16x 2 4y 2 + z x 80 = 0. HOMEWORK (Due 7/23/13): Section 13.1 #12, 16, 23, 27, 29, 33, 35, 38, 42, 45, 49, 57

14 14 MULTIVARIABLE CALCULUS MATH SUMMER 2013 (COHEN) LECTURE NOTES 11 Graphs and Level Curves Definition A function in two variables f(x, y) is a function which assigns a unique (real) output to each input pair (x, y) from a particular set D in R 2. The set D is called the domain of f and is often implicit rather than explicitly described. The range of f is the set of all real numbers z = (x, y) which are assumed as (x, y) ranges over the domain D. The graph of a function in two variables is the set of all triples (x, y, z) in R 3 for which z = f(x, y). Definition 11.2 (Set-Builder Notation). To facilitate the next example and the homework problems, we recall for the student the use of set-builder notation. For convenience we will describe informally using examples, rather than give a formal definition. For an example, suppose we wish to formally describe the set D of all ordered pairs (x, y) for which x is twice y. Then we may write A = {(x, y) : x = 2y}. The above notation should be read as The set of all (x, y) in R 2 such that x = 2y, which clearly and precisely defines our set A. For another example, we could write Y = {x R : 5 x < 10}, which reads The set of all x in R such that 5 is less than or equal to x and x is strictly less than 10. The student should easily verify that, using the interval notation, Y = [5, 10). In general, given a set A and a precise mathematical sentence P (x) about a variable x, the set-builder notation should be read as follows. { x A : P (x)} The set of all elements x in A such that sentence P (x) is true for the element x. Example Let g(x, y) = 4 x 2 y 2. (a) Find the domain and range of g. (b) Sketch the graph of g. Definition Let f(x, y) be a function. Given any fixed real number z 0, the level curve of f at z 0 is the set of all (x, y) in R 2 for which f(x, y) = z 0. Example Sketch some of the level curves of the following functions. (a) f(x, y) = y x 2 1 (b) f(x, y) = e x2 y 2 12 Limits and Continuity Definition Let f(x, y) be a function in two variables. The function f has the limit L as (x, y) approaches (a, b), denoted if given any ɛ > 0, there exists a δ > 0 such that lim f(x, y) = L, (x,y) (a,b) f(x, y) L < ɛ whenever (x, y) (a, b) < δ. A function f(x, y) is continuous at a point (a, b) provided f(a, b) exists, lim f(x, y) = f(a, b). (x,y) (a,b) Example Evaluate Example Evaluate lim (x,y) (2,8) (3x2 y + xy, if it exists. lim (x,y) (0,0) 3x 2 y 2, if it exists. lim f(x, y) exists, and (x,y) (a,b)

15 MULTIVARIABLE CALCULUS MATH SUMMER 2013 (COHEN) LECTURE NOTES 15 Definition A disk in R 2 is the set {(x, y) : (x, y) (a, b) < r} for some fixed point (a, b) and some fixed radius r. Let R be a region of R 2. A point P in R is called an interior point if there is some disk about P which is contained entirely in R. A point Q is called a boundary point if every disk containing Q contains both a point in R and a point not in R. A region R is open if it consists entirely of interior points. A region is closed if it contains all its boundary points. Example Evaluate Example Evaluate lim (x,y) (4,1) lim (x,y) (0,0) xy 4y 2 x 2 y, if it exists. (x + y) 2 x 2, if it exists. + y2 HOMEWORK (Due 7/24/13): Section 13.2 #12, 14, 15, 22, 27, 28, 29, 32, 33, 34; Section 13.3 #11, 12, 16, 25, Partial Derivatives Definition Let f(x, y) be a function in two variables. The partial derivative of f with respect to x is f x (x, y) = δ f(x + h, y) f(x, y) δxf(x, y) = lim, h 0 h provided the limit exists. The partial derivative of f with respect to y is f y (x, y) = δ f(x, y + h) f(x, y) δy f(x, y) = lim, h 0 h provided the limit exists. Example Let f(x, y) = x 2 y (a) Compute δf δf δx and δy. (b) Evaluate each derivative at (2, 4). Example Compute the partial derivatives of the following functions. (a) f(x, y) = sin xy (b) g(x, y) = x 2 e xy Definition Let f(x, y) be a function in two variables. The second-order partial derivatives of f are the following four functions (written with both available notations): ) = δ2 f δx of (f 2 x ) x = f xx ; ( δ δf δx δx ( δ δf δy δy ( δ δf δx δy ( δ δf δx δy ) ) ) = δ2 f δy 2 of (f y ) y = f yy ; = δ2 f δxδy of (f y) x = f yx ; = δ2 f δyδx of (f x) y = f xy. The latter two derivatives above are called mixed partial derivatives. Example Compute the second-order partial derivatives of f(x, y) = 3x 4 y 2xy + 5xy 3. Example 13.6 (Special Example). Let xy(x 2 y 2 ) f(x, y) = x 2 + y 2, if (x, y) (0, 0) 0, if (x, y) = (0, 0).

16 16 MULTIVARIABLE CALCULUS MATH SUMMER 2013 (COHEN) LECTURE NOTES (a) Is f continuous? (b) Compute f xy (0, 0) and f yx (0, 0). Are the mixed partial derivatives equal to one another? The above example shows that in general f xy need not equal f yx. However, the following theorem says that in most cases we care about, the equality f xy = f yx really does hold. The proof is beyond the scope of this course and is omitted. Theorem Assume that f is defined on an open set D of R 2, and f xy and f yx are continuous at every point of D. Then f xy = f yx at every point of D. HOMEWORK (Due 7/25/13): Section 13.3 #19, 20, 22; Section 13.4 #8, 10, 11, 12, 15, 18, 21, 22, 27, 28, 29, 32, 36, 48, The Chain Rule Theorem 14.1 (Chain Rule (One Independent Variable)). Let z be a differentiable function of x and y, where x and y are differentiable functions of t. Then dz = δz dx δx + δz dy δy. Example Let z = x 2 3y , where x = 2 cos t and y = 2 sin t. Find dz t = π 4. and evaluate it at Theorem 14.3 (Chain Rule (Two Independent Variables)). Let z be a differentiable function of x and y, where x and y are differentiable functions of s and t. Then δz δs = δz δx δx δs + δz δy δy δs δz and δt = δz δx δx δt + δz δy δy δt. Example Let z = sin 2x cos 3y, where x = s + t and y = s t. Evaluate δz δs and δz δt. Example 14.5 (Special Exercise - Implicit Differentiation). Use a function in two variables and the chain rule to find dy dx, where sin(xy) + πy2 = x. Fact Let F be a differentiable function in two variables, and suppose a relationship between y and x is defined implicitly by the rule F (x, y) = 0. If F y 0, then dy dx = Fx F y. Example Find dy dx, where x4 + 3x 2 y 2 y = Directional Derivatives and the Gradient Definition Let f be differentiable at (a, b) and let u = (cos θ, sin θ) be a unit vector in R 2. The directional derivative of f at (a, b) in the direction of u is f(a + h cos θ, b + h sin θ) f(a, b) D u f(a, b) = lim, h 0 h provided the limit exists. Theorem Let f be differentiable at (a, b) and let u = (u 1, u 2 ) be a unit vector in R 2. Then D u f(a, b) = (f x (a, b), f y (a, b)) (u 1, u 2 ). Proof. Define a new function g(s) (real inputs and real outputs) by the rule g(s) = f(a + su 1, b + su 2 ). Geometrically, the graph of g is a cross-section of the graph of f passing through the point (a, b) parallel to vector u. It is clear from the definition of the directional derivative that D u f(a, b) = g (0).

17 MULTIVARIABLE CALCULUS MATH SUMMER 2013 (COHEN) LECTURE NOTES 17 Now setting x(s) = a + su 1 and y(s) = b + su 2 (so g(s) = (x(s), y(s)) and applying the chain rule, we get D u f(a, b) = g (0) = f x (x(0), y(0))x (0) + f y (x(0), y(0))y (0) = f x (a, b)u 1 + f y (a, b)u 2 ) = (f x (a, b), f y (a, b)) (u 1, u 2 ). Example Consider the paraboloid z = f(x, y) = 1 4 (x2 +2y 2 )+2 and the unit vectors u = ( 1 2, 1 2 ) and v = ( 1 2, 3 2 ). (a) Find the directional derivative of f at (3, 2) in the directions of u and v. (b) Graph the surface and interpret the directional derivatives. Definition Let f(x, y) be differentiable. The gradient of f at (x, y) is the function f(x, y) = (f x (x, y), f y (x, y)). Example Find f and f(3, 2) for f(x, y) = x 2 + 2xy y 3. Example Let f(x, y) = 3 x xy2 10. (a) Compute f(3, 1). (b) Compute D u f(3, 1) where u = ( 1 2, 1 2 ). (c) Compute the directional derivative of f at (3, 1) in the direction of the vector (3, 4). Theorem Let f be differentiable at (a, b). (1) f has its maximum rate of increase at (a, b) in the direction of the gradient f(a, b). The rate of increase in this direction is f(a, b). (2) f has its maximum rate of decrease at (a, b) in the direction of f(a, b). The rate of decrease in this direction is f(a, b). (3) If u is orthogonal to f(a, b), then D u f(a, b) = 0. Proof. For any unit vector u, we have D u = f(a, b) u = f(a, b) u cos θ = f(a, b) cos θ, where θ is the angle between f(a, b) and u. Then cos θ is maximized when θ = 0 and minimized when θ = π, which proves statements (1) and (2) above. If f(a, b) and u are orthogonal then θ = π 2 and hence cos θ = 0; this shows statement (3). Example Consider the bowl-shaped paraboloid z = f(x, y) = 4 + x 2 + 3y 2. (a) If you are located at the point (2, 1 2, 35 4 ) on the paraboloid, in which direction should you move in order to ascend the surface at the maximum rate? How quickly will you ascend? (b) If you are at the point (3, 1, 16), in which directions may you walk in order to neither gain nor lose height? HOMEWORK (Due 7/29/13): Section 13.5 #8, 10, 12, 18, 20, 22, 28, 30, 32; Section 13.6 #10, 12, 14, 16, 18, 21, 22

18 18 MULTIVARIABLE CALCULUS MATH SUMMER 2013 (COHEN) LECTURE NOTES 16 Tangent Planes Definition Let f(x, y) be differentiable at (a, b). The plane tangent to the graph of f at (a, b) is the graph of the equation z f(a, b) = f x (a, b)(x a) + f y (a, b)(y b). More generally, if F (x, y, z) is a function of three variables and the equation F (x, y, z) = 0 defines a surface, then the tangent plane at a point (a, b, c) on the surface is the graph of the equation F x (a, b, c)(x a) + F y (a, b, c)(y b) + F z (a, b, c)(z c) = 0. Example Find an equation for the plane tangent to the paraboloid f(x, y) = 32 3x 2 4y 2 at the point (2, 1, 16). Example Consider the ellipsoid defined by x2 9 + y z2 = 1. (1) Find the equation of the plane tangent to the ellipsoid at (0, 4, 3 5 ). (2) At what points on the ellipsoid is the tangent plane horizontal? 17 Maximum/Minimum Problems Definition Let f be a function in two variables. We say f has a local maximum at (a, b) is there is some disk D containing (a, b) such that f(a, b) f(x, y) for all (x, y) in D. We say that f has a local minimum at (a, b) if there is some disk D containing (a, b) such that f(a, b) f(x, y) for all (x, y) in D. In either case, we say that f has a local extremum at (a, b). A point (a, b) is a critical point of f if either (1) f x (a, b) = f y (a, b) = 0 or (2) one (or both) of f x and f y does not exist at (a, b). Fact If f has a local maximum or minimum at (a, b), then (a, b) is a critical point of f. Example Find the critical points of f(x, y) = xy(x 2)(y + 3). Definition Let f be a function in two variables. We say that f has a saddle point at (a, b) if (a, b) is a critical point, but for every disk D containing (a, b) there are points (x, y) in D for which f(x, y) > f(a, b) and points (x, y) in D for which f(x, y) < f(a, b) (in other words f has neither a min nor a max at (a, b)). Theorem 17.5 (Second Derivative Test). Suppose that the second partial derivatives of f(x, y) are continuous in a disk containing (a, b), where f x (a, b) = f y (a, b) = 0. Set D(x, y) = [f xx f yy f 2 xy](x, y). (1) If D(a, b) > 0 and f xx (a, b) < 0, then f has a local maximum value at (a, b). (2) If D(a, b) > 0 and f xx (a, b) > 0, then f has a local minimum value at (a, b). (3) If D(a, b) < 0, then f has a saddle point at (a, b). (4) If D(a, b) = 0, the test is inconclusive. Definition The quantity D(x, y) in the above [ Theorem] 17.5 is called the discriminant of f. fxx f D(x, y) is the determinant of the Hessian matrix xy. f yx f yy Example Classify all the critical points of f(x, y) = x 2 + 2y 2 4x + 4y + 6. Example Classify all the critical points of f(x, y) = xy(x 2)(y + 3). Example A shipping company handles rectangular boxes provided the sum of the length, wih, and height of the box does not exceed 96 in. Find the dimensions of the box that meets the condition at has the largest volume. Definition If f(x, y) f(a, b) for all (x, y) in the domain of f, then f has an absolute maximum at (a, b). If f(x, y) f(a, b) for all (x, y) in the domain of f, then f has an absolute minimum at (a, b).

19 MULTIVARIABLE CALCULUS MATH SUMMER 2013 (COHEN) LECTURE NOTES 19 Example Find the absolute maximum and minimum values, if they exist, of f(x, y) = 4 x 2 y 2 on the open disk R = {(x, y) : x 2 + y 2 < 1}. Fact (Extreme Value Theorem). If f is continuous on a closed bounded region R in R 2, then f obtains an absolute maximum and absolute minimum value on R. Example Find the absolute maximum and minimum values of f(x, y) = x 2 + y 2 2x + 2y + 5 on the set R = {(x, y) : x 2 + y 2 4}. Example Find the absolute maximum and minimum values of f(x, y) = 6 x 2 4y 2 on the set R = {(x, y) : 2 x 2, 1 y 1}. HOMEWORK (Due 7/30/13): Section 13.7 #11, 12, 17, 18; Section 13.8 #15, 16, 19, 20, 22, 23, 29, Double Integrals over Rectangular Regions Definition Let f be a function in two variables, and let R = {(x, y) : a x b, c y d} be a rectangular region in R 2. Given any positive integer n, set x = b a d c n and y = n. For each integer k with 0 k n, set x k = a + k y and y k = c + k y. For each pair of integers j, k with 1 j n and 1 k n, let (x j, y k ) be a point chosen arbitrarily from the rectangular region {(x, y) : x j 1 x x j, y k 1 y y k }. We define the double integral, or double definite integral, of f over R to be n n f(x, y)d(x, y) = lim f(x j, y k ) x y, R n j=1 k=1 provided the limit exists and is independent of the choices of (x k, y k ). (Note: Our notation differs slightly from Briggs/Cochran here.) If the limit exists we say that f is integrable over R. If f is non-negative on R, then the double definite integral corresponds to the volume of the solid bounded by the graph of f over R. Theorem 18.2 (Fubini s Theorem). Let f be continuous on the rectangular region R = {(x, y) : a x b, c y d}. Then R f(x, y)d(x, y) = d b c a f(x, y)dxdy = b d f(x, y)dydx. a c Example Find the volume of the solid bounded by the surface z = 4 + 9x 2 y 2 over the region R = {(x, y) : 1 x 1, 0 y 2}. Use both possible orders of integration. Example Evaluate R xexy d(x, y), where R = {(x, y) : 0 x 1, 0 y ln 2}. HOMEWORK (Due 7/31/13): Section 13.8 #37, 38, 39; Section 14.1 #6, 7, 8, 10, 11, 13, 15, 16, 19, 20, 21, Double Integrals over General Regions Fact Let g and h be continuous functions in one variable. Suppose R is a region in R 2 bounded below and above by the graphs of y = g(x) and y = h(x) respectively, and the lines x = a and x = b. If f is continuous on R, then R f(x, y)d(x, y) = b a h(x) f(x, y)dydx. g(x) Alternatively, if R is bounded on the left and right by the graphs of x = g(y) and x = h(y) respectively, and the lines y = c and y = d, and f is continuous on R, then R f(x, y)d(x, y) = d h(y) f(x, y)dxdy. c g(y)

20 20 MULTIVARIABLE CALCULUS MATH SUMMER 2013 (COHEN) LECTURE NOTES Example Compute the integral R 2x2 yd(x, y), where R is the region bounded by the parabolas y = 3x 2 and y = 16 x 2. Example Compute the volume of the solid below the surface f(x, y) = 2+ 1 y and above the region R in the xy-plane bounded by the lines y = x, y = 8 x, and y = 1. Example Evaluate π π sin x 2 dxdy. 0 y HOMEWORK (Due 8/6/13): Section 14.2 #14, 15, 16, 18, 19, 20, 29, 30, 33, 43, 49, 50, 51, Double Integrals in Polar Coordinates Definition To each pair (r, θ) of real numbers with r 0, we (implicitly) associate the pair (x, y) in R 2 where x = r cos θ and y = r sin θ. We refer to (r, θ) as the polar coordinates of (x, y). Notice that any pair (x, y) has infinitely many polar coordinatizations, as (r, θ + 2πn) gives the same (x, y) for any choice of integer n. A polar rectangle is a set of the form {(r, θ) : 0 a r b, α θ β}, where a, b, α, β are real numbers with β α 2π. Theorem Let f be continuous on the region R = {(r, θ) : 0 a r b, α θ} (where (rθ) corresponds to a point (x, y) = (r cos θ, r sin θ) as in the above definition). Then R f(r, θ)d(x, y) = β f(r, θ)rdrdθ. α Example Find the volume of the solid bounded by the paraboloid z = 9 x 2 y 2 and the xy-plane. Example Find the volume of the region bounded beneath the surface z = xy + 10 and above the annular region R = {(r, θ) : 2 r 4, 0 θ 2π}. Example Compute 3 9 x 2 x2 + y dydx. Fact Let f be continuous on the region R = {(r, θ) : 0 g(θ) r h(θ), α θ β}, where g and h are continuous functions of θ. Then R f(r, θ)d(x, y) = β h(θ) f(r, θ)rdrdθ. α g(θ) 21 Triple Integrals Definition Let D = {(x, y, z) : a x b, g(x) y h(x), G(x, y) z H(x, y)}, where g, h, G, H are continuous functions. Then D f(x, y, z)d(x, y, z) = b h(x) H(x,y) f(x, y, z)dzdydx. a g(x) G(x,y) (Other orders of integration are handled similarly.) b a Example Evaluate e 1 xy 2 z dzdxdy. Example Evaluate 1 1 x 2 1 x 2 y 2 2xzdzdydx Example A solid box D is bounded by the planes x = 0, x = 3, y = 0, y = 2, z = 0, and z = 1. The density of the box decreases linearly in the z-direction and is given by f(x, y, z) = 2 z. Find the mass of the box. (Hint: Mass is the integral of density over the box.) Example Compute the volume of the region D bounded by the paraboloids y = x 2 + z 2 and y = 16 3x 2 z 2. HOMEWORK (Due 8/7/13): Section 14.3 #11, 14, 16, 19, 21, 22, 24, 25; Section 14.4 #8, 10, 21, 25, 28, 31