(1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3

Transcription

1 Math 127 Introduction and Review (1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3 MATH 127 Introduction to Calculus III The University of Kansas 1 / 22

2 Differential Calculus Types of problems in Differential Calculus f : R R, y = f (x) Slope of the tangent line (rate of change) at (a, f (a)) is f (a) Area under the graph of y = f (x) between two x values is b a f (x) dx MATH 127 Introduction to Calculus III The University of Kansas 2 / 22

3 Differential Calculus Optimization Arc Length b ( ) dy dx dx a MATH 127 Introduction to Calculus III The University of Kansas 3 / 22

4 Multivariable Calculus Scalar Valued Functions f : R 2 R, z = f (x, y), w = g(x, y, z) Examples 1 z = x 2 + y 2 2 w = e xyz Partial Derivative Line Integral MATH 127 Introduction to Calculus III The University of Kansas 4 / 22

5 Multivariable Calculus Optimization Volume MATH 127 Introduction to Calculus III The University of Kansas 5 / 22

6 Vector Valued Functions f : R R 2, r(t) = x(t), y(t), z(t) f : R 2 R 3, r(u, v) = x(u, v), y(u, v), z(u, v) MATH 127 Introduction to Calculus III The University of Kansas 6 / 22

7 Vector Field f : R 2 R 2 or f : R 3 R 3 MATH 127 Introduction to Calculus III The University of Kansas 7 / 22

8 Vectors Definition A vector is a geometric quantity described by magnitude (length) and direction. A scalar is a constant in R which has no direction and is magnitude only. Example: Force, flux, pressure, velocity, and acceleration. Geometric Representations of Vectors: a segment AB from A to B with an arrow beginning at A and ending at B. A - Initial Point Notation: v = AB or v= AB B - Terminal Point Only the magnitude and direction are relevant to a vector. It s location is NOT relevant. Vectors are invariant under translation. MATH 127 Introduction to Calculus III The University of Kansas 8 / 22

9 Vector Addition and Scalar Multiplication Vector Addition Vectors u and v are summed through the Parallelogram Law. Scalar Multiplication When a vector v and a scalar c are multiplied, the direction remains unchanged and the magnitude of v is multiplied by c. MATH 127 Introduction to Calculus III The University of Kansas 9 / 22

10 Cartesian Representation of Vectors Components of a Vector v Position the initial point of the vector v at the origin of the Cartesian coordinate system. The coordinates of the terminal point of v will be the components of v. v = OP = a, b, c In general, if v = AB where A(x 1, y 1, z 1 ) and B(x 2, y 2, z 2 ) then v = x 2 x 1, y 2 y 1, z 2 z 1 MATH 127 Introduction to Calculus III The University of Kansas 10 / 22

11 Vector Addition and Scalar Multiplication If v = a 1, b 1, c 1 and u = a 2, b 2, c 2 and k is a scalar, then (I) v + u = a 1 + a 2, b 1 + b 2, c 1 + c 2 = u + v (II) k v = ka 1, kb 1, kc 1 (III) v u = a 1 a 2, b 1 b 2, c 1 c 2 = ( u v) The zero vector is 0 = 0, 0, 0. i = 1, 0, 0 j = 0, 1, 0 k = 0, 0, 1 Algebraic Properties a, b, c = a i + b j + c k v v = 0 v + u = u + v k ( v + u) = k v + k v ( v + u) + w = v + ( u + w) (k 1 + k 2 ) v = k 1 v + k 2 v v + 0 = v = 0 + v (k 1 k 2 ) v = k 1 (k 2 v) MATH 127 Introduction to Calculus III The University of Kansas 11 / 22

12 Magnitude of a Vector Magnitude The magnitude (or length) of a vector v = a, b, c is v = a 2 + b 2 + c 2 If v = AB with A(x 1, y 1, z 1 ) and B = (x 2, y 2, z 2 ), then v = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 + (z 2 z 1 ) 2 The zero vector 0 = 0, 0, 0 is the only vector with magnitude zero. Unit Vector A unit vector is a vector of magnitude one. If v = a, b, c 0, then the unit vector with the same direction of v is U v = v v = 1 a, b, c a 2 + b 2 + c2 MATH 127 Introduction to Calculus III The University of Kansas 12 / 22

13 In addition to the vector addition and scalar multiplication, there are other operations among vectors. In this course we will discuss two types of vector products: (I) The Dot Product, (II) The Cross Product. The dot product of two vectors v and u is the number given by v u = u v cos (θ) where θ is the acute angle between the vectors v and u. MATH 127 Introduction to Calculus III The University of Kansas 13 / 22

14 Orthogonal Two vectors v and u are orthogonal to each other, denoted v u, if and only if θ = π 2 if and only if v u = 0. v u is positive if and only if 0 θ < π 2, v u is negative if and only if π 2 < θ π Dot Product in Component Notation Suppose that v = a 1, b 1, c 1 and u = a 2, b 2, c 2. Recall the Cosine Law v u 2 = v 2 + u 2 2 v u cos (θ) = v 2 + u 2 2 v u v u = a 1 a 2 + b 1 b 2 + c 1 c 2 MATH 127 Introduction to Calculus III The University of Kansas 14 / 22

15 Example: Find the angle between v = 1, 2, 0 and u = 3, 1, 7. Solution: v u = v u cos (θ) cos (θ) = v u v u = ( 1)(3) + (2)(1) + (0)(7) ( 1) 2 + (2) = ( ) 1 θ = arccos > π MATH 127 Introduction to Calculus III The University of Kansas 15 / 22

16 Properties of Dot Product v u = u v u ( v ± w) = u v ± u w (k u) v = k ( u v) 0 v = 0 v v = v 2 Example: Suppose that OR PQ and OS PQ, show that RS PQ. Solution: Since the vectors are orthogonal, As RS = OS RS PQ = OR we have OR PQ = 0 = OS PQ ( ) OS OR PQ = OS PQ OR PQ = 0 MATH 127 Introduction to Calculus III The University of Kansas 16 / 22

17 The cross product of two vectors v and u, denoted v u, is the vector where (i) θ is the angle between v and u, v u = ( v u sin (θ)) n (ii) n is the unit vector, so that n v, n u, and the vectors ( v, u, n) satisfies the right-hand rule. MATH 127 Introduction to Calculus III The University of Kansas 17 / 22

18 v u = ( v u sin (θ)) n Properties of the Cross Product If v and u are parallel, then v u = 0. ( v u) v and ( v u) u. v u = u v v u is the area of the parallelogram determined by v and u which is twice the area of the triangle with edges v and u. MATH 127 Introduction to Calculus III The University of Kansas 18 / 22

19 Cross Product in Component Form i j = k j k = i k i = j If v = a 1, b 1, c 1 and u = a 2, b 2, c 2 then we can distribute over the cross product: ) ) v u = (a 1 i + b 1 j + c 1 k (a 2 i + b 2 j + c 2 k = (b 1 c 2 b 2 c 1 ) i (a 1 c 2 a 2 c 1 ) j + (a 1 b 2 a 2 b 1 ) k Alternatively, you can consider the determinant of the matrix: i j k v u = a 1 b 1 c 1 a 2 b 2 c 2 = b 1 c 1 b 2 c 2 i a 1 c 1 a 2 c 2 j + a 1 b 1 a 2 b 2 k = (b 1 c 2 b 2 c 1 ) i (a 1 c 2 a 2 c 1 ) j + (a 1 b 2 a 2 b 1 ) k MATH 127 Introduction to Calculus III The University of Kansas 19 / 22

20 Example: (I) Find a vector which is orthogonal to the plane through the points P(2, 0, 3), Q(3, 1, 0), and R(5, 2, 2). (II) Find the area of the triangle with vertices at P, Q, and R. Solution: Let u = PR = 3, 2, 5 and v = PQ = 1, 1, 3. (I) u v = i j k orthogonal to the plane. (II) The area of the triangle is 1 2 u v = = 1, 4, 1 is MATH 127 Introduction to Calculus III The University of Kansas 20 / 22

21 Example: 1 Find the algebraic equation of the plane containing the points (1,0,1), (0, -1, -1), and (1, -1, 0). 2 Find the parametric equations of the line perpendicular to the plane in (1), passing through the point (1,0,1). Solution: Let P = (1, 0, 1), Q = (0, 1, 1) and R = (1, 1, 0) u = PQ = 1, 1, 2 and v = PR = 1, 0, 1 u and v lie in the required plane. The normal vector of that plane is i j k u v = = 1, 1, The equation of the plane is 1, 1, 1 ( x, y, x 1, 0, 1 ) = 0 (x 1) + (y 0) (z 1) = 0 or x + y z = 0 MATH 127 Introduction to Calculus III The University of Kansas 21 / 22

22 Example continued Vector Equation: r(t) = (1, 0, 1) + t 1, 1, 1 = 1 + t, t, 1 t Parametric Equations: x(t) = 1 + t y(t) = t z(t) = 1 t Symmetric Equations: x 1 = y = 1 z MATH 127 Introduction to Calculus III The University of Kansas 22 / 22