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MULTIPLE INTEGRALS 16

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2 MULTIPLE INTEGRALS 16.9 Change of Variables in Multiple Integrals In this section, we will learn about: The change of variables in double and triple integrals.

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3 CHANGE OF VARIABLES IN SINGLE INTEGRALS In one-dimensional calculus. we often use a change of variable (a substitution) to simplify an integral.

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4 By reversing the roles of x and u, we can write the Substitution Rule (Equation 6 in Section 5.5) as: where x = g(u) and a = g(c), b = g(d). SINGLE INTEGRALS Formula 1

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5 Another way of writing Formula 1 is as follows: SINGLE INTEGRALS Formula 2

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6 A change of variables can also be useful in double integrals. We have already seen one example of this: conversion to polar coordinates. CHANGE OF VARIABLES IN DOUBLE INTEGRALS

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7 The new variables r and θ are related to the old variables x and y by: x = r cos θ y = r sin θ DOUBLE INTEGRALS

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8 The change of variables formula (Formula 2 in Section 16.4) can be written as: where S is the region in the rθ-plane that corresponds to the region R in the xy-plane. DOUBLE INTEGRALS

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9 More generally, we consider a change of variables that is given by a transformation T from the uv-plane to the xy-plane: T(u, v) = (x, y) where x and y are related to u and v by: x = g(u, v)y = h(u, v) We sometimes write these as: x = x(u, v), y = y(u, v) TRANSFORMATION Equations 3

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10 We usually assume that T is a C 1 transformation. This means that g and h have continuous first-order partial derivatives. C 1 TRANSFORMATION

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11 A transformation T is really just a function whose domain and range are both subsets of. TRANSFORMATION

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12 If T(u 1, v 1 ) = (x 1, y 1 ), then the point (x 1, y 1 ) is called the image of the point (u 1, v 1 ). If no two points have the same image, T is called one-to-one. IMAGE & ONE-TO-ONE TRANSFORMATION

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13 The figure shows the effect of a transformation T on a region S in the uv-plane. T transforms S into a region R in the xy-plane called the image of S, consisting of the images of all points in S. CHANGE OF VARIABLES Fig. 16.9.1, p. 1049

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14 If T is a one-to-one transformation, it has an inverse transformation T –1 from the xy–plane to the uv-plane. INVERSE TRANSFORMATION Fig. 16.9.1, p. 1049

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15 Then, it may be possible to solve Equations 3 for u and v in terms of x and y : u = G(x, y) v = H(x, y) INVERSE TRANSFORMATION

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16 A transformation is defined by: x = u 2 – v 2 y = 2uv Find the image of the square S = {(u, v) | 0 ≤ u ≤ 1, 0 ≤ v ≤ 1} TRANSFORMATION Example 1

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17 The transformation maps the boundary of S into the boundary of the image. So, we begin by finding the images of the sides of S. TRANSFORMATION Example 1

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18 The first side, S 1, is given by: v = 0 (0 ≤ u ≤ 1) TRANSFORMATION Example 1 Fig. 16.9.2, p. 1050

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19 From the given equations, we have: x = u 2, y = 0, and so 0 ≤ x ≤ 1. Thus, S 1 is mapped into the line segment from (0, 0) to (1, 0) in the xy-plane. TRANSFORMATION Example 1

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20 The second side, S 2, is: u = 1 (0 ≤ v ≤ 1) Putting u = 1 in the given equations, we get: x = 1 – v 2 y = 2v TRANSFORMATION E. g. 1—Equation 4 Fig. 16.9.2, p. 1050

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21 Eliminating v, we obtain: which is part of a parabola. TRANSFORMATION E. g. 1—Equation 4

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22 Similarly, S 3 is given by: v = 1 (0 ≤ u ≤ 1) Its image is the parabolic arc TRANSFORMATION E. g. 1—Equation 5 Fig. 16.9.2, p. 1050

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23 Finally, S 4 is given by: u = 0(0 ≤ v ≤ 1) Its image is: x = –v 2, y = 0 that is, –1 ≤ x ≤ 0 TRANSFORMATION Example 1 Fig. 16.9.2, p. 1050

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24 Notice that as, we move around the square in the counterclockwise direction, we also move around the parabolic region in the counterclockwise direction. TRANSFORMATION Example 1 Fig. 16.9.2, p. 1050

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25 The image of S is the region R bounded by: The x-axis. The parabolas given by Equations 4 and 5. TRANSFORMATION Example 1 Fig. 16.9.2, p. 1050

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26 Now, let’s see how a change of variables affects a double integral. DOUBLE INTEGRALS

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27 We start with a small rectangle S in the uv-plane whose: Lower left corner is the point (u 0, v 0 ). Dimensions are ∆u and ∆v. DOUBLE INTEGRALS Fig. 16.9.3, p. 1050

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28 The image of S is a region R in the xy-plane, one of whose boundary points is: (x 0, y 0 ) = T(u 0, v 0 ) DOUBLE INTEGRALS Fig. 16.9.3, p. 1050

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29 The vector r(u, v) = g(u, v) i + h(u, v) j is the position vector of the image of the point (u, v). DOUBLE INTEGRALS Fig. 16.9.3, p. 1050

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30 The equation of the lower side of S is: v = v 0 Its image curve is given by the vector function r(u, v 0 ). DOUBLE INTEGRALS Fig. 16.9.3, p. 1050

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31 The tangent vector at (x 0, y 0 ) to this image curve is: DOUBLE INTEGRALS Fig. 16.9.3, p. 1050

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32 Similarly, the tangent vector at (x 0, y 0 ) to the image curve of the left side of S (u = u 0 ) is: DOUBLE INTEGRALS Fig. 16.9.3, p. 1050

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33 We can approximate the image region R = T(S) by a parallelogram determined by the secant vectors DOUBLE INTEGRALS Fig. 16.9.4, p. 1051

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34 However, So, Similarly, DOUBLE INTEGRALS

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35 This means that we can approximate R by a parallelogram determined by the vectors ∆u r u and ∆v r v DOUBLE INTEGRALS Fig. 16.9.5, p. 1051

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36 Thus, we can approximate the area of R by the area of this parallelogram, which, from Section 13.4, is: |(∆u r u ) x (∆v r v )| = |r u x r v | ∆u ∆v DOUBLE INTEGRALS Equation 6

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37 Computing the cross product, we obtain: DOUBLE INTEGRALS

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38 The determinant that arises in this calculation is called the Jacobian of the transformation. It is given a special notation. JACOBIAN

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39 The Jacobian of the transformation T given by x = g(u, v) and y = h(u, v) is: JACOBIAN OF T Definition 7

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40 With this notation, we can use Equation 6 to give an approximation to the area ∆A of R: where the Jacobian is evaluated at (u 0, v 0 ). JACOBIAN OF T Approximation 8

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41 The Jacobian is named after the German mathematician Carl Gustav Jacob Jacobi (1804–1851). The French mathematician Cauchy first used these special determinants involving partial derivatives. Jacobi, though, developed them into a method for evaluating multiple integrals. JACOBIAN

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42 Next, we divide a region S in the uv-plane into rectangles S ij and call their images in the xy-plane R ij. DOUBLE INTEGRALS Fig. 16.9.6, p. 1052

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43 Applying Approximation 8 to each R ij, we approximate the double integral of f over R as follows. DOUBLE INTEGRALS Fig. 16.9.6, p. 1052

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44 where the Jacobian is evaluated at (u i, v j ). DOUBLE INTEGRALS

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45 Notice that this double sum is a Riemann sum for the integral DOUBLE INTEGRALS

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46 The foregoing argument suggests that the following theorem is true. A full proof is given in books on advanced calculus. DOUBLE INTEGRALS

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47 Suppose: T is a C 1 transformation whose Jacobian is nonzero and that maps a region S in the uv-plane onto a region R in the xy-plane. f is continuous on R and that R and S are type I or type II plane regions. T is one-to-one, except perhaps on the boundary of S. CHG. OF VRBLS. (DOUBLE INTEG.) Theorem 9

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48 Then, CHG. OF VRBLS. (DOUBLE INTEG.) Theorem 9

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49 Theorem 9 says that we change from an integral in x and y to an integral in u and v by expressing x and y in terms of u and v and writing: CHG. OF VRBLS. (DOUBLE INTEG.)

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50 Notice the similarity between Theorem 9 and the one-dimensional formula in Equation 2. Instead of the derivative dx/du, we have the absolute value of the Jacobian, that is, |∂(x, y)/∂(u, v)| CHG. OF VRBLS. (DOUBLE INTEG.)

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51 As a first illustration of Theorem 9, we show that the formula for integration in polar coordinates is just a special case. CHG. OF VRBLS. (DOUBLE INTEG.)

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52 Here, the transformation T from the rθ-plane to the xy-plane is given by: x = g(r, θ) = r cos θ y = h(r, θ) = r sin θ CHG. OF VRBLS. (DOUBLE INTEG.)

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53 The geometry of the transformation is shown here. T maps an ordinary rectangle in the rθ-plane to a polar rectangle in the xy-plane. CHG. OF VRBLS. (DOUBLE INTEG.) Fig. 16.9.7, p. 1053

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54 The Jacobian of T is: CHG. OF VRBLS. (DOUBLE INTEG.)

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55 So, Theorem 9 gives: This is the same as Formula 2 in Section 16.4 CHG. OF VRBLS. (DOUBLE INTEG.)

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56 Use the change of variables x = u 2 – v 2, y = 2uv to evaluate the integral where R is the region bounded by: The x-axis. The parabolas y 2 = 4 – 4x and y 2 = 4 + 4x, y ≥ 0. CHG. OF VRBLS. (DOUBLE INTEG.) Example 2

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57 The region R is pictured here. CHG. OF VRBLS. (DOUBLE INTEG.) Example 2 Fig. 16.9.2, p. 1050

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58 In Example 1, we discovered that T(S) = R where S is the square [0, 1] x [0, 1]. Indeed, the reason for making the change of variables to evaluate the integral is that S is a much simpler region than R. CHG. OF VRBLS. (DOUBLE INTEG.) Example 2

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59 First, we need to compute the Jacobian: CHG. OF VRBLS. (DOUBLE INTEG.) Example 2

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60 So, by Theorem 9, CHG. OF VRBLS. (DOUBLE INTEG.) Example 2

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61 Example 2 was not very difficult to solve as we were given a suitable change of variables. If we are not supplied with a transformation, the first step is to think of an appropriate change of variables. CHG. OF VRBLS. (DOUBLE INTEG.) Note

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62 If f(x, y) is difficult to integrate, The form of f(x, y) may suggest a transformation. If the region of integration R is awkward, The transformation should be chosen so that the corresponding region S in the uv-plane has a convenient description. CHG. OF VRBLS. (DOUBLE INTEG.) Note

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63 Evaluate the integral where R is the trapezoidal region with vertices (1, 0), (2, 0), (0, –2), (0,–1) CHG. OF VRBLS. (DOUBLE INTEG.) Example 3

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64 It isn’t easy to integrate e (x+y)/(x–y). So, we make a change of variables suggested by the form of this function: u = x + y v = x – y These equations define a transformation T –1 from the xy-plane to the uv-plane. CHG. OF VRBLS. (DOUBLE INTEG.) E. g. 3—Eqns. 10

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65 Theorem 9 talks about a transformation T from the uv-plane to the xy-plane. It is obtained by solving Equations 10 for x and y: x = ½(u + v) y = ½(u – v) CHG. OF VRBLS. (DOUBLE INTEG.) E. g. 3—Equation 11

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66 The Jacobian of T is: CHG. OF VRBLS. (DOUBLE INTEG.) Example 3

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67 To find the region S in the uv-plane corresponding to R, we note that: The sides of R lie on the lines y = 0 x – y = 2 x = 0 x – y = 1 From either Equations 10 or Equations 11, the image lines in the uv-plane are: u = v v = 2 u = –v v = 1 CHG. OF VRBLS. (DOUBLE INTEG.) Example 3

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68 Thus, the region S is the trapezoidal region with vertices (1, 1), (2, 2), (–2, 2), (–1,1) CHG. OF VRBLS. (DOUBLE INTEG.) Example 3 Fig. 16.9.8, p. 1054

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69 S = {(u, v) | 1 ≤ v ≤ 2, –v ≤ u ≤ v} CHG. OF VRBLS. (DOUBLE INTEG.) Example 3 Fig. 16.9.8, p. 1054

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70 So, Theorem 9 gives: CHG. OF VRBLS. (DOUBLE INTEG.) Example 3

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71 There is a similar change of variables formula for triple integrals. Let T be a transformation that maps a region S in uvw-space onto a region R in xyz-space by means of the equations x = g(u, v, w) y = h(u, v, w) z = k(u, v, w) TRIPLE INTEGRALS

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72 The Jacobian of T is this 3 x 3 determinant: TRIPLE INTEGRALS Equation 12

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73 Under hypotheses similar to those in Theorem 9, we have this formula for triple integrals: TRIPLE INTEGRALS Formula 13

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74 Use Formula 13 to derive the formula for triple integration in spherical coordinates. The change of variables is given by: x = ρ sin Φ cos θ y = ρ sin Φ sin θ z = ρ cos Φ TRIPLE INTEGRALS Example 4

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75 We compute the Jacobian as follows: TRIPLE INTEGRALS Example 4

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76 TRIPLE INTEGRALS Example 4

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77 Since 0 ≤ Φ ≤ π, we have sin Φ ≥ 0. Therefore, TRIPLE INTEGRALS Example 4

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78 Thus, Formula 13 gives: This is equivalent to Formula 3 in Section15.8 TRIPLE INTEGRALS Example 4

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