Copyright © Cengage Learning. All rights reserved. 15 Multiple IntegralsCopyright © Cengage Learning. All rights reserved. 15.10 Change of Variables in Double Integrals3 3 Change of Variables in Double Integrals Learning objective: § simplify double integrals by a change of variables (and then evaluate) § (a similar technique for triple integrals will be discussed later)4 4 Change of Variables in Double Integrals In one-dimensional calculus we often use a change of variable (a substitution) to simplify an integral. By reversing the roles of x and u, we can write where x = g(u) and a = g(c), b = g(d). Another way of writing Formula 1 is as follows:5 5 Change of Variables in Double Integrals A change of variables can also be useful in double integrals. We have already seen one example of this: conversion to polar coordinates. The new variables r and θ are related to the old variables x and y by the equations x = r cos θ y = r sin θ and the change of variables formula can be written as f (x, y) dA = f (r cos θ, r sin θ) r dr dθ where S is the region in the rθ -plane that corresponds to the region R in the xy-plane.6 6 Change of Variables in Double Integrals 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 the equations x = g (u, v) y = h (u, v) or, as we sometimes write, x = x (u, v) y = y (u, v)7 7 Change of Variables in Double Integrals We usually assume that T is a C1 transformation, which means that g and h have continuous first-order partial derivatives. A transformation T is really just a function whose domain and range are both subsets of If T(u1, v1) = (x1, y1), then the point (x1, y1) is called the image of the point (u1, v1). If no two points have the same image, T is called one-to-one.8 8 Change of Variables in Double Integrals Figure 1 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. Figure 19 9 Change of Variables in Double Integrals If T is a one-to-one transformation, then it has an inverse transformation T –1 from the xy-plane to the uv-plane and 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) You have seen the first examples of such transformations on the last worksheet. Here is another example:10 10 Example A transformation is defined by the equations x = u2 – v2 y = 2uv Find the image of the square S = {(u, v) | 0 ≤ u ≤ 1 , 0 ≤ v ≤ 1}. Solution: 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.11 11 Example – Solution The first side, S1, is given by v = 0 (0 ≤ u ≤ 1). (See Figure 2.) Figure 2 continued12 12 Example – Solution From the given equations we have x = u2, y = 0, and so 0 ≤ x ≤ 1. Thus S1 is mapped into the line segment from (0, 0) to (1, 0) in the xy-plane. The second side, S2, is u = 1 (0 ≤ v ≤ 1) and, putting u = 1 in the given equations, we get x = 1 – v2 y = 2v continued13 13 Example – Solution Eliminating v, we obtain x = 1 – 0 ≤ x ≤ 1 which is part of a parabola. Similarly, S3 is given by v = 1 (0 ≤ u ≤ 1) , whose image is the parabolic arc x = – 1 –1 ≤ x ≤ 0 continued14 14 Example – Solution Finally, S4 is given by u = 0 (0 ≤ v ≤ 1) whose image is x = –v2, y = 0, that is, –1 ≤ x ≤ 0. (Notice that as we move around the square in the counterclockwise direction, we also move around the parabolic region in the counterclockwise direction.) The image of S is the region R (shown in Figure 2) bounded by the x-axis and the parabolas given by Equations 4 and 5. Figure 2 continued15 15 Change of Variables in Double Integrals Now let’s see how a change of variables affects a double integral. We start with a small rectangle S in the uv-plane whose lower left corner is the point (u0, v0) and whose dimensions are Δu and Δv. (See Figure 3.) Figure 316 16 Change of Variables in Double Integrals The image of S is a region R in the xy-plane, one of whose boundary points is (x0, y0) = T(u0, v0). 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). The equation of the lower side of S is v = v0, whose image curve is given by the vector function r(u, v0).17 17 Change of Variables in Double Integrals The tangent vector at (x0, y0) to this image curve is ru = gu(u0, v0) i + hu(u0, v0) j Similarly, the tangent vector at (x0, y0) to the image curve of the left side of S (namely, u = u0) is rv = gv(u0, v0) i + hv(u0, v0) j18 18 Change of Variables in Double Integrals We can approximate the image region R = T(S) by a parallelogram determined by the secant vectors a = r(u0 + Δu, v0) – r(u0, v0) b = r(u0, v0 + Δv) – r(u0, v0) shown in Figure 4. Figure 419 19 Change of Variables in Double Integrals But and so r(u0 + Δu, v0) – r(u0, v0) ≈ Δu ru Similarly r(u0, v0 + Δv) – r(u0, v0) ≈ Δv rv This means that we can approximate R by a parallelogram determined by the vectors Δu ru and Δv rv. (See Figure 5.) Figure 520 20 Change of Variables in Double Integrals Therefore we can approximate the area of R by the area of this parallelogram, which is | (Δu ru) × (Δv rv) | = | ru × rv | Δu Δv Computing the cross product, we obtain21 21 Change of Variables in Double Integrals The determinant that arises in this calculation is called the Jacobian of the transformation and is given a special notation. With this notation we can use Equation 6 to give an approximation to the area ΔA of R: where the Jacobian is evaluated at (u0, v0).22 22 Change of Variables in Double Integrals Next we divide a region S in the uv-plane into rectangles Sij and call their images in the xy-plane Rij. (See Figure 6.) Figure 623 23 Change of Variables in …
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