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Chapter 15 – Multiple Integrals 15.10 Change of Variables in Multiple Integrals Objectives: How to change variables for double and triple integrals Carl Gustav Jacob Jacobi 15.10 Change of Variables in Multiple Integrals 1 Change of Variable - Single 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 the Substitution Rule (Equation 6 in Section 5.5) as: b a d f ( x) dx f ( g (u)) g '(u) du c where x = g(u) and a = g(c), b = g(d). 15.10 Change of Variables in Multiple Integrals 2 Change of Variables - Double A change of variables can also be useful in double integrals. ◦ We have already seen one example of this: conversion to polar coordinates where the new variables r and θ are related to the old variables x and y by: x = r cos θ y = r sin θ ◦ 15.10 Change of Variables in Multiple Integrals 3 Change of Variables - Double The change of variables formula (Formula 2 in Section 15.4) can be written as: f ( x, y ) dA f (r cos , r sin ) r dr d R S where S is the region in the rθ-plane that corresponds to the region R in the xy-plane. 15.10 Change of Variables in Multiple Integrals 4 Transformation More generally, we consider a change of variables that is given by a transformation T from the uvplane 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) 15.10 Change of Variables in Multiple Integrals 5 C1 transformation We usually assume that T is a C1 transformation. ◦ This means that g and h have continuous first-order partial derivatives. 15.10 Change of Variables in Multiple Integrals 6 Image & One-to-one Transformation 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. 15.10 Change of Variables in Multiple Integrals 7 Change of Variables 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. 15.10 Change of Variables in Multiple Integrals 8 Inverse Transform If T is a one-to-one transformation, it has an inverse transformation T–1 from the xy–plane to the uv-plane. 15.10 Change of Variables in Multiple Integrals 9 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). ◦ Dimensions are ∆u and ∆v. 15.10 Change of Variables in Multiple Integrals 10 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) 15.10 Change of Variables in Multiple Integrals 11 Double Integrals We can approximate R by a parallelogram determined by the vectors ∆u ru and ∆v rv 15.10 Change of Variables in Multiple Integrals 12 Double Integrals Thus, we can approximate the area of R by the area of this parallelogram, which, from Section 12.4, is: |(∆u ru) x (∆v rv)| = |ru x rv| ∆u ∆v 15.10 Change of Variables in Multiple Integrals 13 Double Integrals Computing the cross product, we obtain: i x ru rv u x v j y u y u x u 0 x v 0 k y u k y u x u y u x v k y v 15.10 Change of Variables in Multiple Integrals 14 Jacobian The determinant that arises in this calculation is called the Jacobian of the transformation. ◦ It is given a special notation. 15.10 Change of Variables in Multiple Integrals 15 Definition - Jacobian of T The Jacobian of the transformation T given by x = g(u, v) and y = h(u, v) is: x ( x, y ) u (u, v) y u x v x y x y y u v v u v 15.10 Change of Variables in Multiple Integrals 16 Jacobian of T With this notation, we can give an approximation to the area ∆A of R: ( x, y) A u v (u, v) where the Jacobian is evaluated at (u0, v0). 15.10 Change of Variables in Multiple Integrals 17 Math Fun Fact 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. 15.10 Change of Variables in Multiple Integrals 18 Example 1 – pg. 1020 Find the Jacobian of the transformation. u 2. x uv, y v s t 4. x e , ye s t 15.10 Change of Variables in Multiple Integrals 19 Change of Variables in a Double Integral – Theorem 9 Suppose: ◦ T is a C1 transformation whose Jacobian is nonzero and that maps a region S in the uvplane 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. Then, f ( x, y) dA R S ( x, y) f ( x(u, v), y(u, v)) du dv (u, v) 15.10 Change of Variables in Multiple Integrals 20 Example 2 – pg. 1020 # 12 Use the given transformation to evaluate the integral. 4 x 8 y dA, where R is the parallelogram R with vertices (-1, 3), (1, - 3), (3, -1), and (1, 5); 1 1 x u v , y v 3u 4 4 15.10 Change of Variables in Multiple Integrals 21 Example 3 – pg. 1020 # 20 Evaluate the integral by making the appropriate change of variables. x y e x2 y 2 dA, where R is the rectangle enclosed R by the lines x y 0, x y 2, x y 0, and x y 3. 15.10 Change of Variables in Multiple Integrals 22 Triple Integrals 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) 15.10 Change of Variables in Multiple Integrals 23 Triple Integrals - Equation 12 The Jacobian of T is this 3 x 3 determinant: x u ( x, y , z ) y (u, v, w) u z u x v y v z v x w y w z w 15.10 Change of Variables in Multiple Integrals 24 Triple Integrals Under hypotheses similar to those in Theorem 9, we have this formula for triple integrals: f ( x, y, z) dV R S ( x, y , z ) f ( x(u, v, w), y (u, v, w), z (u, v, w)) du dv dw (u, v, w) 15.10 Change of Variables in Multiple Integrals 25 Example 5 – pg. 1020 #5 Find the Jacobian of the transformation. u x , v v w y , z w u 15.10 Change of Variables in Multiple Integrals 26