Chapter 15 – Multiple Integrals

1. 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

2. 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: where x =g(u) and a =g(c), b =g(d). 15.10 Change of Variables in Multiple Integrals

3. 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 cosθy =r sin θ 15.10 Change of Variables in Multiple Integrals

4. Change of Variables - Double • The change of variables formula (Formula 2 in Section 15.4) can be written as: 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

5. Transformation • 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) 15.10 Change of Variables in Multiple Integrals

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. Double Integrals • We can approximate R by a parallelogram determined by the vectors ∆u ruand ∆v rv 15.10 Change of Variables in Multiple Integrals

13. 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)| = |rux rv|∆u ∆v 15.10 Change of Variables in Multiple Integrals

14. Double Integrals • Computing the cross product, we obtain: 15.10 Change of Variables in Multiple Integrals

15. 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

16. Definition - Jacobian of T • The Jacobian of the transformation T given by x =g(u, v) and y =h(u, v) is: 15.10 Change of Variables in Multiple Integrals

17. Jacobian of T • With this notation, we can give an approximation to the area ∆A of R: • where the Jacobian is evaluated at (u0, v0). 15.10 Change of Variables in Multiple Integrals

18. 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

19. Example 1 – pg. 1020 • Find the Jacobian of the transformation. 15.10 Change of Variables in Multiple Integrals

20. 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 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. Then, 15.10 Change of Variables in Multiple Integrals

21. Example 2 – pg. 1020 # 12 • Use the given transformation to evaluate the integral. 15.10 Change of Variables in Multiple Integrals

22. Example 3 – pg. 1020 # 20 • Evaluate the integral by making the appropriate change of variables. 15.10 Change of Variables in Multiple Integrals

23. Triple Integrals • There is a similar change of variables formula for triple integrals. • Let T be a transformation that maps a region Sin 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

24. Triple Integrals - Equation 12 • The Jacobian of T is this 3 x 3 determinant: 15.10 Change of Variables in Multiple Integrals

25. Triple Integrals • Under hypotheses similar to those in Theorem 9, we have this formula for triple integrals: 15.10 Change of Variables in Multiple Integrals

26. Example 5 – pg. 1020 #5 • Find the Jacobian of the transformation. 15.10 Change of Variables in Multiple Integrals