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 cos θ y = 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 ru and ∆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)| = |ru x 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 # 12
Use the given transformation to evaluate the integral.
15.10 Change of Variables in Multiple Integrals

22. Example 3 – pg # 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 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

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