1. 14.8 Jacobians For more interesting illustrations of this topic, visit http://www.math.umn.edu/~rogness/multivar/nonlineartransformation.html

2. Definition of the Jacobian

3. Example 1 Find the Jacobian for the change of variables x = r cosө and y = r sinө

4. Example 1 Solution Find the Jacobian for the change of variables x = r cosө and y = r sinө

5. Why would we change variables?

6. Example 2 Let R be the region bounded by the lines x - 2y = 0, x – 2y = -4, x + y =4 and x + y = 1 Find a transformation T from region R to region S such that S is a rectangular region.

7. Example: 2 Solution

8. Example 2 Solution We can convert individual points between coordinate systems Similarly, we could use these formulas to convert in the other direction

10. Change of variables

11. Example 3 use a change of variables to simplify a region Let R be the region bounded by the lines x - 2y = 0, x – 2y = -4, x + y =4 and x + y = 1 as shown below. Evaluate the double integral.

12. Example 3 Solution slide 1

13. Example 3 Solution slide 2

14. Example 4 Let R be the region bounded by vertices (0,1),(1,2) (2,1), (1,0) a) Sketch the transformed region b) Evaluate the integral

15. Example 4 a Let u = x + y Let v = x- y

16. Example 4 solution Let u = x + y Let v = x- y

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