14.8 JacobiansFor more interesting illustrations of this topic, visit

http://www.math.umn.edu/~rogness/multivar/nonlineartransformation.html

Definition of the Jacobian

Example 1

Find the Jacobian for the change of variables

x = r cosө and

y = r sinө

Example 1 SolutionFind the Jacobian for the change of variablesx = r cosө and y = r sinө

Why would we change variables?

Example 2Let 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.

Example: 2 Solution

Example 2 SolutionWe can convert individual points between coordinate systems

Similarly, we could use these formulas to convert in the other direction

Change of variables

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.

Example 3 Solution slide 1

Example 3 Solution slide 2

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

Example 4 a Let u = x + yLet v = x- y

Example 4 solution

Let u = x + yLet v = x- y

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