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On page 1103 is the derivation of the Jacobian. It is very similar to the derivation for surface area where we use the area
of a parallelogram to approximate the area of a subregion that is not rectangular.
Definition of the Jacobian: For: (Note: we have the Absolute Value of the Jacobian as an added factor when we change the variables) Let’s show how this works when we change from rectangular to polar coordinates:
Example: This method also works for 3D. For an example from rectangular to spherical coordinates read page 1109.
Notice we end up with the Jacobian: Example:
Show the area of the ellipse and the area of the circle are connected through the Jacobian. Example: I used a shortcut in the last example:
My equations were for u and v:
Instead of solving these for x and y so I can
calculate the Jacobian J(x,y), I calculated theJacobian for J(u,v) and used the above fact to get J(x,y). This is especially useful when the Jacobian is a constant.
This example uses the vertice...
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