This is notated by where the result of substitution

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ntegrand. 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. Example: This example uses the vertice...
View Full Document

Ask a homework question - tutors are online