A Visual Introduction to Differential Forms and Calculus on Manifolds by Jon Pierre Fortney

Author:Jon Pierre Fortney
Language: eng
Format: epub
ISBN: 9783319969923
Publisher: Springer International Publishing

which means that g(r, θ) = −r. Hence

We had of course already found in the previous chapter.

Now we have all the ingredients we need to rewrite the integral in polar coordinates. Of course, we have gone through a lot of trouble to show all the steps in excruciating detail. Once you have done this a few times the end results are available and you can simply write down the integral in polar coordinates,

This is an altogether easier integral to compute than the one we started with. The only caveat is the negative sign in front of the r in the pull-back of the area form dx ∧ dy, which reflects the change in orientation.

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