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Change of Variables and the Jacobian

Changing variables can sometimes make double integrals way easier to compute, but fully converting over from one coordinate system to another can be tricky, because in addition to changing the integrand and the integration bounds, you also have to throw in this additional factor called the Jacobian. But why? And what is the Jacobian doing? ~~~ This video was funded by Texas A&M University as part of the Enhancing Online Courses grant. ~~~ The animations in this video were mostly made with a homemade Python library called "Morpho". You can find the project here: https://github.com/morpho-matters/mor...