November 23, 2017

For those of you wondering about the assumptions of the Change of variables theorem, here is the theorem in its full glory:

Change of variables theorem. Let \( D \subset \mathbb{R}^n \) be a bounded and (Peano-Jordan) measurable domain, and let \( \varphi : T \rightarrow D \) be an invertible function of class \( \mathcal{C}^1\) such that the Jacobian determinant of \( \varphi \) is non-zero in the interior of T. If T is (Peano-Jordan) measurable, then for every continuous function \( f : D \rightarrow \mathbb{R} \)\[  \int_D f(\vec{x}) \, \mathrm{d}^n \vec{x} = \int_T f(\varphi(\vec{u})) \, \lvert \det{D\varphi(\vec{u})} \rvert \, \mathrm{d}^n \vec{u}. \]

Each of these assumptions would take a comic (or two) to unpack, so we left them out of this comic for simplicity. Also, the point of this comic is not to provide a comprehensive explanation of the theorem, but to highlight why it's useful and what the Jacobian determinant's role is.

We're soon going to branch out from calculus topics, which means new characters and a bit of comic experimentation. But don't worry, we'll have one more Cathy and Doug comic before the holiday break!