\newsectionNoPB
\subsection{Change of Variable Formula}
\compacttheorem{Change of variable formula} $\overline{X}, \overline{Y} \subseteq \R^n$ compact, $\varphi : \overline{X} \rightarrow \overline{Y}$ continuous.
For the open sets $X, Y$, negligible sets $B, C$ and restriction of $\varphi : X \rightarrow Y$ to open set $X$ is a $C^1$ bijection,
we can write $\overline{X} = X \cup B$ and $\overline{Y} = Y \cup C$.
The Jacobian $J_\varphi(x)$ is invertible at all $x \in X$.
For any cont. func. $f$ on $\overline{Y}$ we have $\displaystyle \int_{\overline{X}} f(\varphi(x)) |\det(J_\varphi(x))| \dx x = \int_{\overline{Y}} f(y) \dx y$
% TODO: Add notes from TA's notes for how to apply it
\rmvspace
