# The Unapologetic Mathematician

## The Geometric Interpretation of the Jacobian Determinant

We first defined the Jacobian determinant as measuring the factor by which a transformation scales infinitesimal pieces of $n$-dimensional volume. Now, with the change of variables formula and the mean value theorem in hand, we can pull out a macroscopic result.

If $g:S\rightarrow\mathbb{R}^n$ is injective and continuously differentiable on an open region $S\subseteq\mathbb{R}^n$, and $X$ is a compact, connected, Jordan measurable subset of $S$, then we have

$\displaystyle c(Y)=\lvert J_h(x_0)\rvert c(X)$

for some $x_0\in X$, where $Y=g(X)$.

Simply take $f(y)=1$ in the change of variables formula

$\displaystyle c(Y)=\int\limits_Y\,dy=\int\limits_X\lvert J_g(x)\rvert\,dx$

The mean value theorem now tells us that

$\displaystyle c(Y)=\int\limits_X\lvert J_g(x)\rvert\,dx=\lambda\int\limits_X\,dx=\lambda c(X)$

for some $\lambda$ between the maximum and minimum of $\lvert J_g(x)\rvert$ on $X$. But then since $X$ is connected, we know that there is some $x_0\in X$ so that $\lambda=\lvert J_g(x_0)\rvert$, as we asserted.

January 8, 2010 Posted by | Analysis, Calculus | 10 Comments