The Unapologetic Mathematician

Mathematics for the interested outsider

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.

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