Another Lemma on Nonzero Jacobians
Sorry for the late post. I didn’t get a chance to get it up this morning before my flight.
Brace yourself. Just like last time we’ve got a messy technical lemma about what happens when the Jacobian determinant of a function is nonzero.
This time we’ll assume that is not only continuous, but continuously differentiable on a region . We also assume that the Jacobian at some point . Then I say that there is some neighborhood of so that is injective on .
First, we take points in and make a function of them
That is, we take the th partial derivative of the th component function and evaluate it at the th sample point to make a matrix , and then we take the determinant of this matrix. As a particular value, we have
Since each partial derivative is continuous, and the determinant is a polynomial in its entries, this function is continuous where it’s defined. And so there’s some ball of so that if all the are in we have . We want to show that is injective on .
So, let’s take two points and in so that . Since the ball is convex, the line segment is completely contained within , and so we can bring the mean value theorem to bear. For each component function we can write
for some in (no summation here on ). But like last time we now have a linear system of equations described by an invertible matrix. Here the matrix has determinant
which is nonzero because all the are inside the ball . Thus the only possible solution to the system of equations is . And so if for points within the ball , we must have , and thus is injective.