The Jacobian of a Composition
Let’s start today by introducing some notation for the Jacobian determinant which we introduced yesterday. We’ll write the Jacobian determinant of a differentiable function at a point as . Or, in more of a Leibnizean style:
We’re interested in determining the Jacobian of the composite of two differentiable functions. To which end, suppose and are differentiable functions on two open regions and in , with , and let be their composite. Then the chain rule tells us that
where each differential is an matrix, and the right-hand side is a matrix multiplication.
But these matrices are exactly the Jacobian matrices of the functions! And since the by definition, the determinant of the product of two matrices is the product of their determinants. That is, we find the equation
Or, we could define and use the Leibniz notation to write
As a special case, let’s assume that the differentiable function is injective in some open neighborhood of a point . That is, every is sent to a distinct point by , making up the whole image . Further, let’s suppose that the function which sends each point back to the point in from which it came — if and only if — is also differentiable. Then we have the composition , and thus we find
Thus, if a differentiable function has a differentiable inverse function defined in some neighborhood of a point , then the Jacobian determinant of the function must be nonzero at that point. A fair bit of work will now be put to turning this statement around. That is, we seek to show that if the Jacobian determinant , then has a differentiable inverse in some neighborhood of .