Why is the Jacobian matrix so useful?

Question

I am a first year undergraduate and I always see the Jacobian crop up in some many places, e.g., integration, solving systems of equations, analysis and so many more places. I was wondering, what makes it so useful?

Answer 1

Personally, I think that's the same as asking why the derivative is useful, which I'd assume you'd have an easier time intuiting.

In some sense, the Jacobian is the derivative of a multivariate vector function. I.e. if $f:\mathbb{R}^m \rightarrow \mathbb{R}^n$, what is the derivative of $f$? The Jacobian encodes the derivatives of all components of $f$ with respect to all variables, and thus serves as its derivative. Notice that we recover the classic derivative when $n=m=1$ and the gradient when $n=1$. Dynamical systems, for example, can be treated as such a function, and linearized that way, using the property of the derivative (and Jacobian) as providing the best local linear approximation to a function.

The fact that it is structured as an array is also useful, as it lets it be used very naturally with tensors or matrices and vectors, so that useful equations and identities in $\mathbb{R}$ generalize in notationally pleasant ways to higher dimensions (and even to manifolds) via the Jacobian.