Question

What is a Jacobian matrix?

Answer 1

The Jacobian matrix of a vector-valued function f : R^n → R^m is the m×n matrix of all first-order partial derivatives of f. It represents the best linear approximation (derivative) of f at a point.

Notation and formula:
If f(x) = (f1(x), f2(x), …, fm(x)) with x = (x1,…,xn), the Jacobian Jf(x) has entries
Jf(x)_{ij} = ∂fi/∂xj.
So
Jf(x) = [∂fi/∂xj]_{i=1..m, j=1..n}.

Key facts and uses
- The Jacobian is the matrix of the differential Df(x): R^n → R^m.
- Chain rule: J(g∘f)(x) = Jg(f(x)) · Jf(x).
- If m = n, the determinant det Jf(x) (the Jacobian determinant) gives the local volume-scaling factor and appears in change-of-variables for integrals.
- Special cases: for m = 1, Jf is the row gradient ∇f; for scalar functions one often uses the Hessian (matrix of second derivatives) for curvature.
- Applications: nonlinear system linearization, stability and bifurcation analysis, Newton’s method, coordinate changes in integrals, sensitivity/uncertainty propagation.

Example
f(x,y) = (x^2 y, sin x + y). Then
Jf(x,y) = [ [2x y, x^2], [cos x, 1] ].

That is the Jacobian in a nutshell.