Jacobian matrix

When a map sends one space into another — say it takes (u, v) and produces (x, y) — it stretches, rotates, and skews little patches near each point. The Jacobian matrix is the bookkeeping device that records exactly how: it collects every first partial derivative of every output with respect to every input into one rectangular array, the local linear blueprint of the map.

For f: R^n -> R^m with components f_1, ..., f_m, the Jacobian at a is the m-by-n matrix J whose (i, j) entry is the partial derivative of f_i with respect to x_j, evaluated at a. When f is differentiable, J is precisely the matrix of the total derivative Df(a) in the standard bases. For a scalar function (m = 1) the Jacobian is a single row, the transpose of the gradient.

When m = n the determinant of J — the Jacobian determinant — measures the local factor by which the map scales n-dimensional volume (and its sign records orientation). This determinant is the linchpin of the change-of-variables formula for multiple integrals and the invertibility hypothesis of the inverse function theorem.