The Jacobian tells you how a smooth map stretches and twists space
Warp a photograph smoothly and each tiny patch gets stretched, rotated or sheared. The Jacobian matrix, named after the German mathematician Carl Gustav Jacob Jacobi, records exactly how, point by point. Its determinant then says whether areas grow or shrink there, and whether the image has been flipped into its mirror version.
The Jacobian gathers every first-order partial derivative of a function that takes several inputs and returns several outputs, arranged so that each row corresponds to one output. It is the natural next step in a chain of generalisations. For a function of one variable it is simply the ordinary derivative; for a function returning a single number from several inputs it becomes the gradient; and for vector-valued functions it becomes the full matrix. Some authors write it transposed, and it goes by several notations.
Its central job is approximation. Near any point where the function is differentiable, the Jacobian gives the best linear stand-in for the function, just as a tangent line approximates a curve. Chain two such functions together and their Jacobians simply multiply, which is the chain rule in matrix form. Take the Jacobian of a gradient and you get the Hessian matrix, a kind of second derivative.
When the numbers of inputs and outputs match, the matrix is square and has a determinant. A nonzero value at a point means the function can be smoothly inverted nearby, a result known as the inverse function theorem, and the Jacobian of that inverse is the inverse of the original matrix. A positive determinant preserves orientation, a negative one reverses it, and its absolute value gives the local factor by which volumes expand or contract.
That volume factor is why the determinant appears whenever variables are changed inside a multiple integral: a tiny box in one coordinate system becomes a slanted parallelepiped in another, and the determinant measures its volume. Engineers and physicists also use Jacobians to judge whether an equilibrium in a system of differential equations is stable, by studying behaviour close to it.
Source: Jacobian matrix and determinant
