Jacobian determinant

When a change of coordinates bends and stretches the plane, a tiny square in the new coordinates does not map to a tiny square of the same size in the old ones — it maps to a tiny tilted parallelogram, larger or smaller. The Jacobian determinant is the single number that tells you the local area-scaling factor of that distortion at each point. In one dimension it is just the derivative dx/du; in two or more it is the determinant of the matrix of all the first partial derivatives.

For a map sending (u, v) to (x(u, v), y(u, v)), the Jacobian matrix is the 2-by-2 array of partials [x_u, x_v; y_u, y_v], and the Jacobian determinant J is x_u times y_v minus x_v times y_u. Geometrically, the two columns of that matrix are the image vectors of the unit steps in u and in v; the absolute value of their determinant is the area of the parallelogram they span — exactly the factor by which a unit uv-area is rescaled into xy-area. In three dimensions it is a 3-by-3 determinant whose absolute value gives the volume-scaling factor. The sign of J records whether the map preserves or flips orientation; for integration you take its absolute value.

The Jacobian is the workhorse of the change-of-variables theorem: dA in old coordinates equals the absolute value of J times the new-coordinate area element, and likewise for dV. This is the source of the r in polar coordinates (J = r), the r in cylindrical (J = r), and the rho^2 sin(phi) in spherical (J = rho^2 sin(phi)). It also appears in probability (transforming densities of random variables), in mechanics (changing configuration variables), and in the inverse function theorem, where a non-zero Jacobian at a point certifies that the map is locally invertible there. A useful shortcut: the Jacobian of an inverse map is the reciprocal of the original Jacobian, so you can compute whichever direction is easier.