The Jacobian

Calculus 3 · Axiom Academy

Change variables in a double integral and area itself gets stretched — the Jacobian |J| is the exact factor that measures the stretch. 1. The Jacobian Is an Area-Scaling Factor A transformation T sends the uv -plane to the xy -plane through x=x(u,v) and y=y(u,v) . Watch what it does to a single unit cell: the little square in uv lands as a slanted parallelogram in xy . The Jacobian matrix collects the four partial derivatives; its determinant is the area of that parallelogram — the factor by which T scales area right there. The Jacobian matrix and its determinant 2. Change of Variables, and Why Polar Has an r Because a cell becomes a patch of area, we simply attach |J| when we rewrite an integral in new variables. Run polar coordinates through the definition and the famous factor of r falls right out: the cells are annular wedges, and they grow in area as you move outward. The change-of-variables formula Polar coordinates: the Jacobian is exactly r A uniform cell in the -rectangle maps to a wedge of area . Near the origin the wedges are tiny; far out they are large — the area scales in direct proportion to r , which is why . 3. Curved Maps: a Local Factor, and Triple Integrals For a linear map the stretch is the same everywhere. For a curved map it is not — the Jacobian is a function |J(u,v)| that changes from point to point, so it must live inside the integral. Watch a uniform grid fan out under : cells on the right stretch more because |J|=1+u grows.

This is the written version of the interactive lesson above. See the full Calculus 3 course.