Jacobian
A determinant of partial derivatives measuring how much a coordinate change stretches or shrinks area (or volume) at a point.
The Jacobian \dfrac{\partial(x,y)}{\partial(u,v)} packages the four partials of a coordinate change x=x(u,v),\ y=y(u,v) into one determinant, because a determinant is exactly the tool that measures how much a linear map scales area, and near any point the transformation is well-approximated by the linear map built from those four numbers. It produces automatically every extra factor a named coordinate system carries — polar's r, spherical's \rho^2\sin\phi — for any change of variables, not just those two named systems. The three-variable version \dfrac{\partial(x,y,z)}{\partial(u,v,w)} is the same idea one dimension up, scaling volume instead of area.
Examples
For x=r\cos\theta,\ y=r\sin\theta: \dfrac{\partial(x,y)}{\partial(r,\theta)}=r\cos^2\theta+r\sin^2\theta=r, exactly the extra factor polar coordinates carry.
For x=au,\ y=bv,\ z=cw, the Jacobian matrix is \text{diag}(a,b,c), with determinant abc.