Suppose S is a rectangle in the uv plane with vertices O(0,0), P(delta u, 0), (delta u, delta v), Q(0, delta v). The image of S under the transformation x=g(u,v), y=h(u,v) is a region R in the xy plane. Let  O', P' and Q' be the images of O,P,Q, respectively, in the xy plane where O',P',Q' do not all lie on the same line. The coordinates of O' , P', and Q' are (g(0,0),h(0,0)), (g(delta u,0),h(delta u,0)), and (g(0,delta v),h(0,delta v)), respectively. Consider the parallelogram determined by the vectors O'P' and O'Q'. Use the cross product to show that the area of the parallelogram is approximately |J(u,v)|delta(u)delta(v). |J(u,v)| is the jacobian determinant.

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