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Consider a domain
in the xy-plane. Suppose now that there exists an invertible function
g=(g1,g2), defined over a domain
in the uv-plane such that
(x,y)=( g1(u,v), g2(u,v)). Moreover suppose that g is differentiable over
(i.e. each gi is differentiable over
Thus, we have:
Suppose that f
have continuous first order partial derivatives. The determinant
is called the Jacobian (determinant)
of the coordinate substitution