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Consider a domain
in the xyz-space. Suppose now that there exists an invertible function
g=(g1,g2,g3), defined over a domain
in the uvw-plane such that
(x,y,z)=( g1(u,v,w), g2(u,v,w), g3(u,v,w)). Moreover suppose that g is differentiable over
(i.e. each gi is differentiable over
Thus, we have:
f(x,y,z)=f(g1(u,v,w), g2(u,v,w),g3(u,v,w) ).
Suppose that f
have continuous first order partial derivatives. The determinant
is called the Jacobian (determinant)
), i.e. of the coordinate substitution