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## Substitution in triple integrals.

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) ).

Definition 7.25   Suppose that f, g1, g2 and g3 have continuous first order partial derivatives. The determinant

is called the Jacobian (determinant) of g=(g1,g2,g3), i.e. of the coordinate substitution .

Noah Dana-Picard
2001-05-30