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Substitution in triple integrals.

Consider a domain $\mathcal{D}$ in the xyz-space. Suppose now that there exists an invertible function g=(g1,g2,g3), defined over a domain $\mathcal{R}$ 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 $\mathcal{R}$ (i.e. each gi is differentiable over $\mathcal{R}$). 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

\begin{displaymath}J(u,v) =
\frac {\partial x}{\partial u} & \f...
...l z}{\partial v} & \frac {\partial z}{\partial w}

is called the Jacobian (determinant) of g=(g1,g2,g3), i.e. of the coordinate substitution $(x,y,z) \longrightarrow (u,v,w)$.

Theorem 7.26  

Noah Dana-Picard