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 g_i 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

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

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