Multivariate Calculus - Bruce E. Shapiro

168 LECTURE 20. TRIPLE INTEGRALS
Triple Integrals in Any Coordinate Frame
Suppose we have a coordinate frame (u, v, w) where
x = X(u, v, w)
y = Y (u, v, w)
z = Z(u, v, w)
Then the volume element is
dV = J(u, v, w)dudvdw
where
∂x ∂x
∂u ∂v
J(u, v, w) =
∂y ∂y
∂u ∂v
∂z ∂a
∣
∂u ∂v
is called the Jacobian of the transformation.
∂x
∂w
∂y
∂w
∂a
∣
∂w
Revised December 6, 2006. Math 250, Fall 2006