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For the printed integration formula, $J$ must mean the Jacobian determinant of the inverse coordinate map:If instead one calls $\partial (u,v,w)/\partial (x,y,z)$ the forward Jacobian, its determinant is $1/J$ where the derivative is invertible. This convention distinction resolves the direction of the arrow in the printed wording.

To derive the change of variables formula, a small rectangular cell of side lengths $du,dv,dw$ maps under the inverse transformation, to first order, to a parallelepiped with edge vectors ${\mathbf{r}}_{u}du$, ${\mathbf{r}}_{v}dv$, ${\mathbf{r}}_{w}dw$. Its volume is the absolute scalar triple product of those vectors:For a one-to-one continuously differentiable coordinate map with continuously differentiable inverse, summing these local volume contributions and taking the partition limit givesThe absolute value removes an orientation reversal; injectivity ensures that each volume element is counted once.

For positive semi-axes, take $u=x/a$, $v=y/b$, $w=z/c$. The solid ellipsoid becomes the unit ball and the inverse Jacobian determinant is $abc$. ThusRotational symmetry gives equal integrals of $u^2,v^2,w^2$ over the ball. Their sum is the integral of $r^2$, so spherical coordinates give $\int u^{2}dV=\frac{1}{3}{\int }_0^{1}4\pi r^{4}dr=4\pi /15$. The Cartesian second moment of a solid ellipsoid is therefore