Prove that under an affine transformation of space the ratio…

Prove that under an affine transformation of space the ratio of the volumes of any two bodies is invariant.

Step 1: Parallelepipeds (lemma)

A parallelepiped with edge vectors $\vec u,\vec v,\vec w$ has volume $V=|\det(\vec u\,|\,\vec v\,|\,\vec w)|$. Its image has edges $M\vec u, M\vec v, M\vec w$ (Problem 171, step 1), so

\[ V'=|\det\big(M(\vec u\,|\,\vec v\,|\,\vec w)\big)|=|\det M|\cdot V . \]

Step 2: Arbitrary bodies

Via the change of variables. The Jacobian of $\vec r\,'=M\vec r+\vec b$ is the constant $\det M$, so

\[ V(\Omega')=\iiint_{\Omega'}dx'\,dy'\,dz'=\iiint_{\Omega}|\det M|\,dx\,dy\,dz=|\det M|\,V(\Omega). \]

Elementary version. Approximate $\Omega$ by small cubes. Each cube's volume is multiplied by the same factor $|\det M|$, by the lemma. Passing to the limit gives the same formula.

\[V(\Omega')=|\det M|\cdot V(\Omega)\quad\text{for every body }\Omega\]

Step 3: The ratio

\[ \frac{V(\Omega_1')}{V(\Omega_2')}=\frac{|\det M|\,V(\Omega_1)}{|\det M|\,V(\Omega_2)}=\frac{V(\Omega_1)}{V(\Omega_2)} . \]

\[\frac{V_1'}{V_2'}=\frac{V_1}{V_2}\qquad\blacksquare\]

The key point is that the scaling factor is the same constant for all bodies. For a non-affine map the Jacobian varies from point to point, and ratios of volumes change. The plane analogue is that affine maps preserve ratios of areas, with factor $|\det M|$ for $2\times2$ matrices.