Theorem: If $T : G \to D$ is a $C^1$ diffeomorphism between bounded open sets in $\mathbb{R}^n$, then $\int_D f(\mathbf{x})\,d\mathbf{x} = \int_G f(T(\mathbf{u})) |\det J_T(\mathbf{u})|\,d\mathbf{u}$.

Step 1: The linear case.

For a linear transformation $T(\mathbf{u}) = A\mathbf{u}$ with $A$ an invertible $n \times n$ matrix, we show $\operatorname{vol}(T(G)) = |\det A| \cdot \operatorname{vol}(G)$ for any measurable set $G$.

First consider elementary row operations:

• Scaling row $i$ by $c$: multiplies volume by $|c|$, and $\det$ by $c$
• Adding a multiple of one row to another: preserves volume (shearing), and $\det$ is unchanged
• Swapping rows: preserves volume, changes sign of $\det$

Since any invertible matrix is a product of elementary matrices, $\operatorname{vol}(A(G)) = |\det A| \cdot \operatorname{vol}(G)$ follows by multiplicativity.

For a linear $T$: $\int_D f(\mathbf{x})\,d\mathbf{x} = \int_G f(A\mathbf{u}) |\det A|\,d\mathbf{u}$ follows from the volume scaling and the substitution in Riemann sums.

Step 2: Infinitesimal linearization.

For a general $C^1$ diffeomorphism $T$, the key idea is that $T$ is approximately linear near each point. At $\mathbf{u}_0 \in G$: $T(\mathbf{u}_0 + \mathbf{h}) \approx T(\mathbf{u}_0) + J_T(\mathbf{u}_0)\mathbf{h}$

A small box $B$ near $\mathbf{u}_0$ with volume $\Delta V$ is mapped approximately to a parallelepiped with volume $\approx |\det J_T(\mathbf{u}_0)| \cdot \Delta V$.

Step 3: Riemann sum argument.

Partition $G$ into small boxes $B_k$ with volumes $\Delta V_k$ and sample points $\mathbf{u}_k$. The Riemann sum for the right side is: $\sum_k f(T(\mathbf{u}_k)) |\det J_T(\mathbf{u}_k)| \Delta V_k$

The images $T(B_k)$ approximately partition $D$, and $\operatorname{vol}(T(B_k)) \approx |\det J_T(\mathbf{u}_k)| \Delta V_k$. So the above sum approximates: $\sum_k f(T(\mathbf{u}_k)) \operatorname{vol}(T(B_k))$ which is a Riemann sum for $\int_D f(\mathbf{x})\,d\mathbf{x}$ with the partition $\{T(B_k)\}$.

Step 4: Making this rigorous.

The rigorous proof uses the fact that $T$ is a $C^1$ diffeomorphism to control the error in the linear approximation uniformly on compact subsets. Specifically, for any $\epsilon > 0$ and sufficiently fine partition: $\left|\operatorname{vol}(T(B_k)) - |\det J_T(\mathbf{u}_k)| \cdot \operatorname{vol}(B_k)\right| \leq \epsilon \cdot \operatorname{vol}(B_k)$

This follows from the uniform continuity of $J_T$ on compact sets. Summing over all boxes and taking the limit as the partition becomes finer gives the change of variables formula.

Alternatively, one can prove the result first for $C^2$ maps using the inverse function theorem and a partition-of-unity argument, then extend to $C^1$ by approximation. $\square$