1 Change-of-Variables Formula
The change-of-variables (or transformation) formula is one of the central tools in multivariate calculus and probability theory. It tells us how integrals - and in particular probability densities - transform under smooth, invertible maps.
It is the multivariate analogue of the familiar one-dimensional substitution rule \[ \int_a^b f(x)\,dx = \int_{\phi(a)}^{\phi(b)} f(\phi^{-1}(y))\,\bigl|(\phi^{-1})'(y)\bigr|\,dy. \] In higher dimensions, the derivative is replaced by the determinant of the Jacobian Matrix.
The change-of-variables formula is the mechanism that makes all such transformations precise: it guarantees that probability mass and integrals are preserved under smooth, invertible maps, with the Jacobian encoding how volume is distorted.
1.1 Statement (Deterministic Form)
Let \(U, V \subset \mathbb{R}^d\) be open sets and let \[ T : U \to V \] be a \(C^1\) bijection with \(C^1\) inverse. Write \(y = T(x)\).
For any integrable function \(f : V \to \mathbb{R}\), \[ \int_V f(y)\,dy = \int_U f(T(x))\,\bigl|\det J_{T}(x)\bigr|\,dx, \] where \(J_{T}(x)\) is the Jacobian Matrix of \(T\) at \(x\). Equivalently, \[ \int_U g(x)\,dx = \int_V g(T^{-1}(y))\,\bigl|\det J_{T}^{-1}(y)\bigr|\,dy. \] ## Probabilistic Form
Let \(X\) be a random vector in \(\mathbb{R}^d\) with density \(f_X\), and let \[Y = T(X),\]where \(T\) satisfies the assumptions above. Then \(Y\) has density\[f_Y(y) = f_X(T^{-1}(y))\,\bigl|\det J_{T}^{-1}(y)\bigr|.\]This ensures that probabilities are preserved:\[\mathbb{P}(Y \in A) = \mathbb{P}(X \in T^{-1}(A)).\]
1.2 Intuition
Locally, a smooth map \(T\) behaves like a linear transformation: \[ T(x + h) \approx T(x) + J_{T}(x)h. \] A small volume element \(dx\) near \(x\) is sent to a distorted parallelepiped near \(T(x)\) whose volume is \[ |\det J_{T}(x)|\,dx. \] The Jacobian determinant is precisely the local volume-scaling factor.
1.3 Proof (Sketch)
We prove the deterministic version.
Linear case. If \(T(x) = Ax\) with \(A \in \mathbb{R}^{d \times d}\) invertible, then for any measurable set \(E\), \[\text{Vol}(A E) = |\det A|\,\text{Vol}(E).\]Hence, \[\int_{A U} f(y)\,dy=\int_U f(Ax)\,|\det A|\,dx,\]as required.
Local linearization. For general \(C^1\) maps, near each \(x\), \[T(x+h) = T(x) + DT(x)h + o(\|h\|).\]On a sufficiently fine partition of \(U\) into small cells \(C_k\), \(T\) is well approximated on each \(C_k\) by its linearization at some point \(x_k \in C_k\).
Riemann sum argument. Writing the integral as a limit of Riemann sums, \[\int_V f(y)\,dy \approx \sum_k f(T(x_k))\,\text{Vol}(T(C_k)) \approx \sum_k f(T(x_k))\,|\det DT(x_k)|\,\text{Vol}(C_k).\]Taking the limit as the mesh goes to zero yields \[\int_V f(y)\,dy=\int_U f(T(x))\,|\det DT(x)|\,dx,\]as required.
A fully rigorous proof uses measure-theoretic arguments and the inverse function theorem.
1.4 Examples
1.4.1 Polar Coordinates
Let \(T(r,\theta) = (r\cos\theta, r\sin\theta)\). Then \[ J_{T}(r,\theta) = \begin{pmatrix} \cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta \end{pmatrix}, \qquad \det J_{T}(r,\theta) = r. \] Hence, \[ \int_{\mathbb{R}^2} f(x,y)\,dx\,dy = \int_0^\infty \int_0^{2\pi} f(r\cos\theta, r\sin\theta)\, r\, d\theta\, dr. \] In probability, if \((R,\Theta)\) has joint density \(g(r,\theta)\), then \[ f_{X,Y}(x,y) = g(r,\theta)\,\frac{1}{r}, \quad (x,y) = (r\cos\theta, r\sin\theta). \] ### Linear Transformation of a Gaussian
Let \(X \sim \mathcal{N}(0, I_d)\) and define \(Y = AX\) with \(A\) invertible. Since \(T(x)=Ax\), we have \(T^{-1}(y)=A^{-1}y\) and \[ |\det J_{T}^{-1}(y)| = |\det A^{-1}| = \frac{1}{|\det A|}. \] Thus, \[ f_Y(y) = (2\pi)^{-d/2} \exp\!\left(-\tfrac12 \|A^{-1}y\|^2\right) \frac{1}{|\det A|}, \] which is exactly the density of \(\mathcal{N}(0, AA^\top)\). ### Ratio of Two Independent Variables
Let \(X,Y\) be independent with joint density \(f_{X,Y}(x,y)\). Define \[ U = \frac{X}{Y}, \qquad V = Y. \] Then \(T(x,y) = (x/y, y)\) and \[ T^{-1}(u,v) = (uv, v). \] The Jacobian of \(T^{-1}\) is \[ J_{T}^{-1}(u,v) = \begin{pmatrix} v & u \\ 0 & 1 \end{pmatrix}, \qquad \det = v. \] Hence, \[ f_{U,V}(u,v) = f_{X,Y}(uv, v)\,|v|. \] Marginalizing over \(v\) yields the density of \(U = X/Y\).