Substitution Formula for Hausdorff Integrals — Statement & Proof

Substitution Formula for Hausdorff Integrals (Theorem # 868)

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The substitution formula is the change-of-variables theorem for [integrals](/page/Integral) with respect to [Hausdorff measure](/page/Hausdorff%20Measure) on surfaces. It answers the question: if $M$ is an $m$-dimensional surface in $\mathbb{R}^n$ and $\Psi$ deforms $M$ into another surface $\Psi(M)$, how does an $\mathcal{H}^m$-integral over $\Psi(M)$ relate to one over $M$? ### The role of $\sigma$ The surface $M$ sits inside $\mathbb{R}^n$ but is only $m$-dimensional, so the $n$ ambient coordinates are redundant for describing $M$. To do calculus on $M$ — compute Jacobians, determinants, tangent spaces — one needs a **parametrisation** $\sigma: A \to \mathbb{R}^n$ that assigns $m$ coordinates (the variables $y \in A \subseteq \mathbb{R}^m$) to each point of $M = \sigma(A)$. The condition that $\sigma$ is an **immersion** ($\operatorname{rank}(J\sigma(y)) = m$ at every $y \in A$) means the [derivative](/page/Derivative) $D\sigma(y): \mathbb{R}^m \to \mathbb{R}^n$ is injective: no nonzero tangent direction in parameter space is collapsed to zero. Geometrically, this ensures $M$ looks like a smooth $m$-dimensional surface near every point — no cusps or degeneracies. Analytically, it ensures the Gram matrix $J\sigma(y)^T J\sigma(y) \in \mathbb{R}^{m \times m}$ is positive definite, so its determinant is strictly positive and can appear in the denominator of the tangential Jacobian $J_M\Psi$. The parametrisation $\sigma$ is an auxiliary object: it is needed to *compute* $J_M\Psi$ (the formula involves $J\sigma$ and $J(\Psi \circ \sigma)$), but the result is **independent of the choice of $\sigma$**. This is verified in the proof (Claim: Parametrisation Independence): switching from $\sigma$ to another parametrisation $\tilde{\sigma}$ introduces the transition map $\varphi = \sigma^{-1} \circ \tilde{\sigma}$, whose Jacobian determinant appears identically in numerator and denominator and cancels. The tangential Jacobian $J_M\Psi(x)$ is therefore an intrinsic quantity associated to the surface $M$ and the map $\Psi$, not to the coordinates chosen on $M$. ### Why the theorem is essential Without the substitution formula, there is no systematic way to transform $\mathcal{H}^m$-integrals from one surface to another. The standard Lebesgue change-of-variables theorem applies to maps between open subsets of $\mathbb{R}^m$, not to maps between surfaces embedded in $\mathbb{R}^n$. The substitution formula bridges this gap: it lets one pull back an integral over $\Psi(M)$ to an integral over $M$, with the tangential Jacobian $J_M\Psi$ playing the same role that $|\det D\Psi|$ plays in the flat (Lebesgue) change of variables. This appears throughout analysis. In the proof of the [mean-value formula for Laplace's equation](/theorems/31), the affine scaling $\Psi(\omega) = z + r\omega$ maps $\partial B(0,1)$ to $\partial B(z,r)$, and the substitution formula with $J_M\Psi = r^{n-1}$ converts integrals over the general sphere to integrals over the unit sphere. In the proof of the [Rankine-Hugoniot condition](/theorems/578), the parametrisation of a shock curve $\Gamma$ by $\gamma(\tau) = (s(\tau), \tau)$ produces the Jacobian factor $\sqrt{1 + \dot{s}(\tau)^2}$, which cancels with the same factor in the outward normal. ### The case $m = 1$: line integrals and the "standard formula" The simplest and most instructive case is $m = 1$, where $M$ is a curve. Consider $\Psi = \mathrm{id}$ (no deformation — we are just computing an integral on $M$ itself). The substitution formula with $\Psi = \mathrm{id}$ is trivially $\int_M f\,d\mathcal{H}^1 = \int_M f\,d\mathcal{H}^1$, but the *proof* of the substitution formula passes through the [Area Formula (Classical)](/theorems/25), which for a parametrisation $\gamma: (a, b) \to \mathbb{R}^n$ of $M$ gives \begin{align*} \int_M f\,d\mathcal{H}^1 &= \int_a^b f(\gamma(t))\,|\gamma'(t)|\,d\mathcal{L}^1(t). \end{align*} This is the formula that many textbooks *define* as the line integral $\int_\gamma f\,ds$. From the Hausdorff measure perspective, however, $\int_M f\,d\mathcal{H}^1$ is defined **intrinsically** — it is the [Lebesgue integral](/page/Lebesgue%20Integral) of $f$ against the measure $\mathcal{H}^1$ restricted to the [set](/page/Set) $M \subseteq \mathbb{R}^n$, with no parametrisation involved. The formula $\int_a^b f(\gamma(t))\,|\gamma'(t)|\,d\mathcal{L}^1(t)$ is then a **theorem** (the area formula), not a definition. This distinction matters: the definition-based approach must separately verify that the integral is independent of the parametrisation $\gamma$, a non-trivial check requiring the substitution $t \mapsto \varphi(t)$ and the chain rule $|(\gamma \circ \varphi)'| = |\gamma'(\varphi(t))|\,|\varphi'(t)|$. In the Hausdorff framework, parametrisation independence is *automatic* — the measure $\mathcal{H}^1$ is defined on subsets of $\mathbb{R}^n$ without reference to any parametrisation, and the area formula merely provides a computational tool for evaluating the intrinsic integral. The same principle applies in higher dimensions: for an $m$-dimensional surface $M$ parametrised by $\sigma: A \to \mathbb{R}^n$, the area formula gives \begin{align*} \int_M f\,d\mathcal{H}^m &= \int_A f(\sigma(y))\,\sqrt{\det(J\sigma(y)^T\,J\sigma(y))}\,d\mathcal{L}^m(y), \end{align*} which reduces the $\mathcal{H}^m$-integral to an ordinary $\mathcal{L}^m$-integral over the parameter domain. The factor $\sqrt{\det(J\sigma^T J\sigma)}$ — the Gram determinant — accounts for how much $\sigma$ stretches $m$-dimensional volume. For curves ($m = 1$), $J\sigma = \gamma' \in \mathbb{R}^{n \times 1}$, so $\sqrt{\det(\gamma'^T \gamma')} = |\gamma'|$, recovering the scalar speed.

Proof

**Proof plan.** The strategy is to reduce both sides to $\mathcal{L}^m$-[integrals](/page/Integral) over the parameter domain $A$ by applying the [Area Formula (Classical)](/theorems/25) twice — once to $\sigma$ (which parametrizes $M$) and once to $\Psi \circ \sigma$ (which parametrizes $\Psi(M)$). The chain rule relates the two Jacobians, and the tangential Jacobian emerges as their ratio. A separate argument using the QR decomposition shows that this ratio is independent of the parametrization. **Step 1: Apply the area formula to $\Psi \circ \sigma$.** The composition $\Psi \circ \sigma: A \to \mathbb{R}^n$ is $C^1$ (as a composition of $C^1$ maps) and injective (since both $\sigma$ and $\Psi|_M$ are injective). By the [Area Formula (Classical)](/theorems/25) applied to the injective $C^1$ map $\Psi \circ \sigma$ on the [open set](/page/Open%20Set) $A \subseteq \mathbb{R}^m$: \begin{align*} \int_{\Psi(M)} f(z) \, d\mathcal{H}^m(z) = \int_A f(\Psi(\sigma(y))) \, \sqrt{\det\bigl(J(\Psi \circ \sigma)(y)^T \, J(\Psi \circ \sigma)(y)\bigr)} \, d\mathcal{L}^m(y). \tag{1} \end{align*} **Step 2: Factor the integrand using the tangential Jacobian.** By definition of $J_M \Psi$, for each $y \in A$: \begin{align*} \sqrt{\det\bigl(J(\Psi \circ \sigma)(y)^T \, J(\Psi \circ \sigma)(y)\bigr)} = J_M \Psi(\sigma(y)) \cdot \sqrt{\det\bigl(J\sigma(y)^T \, J\sigma(y)\bigr)}. \end{align*} Substituting into (1): \begin{align*} \int_{\Psi(M)} f(z) \, d\mathcal{H}^m(z) = \int_A f(\Psi(\sigma(y))) \cdot J_M \Psi(\sigma(y)) \cdot \sqrt{\det\bigl(J\sigma(y)^T \, J\sigma(y)\bigr)} \, d\mathcal{L}^m(y). \tag{2} \end{align*} **Step 3: Apply the area formula to $\sigma$ in the reverse direction.** By the [Area Formula (Classical)](/theorems/25) applied to the injective $C^1$ immersion $\sigma: A \to \mathbb{R}^n$, for any Borel measurable function $g: M \to [0, \infty]$: \begin{align*} \int_A g(\sigma(y)) \, \sqrt{\det\bigl(J\sigma(y)^T \, J\sigma(y)\bigr)} \, d\mathcal{L}^m(y) = \int_M g(x) \, d\mathcal{H}^m(x). \tag{3} \end{align*} The integrand in (2) has the form of the left-hand side of (3) with $g(x) := f(\Psi(x)) \cdot J_M \Psi(x)$. The function $g$ is Borel measurable: $f \circ \Psi$ is Borel measurable as a composition of a Borel measurable function with a [continuous](/page/Continuity) function, and $J_M \Psi$ is continuous on $M$ (as a ratio of continuous, strictly positive [functions](/page/Function)). Applying (3): \begin{align*} \int_{\Psi(M)} f(z) \, d\mathcal{H}^m(z) = \int_M f(\Psi(x)) \cdot J_M \Psi(x) \, d\mathcal{H}^m(x). \end{align*} **Step 4: Verify parametrization independence of $J_M \Psi$.** [claim:Parametrisation Independence] Let $\tilde{\sigma}: \tilde{A} \to \mathbb{R}^n$ be another injective $C^1$ immersion with $\tilde{\sigma}(\tilde{A}) = M$. Then the tangential Jacobian computed via $\tilde{\sigma}$ agrees with the one computed via $\sigma$ at every point of $M$. [/claim] [proof] **Step 4a.** Fix $x \in M$ and let $y = \sigma^{-1}(x) \in A$, $\tilde{y} = \tilde{\sigma}^{-1}(x) \in \tilde{A}$. Since $\sigma$ and $\tilde{\sigma}$ are injective $C^1$ immersions with the same image, the transition map \begin{align*} \varphi := \sigma^{-1} \circ \tilde{\sigma}: \tilde{A} \to A \end{align*} is a $C^1$ diffeomorphism (by the [implicit function theorem](/page/Implicit%20Function%20Theorem) applied to the immersion $\sigma$), and $\sigma \circ \varphi = \tilde{\sigma}$. **Step 4b.** By the chain rule, $J\tilde{\sigma}(\tilde{y}) = J\sigma(y) \cdot J\varphi(\tilde{y})$, where $J\varphi(\tilde{y}) \in \mathbb{R}^{m \times m}$ is invertible. Set $P := J\varphi(\tilde{y})$. Then: \begin{align*} \det\bigl(J\tilde{\sigma}^T J\tilde{\sigma}\bigr) = \det\bigl(P^T J\sigma^T J\sigma \, P\bigr) = (\det P)^2 \cdot \det\bigl(J\sigma^T J\sigma\bigr). \end{align*} Similarly, $J(\Psi \circ \tilde{\sigma})(\tilde{y}) = J(\Psi \circ \sigma)(y) \cdot P$, so: \begin{align*} \det\bigl(J(\Psi \circ \tilde{\sigma})^T J(\Psi \circ \tilde{\sigma})\bigr) = (\det P)^2 \cdot \det\bigl(J(\Psi \circ \sigma)^T J(\Psi \circ \sigma)\bigr). \end{align*} Taking square roots and forming the ratio, the factors $|\det P|$ cancel: \begin{align*} \frac{\sqrt{\det\bigl(J(\Psi \circ \tilde{\sigma})^T J(\Psi \circ \tilde{\sigma})\bigr)}}{\sqrt{\det\bigl(J\tilde{\sigma}^T J\tilde{\sigma}\bigr)}} = \frac{\sqrt{\det\bigl(J(\Psi \circ \sigma)^T J(\Psi \circ \sigma)\bigr)}}{\sqrt{\det\bigl(J\sigma^T J\sigma\bigr)}}. \end{align*} [/proof] [remark:Application To Submanifolds Without A Global Chart] If $M$ is a compact $C^1$ submanifold of $\mathbb{R}^n$ that cannot be covered by a single chart (e.g. $M = \partial B(0,1) \subseteq \mathbb{R}^n$), one partitions $M$ into finitely many $\mathcal{H}^m$-measurable pieces $M_1, \ldots, M_k$, each contained in the image of a chart $\sigma_\alpha: A_\alpha \to \mathbb{R}^n$. The formula is applied to each piece, and the results are summed by $\sigma$-additivity of $\mathcal{H}^m$. The conclusion $\int_{\Psi(M)} f \, d\mathcal{H}^m = \int_M (f \circ \Psi) \cdot J_M \Psi \, d\mathcal{H}^m$ follows, with $J_M \Psi$ well-defined by parametrisation independence. [/remark] [remark:Affine Maps And Spheres] When $\Psi: \mathbb{R}^n \to \mathbb{R}^n$ is the affine map $\Psi(x) = z + rx$ for fixed $z \in \mathbb{R}^n$ and $r > 0$, the [derivative](/page/Derivative) is $D\Psi(x) = rI_n$ for all $x$. For any $C^1$ parametrisation $\sigma: A \to \mathbb{R}^n$ of an $m$-dimensional surface $M$: \begin{align*} J(\Psi \circ \sigma)(y) = r \cdot J\sigma(y), \end{align*} so $\det(J(\Psi \circ \sigma)^T J(\Psi \circ \sigma)) = r^{2m} \det(J\sigma^T J\sigma)$, giving $J_M \Psi = r^m$. The formula becomes \begin{align*} \int_{\Psi(M)} f \, d\mathcal{H}^m = r^m \int_M f(z + rx) \, d\mathcal{H}^m(x). \end{align*} In particular, for $M = \partial B(0,1)$, $\Psi(M) = \partial B(z, r)$, and $m = n - 1$: \begin{align*} \int_{\partial B(z, r)} f(y) \, d\mathcal{H}^{n-1}(y) = r^{n-1} \int_{\partial B(0, 1)} f(z + r\omega) \, d\mathcal{H}^{n-1}(\omega), \end{align*} which is the change-of-variables identity for spherical integrals. [/remark]

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Hausdorff measure Definition Lebesgue integral Definition immersion Definition Area Formula (Classical) Theorem #25 Mean-value formulas for Laplace's equation Theorem #31 Rankine-Hugoniot Jump Condition Theorem #578 Stone-Weierstrass Theorem Functional Analysis Continuity of Measures Analysis Compact Subspaces and Hausdorff Spaces Topology Lebesgue Fundamental Theorem of Calculus Measure Theory Hilbert-Schmidt Integral Operators are Compact Analysis Fundamental Theorem of Calculus for Lebesgue Integration Measure Theory Bandwidth Preservation in LU Factorisation Numerical Analysis Reverse Hölder Inequality Analysis Analysis Area

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Substitution Formula for Hausdorff Integrals (Theorem # 868)

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