The Area and Coarea Formulas

The change-of-variables formula must be modified when a map changes dimension or is not injective. The area formula measures the image with multiplicity; the coarea formula integrates over fibres. Both use a Jacobian that records the appropriate volume distortion.

We first define this Jacobian for linear maps using polar decomposition. Rademacher’s theorem then supplies the derivative of a Lipschitz map almost everywhere, allowing the same definition pointwise.

Definition 1. A linear map

\(O: \mathbb R^n \to \mathbb R^m\) is orthogonal if \((Ox) \cdot (Oy) = x \cdot y\) for all \(x,y \in \mathbb R^n\).
\(S: \mathbb R^n \to \mathbb R^n\) is symmetric if \(x \cdot (Sy) = (Sx) \cdot y\) for all \(x,y \in \mathbb R^n\).
\(D: \mathbb R^n \to \mathbb R^n\) is diagonal if there exists some vector \(d \in \mathbb R^n\) such that \(Dx = (d_1 x_1, \cdots, d_n x_n)\) for all \(x \in \mathbb R^n\).
\(A: \mathbb R^n \to \mathbb R^m\) has the adjoint \(A^* : \mathbb R^m \to \mathbb R^n\) defined through \(x \cdot (A^* y) = (Ax) \cdot y\) for all \(x \in \mathbb R^n\) and \(y \in \mathbb R^m\).

Theorem 2 (Polar decomposition). Let \(L: \mathbb R^n \to \mathbb R^m\) be a linear map.

If \(n \leq m\), there exists a symmetric map \(S: \mathbb R^n \to \mathbb R^n\) and an orthogonal map \(O : \mathbb R^n \to \mathbb R^m\) such that \(L = O \circ S\).
If \(n \geq m\), there exists a symmetric map \(S: \mathbb R^m \to \mathbb R^m\) and an orthogonal map \(O : \mathbb R^m \to \mathbb R^n\) such that \(L = S \circ O^*\).

Proof of Theorem 2. For \(n\leq m\), the positive semidefinite matrix \(L^*L\) has a positive semidefinite square root \(S\). On \((\ker S)^\perp\), define \(O(Sx)=Lx\). This is well-defined and preserves inner products because \(\langle Lx,Ly\rangle=\langle Sx,Sy\rangle\). Extend an orthonormal basis of its domain to one of \(\mathbb R^n\), and extend the corresponding image basis to \(n\) orthonormal vectors in \(\mathbb R^m\). The extended isometry satisfies \(L=OS\), including on \(\ker S\). For \(n\geq m\), apply this construction to \(L^*\) and take adjoints. Thus \(S\) can always be chosen positive semidefinite, and \(J(L)\) is the product of the \(\min(m,n)\) singular values. ◻

With the polar decomposition, there is no ambiguity in defining the Jacobian of a linear map \(L\) by \(\operatorname{Jac} (L) \triangleq |\det (S)|\) (regardless of \(n\) and \(m\)). A straightforward calculation also shows that if \(n \leq m\), \(L = O \circ S\) (by the polar decomposition), \(L^* = S^* \circ O^* = S \circ O^*\) (by the properties of the adjoint and the definition of a symmetric map), so \(L^* \circ L = S \circ O^* \circ O \circ S = S^2\) by the definition of an orthogonal map. As such, \(\det (L^* \circ L) = \det (S)^2 = (\det (S))^2 = \operatorname{Jac} (L)^2\). A similar calculation applies to other case where \(n \geq m\).

To define the area formula, it is instructive to use the Binet-Cauchy formula. Define

\[\begin{aligned} \Lambda (m,n) \triangleq \{ \ell: \llbracket n \rrbracket \to \llbracket m \rrbracket : \ell \text{ is an increasing map} \} \end{aligned}\]

for \(n \leq m\). Then, for each \(\ell \in \Lambda (m,n)\), define the “projection map” \(P_\ell : \mathbb R^m \to \mathbb R^n\) via the formula \(P_\ell (x_1, \cdots, x_m) = (x_{\ell(1)}, \cdots, x_{\ell(n)})\). Then, for any linear map \(L : \mathbb R^n \to \mathbb R^m\) where \(n \leq m\),

\[\begin{aligned} \operatorname{Jac} (L)^2 = \sum_{\ell \in \Lambda(m,n)} (\det(P_\ell \circ L))^2 ~. \end{aligned}\]

Interpreting determinants as oriented volumes, the identity expresses squared volume as the sum of its squared coordinate projections. The proof of the Binet-Cauchy formula proceeds by a series of direct calculations, and can be found in introductory linear algebra texts.

For completeness, let \(v_1,\ldots,v_n\) be the columns of \(L\). Their exterior product expands in the orthonormal basis \(e_{\ell(1)}\wedge\cdots\wedge e_{\ell(n)}\), with coefficient \(\det(P_\ell L)\). Its squared norm is the sum of the squared coefficients. The same norm is the Gram determinant \(\det(\langle v_i,v_j\rangle)=\det(L^*L)\). This proves the displayed identity and also handles rank-deficient \(L\).

Finally, in the view of the Rademacher theorem, the gradient matrix of a Lipschitz map \(f: \mathbb R^n \to \mathbb R^m\) is defined as

\[\begin{aligned} \Dif f = \begin{bmatrix} \dpd{{f^1}}{{x_1}} & \cdots & \dpd{{f^1}}{{x_n}} \\ \vdots & \ddots & \vdots \\ \dpd{{f^m}}{{x_1}} & \cdots & \dpd{{f^m}}{{x_n}} \end{bmatrix} ~. \end{aligned}\]

The Jacobian of a Lipschitz map is therefore defined via

\[\begin{aligned} J f(x) = \operatorname{Jac} (\Dif f(x)) ~, \end{aligned}\]

which is well-defined \(\lambda_n\)-a.e. \(x\). The two definitions are related directly: at a differentiability point, take the Jacobian of the linear derivative. Rademacher’s theorem guarantees that this defines the Jacobian almost everywhere.

The area formula equates the integral of this local volume distortion with the measure of the image, counting how many points of the domain map to each image point.

Theorem 3 (Area formula). Let \(f : \mathbb R^n \to \mathbb R^m\) be a Lipschitz function, where \(n \leq m\). Then, for each \(\lambda_n\)-measurable subset \(A \subset \mathbb R^n\),

\[\begin{aligned} \int_A J f \dif x = \int_{\mathbb R^m} \mathcal H^0 (A \cap f^{-1} (\{y\})) \dif \mathcal H^n (y) ~. \end{aligned}\]

Proof. Set \(J_nf=0\) on its null nondifferentiability set. That set has a null image by the Lipschitz estimate, so it contributes zero to the multiplicity integral. Exhaust \(A\) by bounded pieces to reduce to \(\lambda_n(A)<\infty\).

First suppose \(J_nf>0\) on \(A\). For a fixed \(t>1\), Lemma 14 of the preceding note gives disjoint Borel pieces \(E_k\) and linear maps \(T_k\). Put \(G_k=A\cap E_k\). On each piece the two Lipschitz estimates and linear change of variables give

\[t^{-n}\mathcal H^n(f(G_k))\leq\lambda_n(T_kG_k)=\det(T_k)\lambda_n(G_k)\leq t^n\mathcal H^n(f(G_k)).\]

Together with the pointwise Jacobian bounds this yields

\[t^{-2n}\mathcal H^n(f(G_k))\leq\int_{G_k}J_nf\,dx\leq t^{2n}\mathcal H^n(f(G_k)).\]

Each restriction is injective, and the pieces are disjoint in the domain. Consequently, at every \(y\),

\[N(f,A,y)=\sum_k\mathbb1_{f(G_k)}(y).\]

There is no further limiting cube argument on these injective pieces. Tonelli gives \(\int N\,d\mathcal H^n=\sum_k\mathcal H^n(f(G_k))\). Sum the inequalities and let \(t\downarrow1\). Lemma 13 bounds the multiplicity integral when \(A\) has finite measure, so the limiting comparison is justified.

Now suppose \(J_nf=0\) on \(A\). This does not imply that \(A\) has measure zero. For \(0<\varepsilon\leq1\), introduce the injective map

\[g_\varepsilon(x)=(f(x),\varepsilon x)\in\mathbb R^{m+n}.\]

If \(\sigma_1(x),\ldots,\sigma_n(x)\) are the singular values of \(Df(x)\), then

\[J_ng_\varepsilon(x)=\prod_{i=1}^n\sqrt{\sigma_i(x)^2+\varepsilon^2}.\]

At least one \(\sigma_i\) vanishes on \(A\) and all are at most \(L=\operatorname{Lip}(f)\). Hence

\[0<\varepsilon^n\leq J_ng_\varepsilon(x)\leq\varepsilon(L^2+1)^{(n-1)/2}.\]

The full-rank case already proved applies to \(g_\varepsilon\). Its injectivity and the 1-Lipschitz projection onto the first \(m\) coordinates give

\[\mathcal H^n(f(A))\leq\mathcal H^n(g_\varepsilon(A))=\int_AJ_ng_\varepsilon\,dx\leq\varepsilon(L^2+1)^{(n-1)/2}\lambda_n(A).\]

Let \(\varepsilon\downarrow0\). The multiplicity function is supported on the resulting null image, so its integral is zero, matching \(\int_AJ_nf=0\). Splitting a general \(A\) into these parts, and then exhausting it, proves the formula. ◻

Applying the area formula first to simple functions and then passing to limits gives the following weighted form.

Theorem 4. Let \(f : \mathbb R^n \to \mathbb R^m\) be a Lipschitz function with \(n \leq m\). Then, for each \(\lambda_n\) summable function \(g: \mathbb R^n \to \mathbb R\),

\[\begin{aligned} \int_{\mathbb R^n} g(x) J f(x) \dif x = \int_{\mathbb R^m} \left( \sum_{x \in f^{-1}(y)} g(x) \right) \dif \mathcal H^n (y) ~. \end{aligned}\]

Proof of Theorem 4. Apply Theorem 3 to indicators, then to nonnegative simple functions, and use monotone convergence. For signed \(g\), apply this separately to \(g^+\) and \(g^-\). The condition \(g\in L^1\) suffices because \(J_nf\leq L^n\), so \(\int|g|J_nf<\infty\) and the fibre sum of \(|g|\) is finite for almost every image point. More generally the formula holds for nonnegative measurable \(g\), allowing infinite integrals. ◻

1. Coarea Formula

The coarea formula extends the slicing principle in Fubini’s theorem. It integrates over level sets of a Lipschitz map, with the Jacobian compensating for the distortion transverse to the fibres.

Theorem 5. Let \(f:\mathbb R^n\to\mathbb R^m\) be Lipschitz, with \(n\geq m\geq1\). Define \(J_mf=\sqrt{\det(Df\,Df^*)}\) at differentiability points and zero elsewhere. For every Lebesgue measurable \(A\),

\[\int_AJ_mf(x)\,dx=\int_{\mathbb R^m}\mathcal H^{n-m}(A\cap f^{-1}(y))\,dy.\]

Proof. Lemma 16 of the preceding note shows that a null set in the domain has null \((n-m)\)-dimensional fibres for almost every \(y\). We may therefore discard the nondifferentiability set and work first with \(\lambda_n(A)<\infty\).

Full rank. Suppose \(J_mf>0\) on \(A\). At each such point choose a coordinate projection \(P_\ell:\mathbb R^n\to\mathbb R^{n-m}\) that is injective on \(\ker Df\). Then

\[h_\ell(x)=(f(x),P_\ell x)\in\mathbb R^n\]

has invertible derivative: a vector in its kernel lies both in \(\ker Df\) and \(\ker P_\ell\), and so is zero. There are finitely many choices of \(\ell\); partition \(A\) according to the first one that works.

Fix \(t>1\). Apply the square-map decomposition, Lemma 17, to each \(h_\ell\). This gives disjoint pieces \(G_k\) covering \(A\), invertible \(S_k\), and \(t\)-bi-Lipschitz maps \(F_k=S_k^{-1}h_\ell\) on \(G_k\). Put \(q(y,z)=y\) and \(B_k=qS_k\). On \(G_k\),

\[f=B_kF_k,\qquad Df=B_kDF_k.\]

Every singular value of \(DF_k\) lies in \([t^{-1},t]\). In the order of symmetric matrices,

\[t^{-2}B_kB_k^*\leq B_kDF_kDF_k^*B_k^*\leq t^2B_kB_k^*.\]

Taking determinants on the \(m\)-dimensional target yields

\[t^{-m}J_mB_k\leq J_mf\leq t^mJ_mB_k.\]

All Jacobians here are Gram determinants; \(B_k\) is rectangular and has no ordinary determinant when \(n>m\).

Write \(Z_k=F_k(G_k)\) and

\[I_k=\int\mathcal H^{n-m}(G_k\cap f^{-1}(y))\,dy.\]

On every fibre, \(F_k\) and its inverse have Lipschitz constant at most \(t\), so

\[t^{-(n-m)}I_k\leq\int\mathcal H^{n-m}(Z_k\cap B_k^{-1}(y))\,dy\leq t^{n-m}I_k.\]

The middle term is \(J_mB_k\lambda_n(Z_k)\) by linear coarea. Also \(t^{-n}\lambda_n(G_k)\leq\lambda_n(Z_k)\leq t^n\lambda_n(G_k)\). Combining these two comparisons with the pointwise Jacobian bounds gives

\[t^{-2n}I_k\leq\int_{G_k}J_mf\,dx\leq t^{2n}I_k.\]

Tonelli sums the fibre measures over the disjoint domain pieces. Letting \(t\downarrow1\) proves the full-rank case in every pair of dimensions \(n\geq m\).

Rank deficient. Now suppose \(J_mf=0\) on \(A\). Set \(E=A\times B_1^m(0)\) and

\[g_\varepsilon(x,w)=f(x)+\varepsilon w,\qquad p(x,w)=w.\]

The derivative of \(g_\varepsilon\) has full target rank and

\[J_mg_\varepsilon=\sqrt{\det(DfDf^*+\varepsilon^2I_m)}\leq C_L\varepsilon\quad\text{on }E,\]

by the same singular-value calculation as in the area proof. Applying the full-rank case in domain dimension \(n+m\) gives

\[\int_{\mathbb R^m}\mathcal H^n(E\cap g_\varepsilon^{-1}(y))\,dy=\int_EJ_mg_\varepsilon\leq C_L\varepsilon\omega_m\lambda_n(A).\]

For almost every \(y\) these fibres have finite \(\mathcal H^n\) measure. Apply the covering inequality of Lemma 16 to the 1-Lipschitz map \(p\) on each such fibre. Its fibre over \(w\in B_1^m(0)\) is isometric to \(A\cap f^{-1}(y-\varepsilon w)\). Tonelli and translation invariance in \(y\) therefore yield

\[\begin{aligned} \omega_m\int\mathcal H^{n-m}(A\cap f^{-1}(y))\,dy &=\int\!\int_{B_1^m}\mathcal H^{n-m}(A\cap f^{-1}(y-\varepsilon w))\,dw\,dy\\ &\leq\frac{\omega_{n-m}\omega_m}{\omega_n}\int\mathcal H^n(E\cap g_\varepsilon^{-1}(y))\,dy\\ &\leq C\varepsilon\lambda_n(A). \end{aligned}\]

All constants are independent of \(\varepsilon\in(0,1]\). Letting \(\varepsilon\downarrow0\) proves zero contribution from the rank-deficient set. Split a general \(A\) and then exhaust by finite-measure pieces. ◻

The argument uses the measurable-fibre covering inequality, not the coarea formula itself, on the augmented fibres, so the rank-deficient step is not circular.

As an application or restatement: the coarea formula is a way to write an integral on a Riemannian manifold in terms of integrals over level sets of a function. By applications of symmetrization arguments, the coarea formula is widely used for proving functional inequalities, such as Sobolev inequalities. For a scalar function, the formula can be read by separating the area of each level set from the distance between nearby levels.

Theorem 6. Let \(\Sigma\) be a smooth \(n\)-dimensional Riemannian manifold and \(h:\Sigma\to\mathbb R\) Lipschitz. Suppose the sublevel set \(\{h\leq t\}\) is compact and \(f\in L^1_{\mathrm{loc}}(\Sigma)\). Then

\[\int_{\{h\leq t\}}f|\nabla_\Sigma h|\,d\operatorname{Vol}_\Sigma=\int_{-\infty}^t\left(\int_{\{h=\tau\}}f\,d\mathcal H^{n-1}_\Sigma\right)d\tau.\]

One may instead assume integrability of \(|f||\nabla_\Sigma h|\) on the sublevel set. Compactness of every sublevel is stronger than the usual definition of properness, which only requires inverse images of compact sets to be compact.

Derivation. The Euclidean formula applies in coordinate patches, with the Riemannian volume and hypersurface Gram factors. These factors transform \(J_1h\) into the metric norm \(|\nabla_\Sigma h|\). A countable partition of unity sums the patch identities. First use nonnegative simple functions and monotone convergence; for signed \(f\), apply this to its positive and negative parts. Compactness and the Lipschitz bound make the absolute integral finite under the stated local-integrability assumption. The embedded-manifold version is given in Simon, Chapter 2, §6. ◻

Theorem 7. Let \(O\subseteq\mathbb R^n\) be open, let \(h:O\to\mathbb R^k\) be Lipschitz with \(1\leq k\leq n\), and let \(A\subseteq O\) be Lebesgue measurable. For nonnegative measurable \(f\), or signed \(f\) with \(\int_A|f|J_kh<\infty\),

\[\int_Af(x)J_kh(x)\,dx=\int_{\mathbb R^k}\left(\int_{A\cap h^{-1}(\tau)}f(x)\,d\mathcal H^{n-k}(x)\right)d\tau.\]

In particular \(f\in L^1(A)\) suffices for a globally Lipschitz \(h\).

Proof. Extend the Lipschitz map from \(O\) componentwise, or apply the preceding theorem on a countable open cover with disjoint measurable pieces. Its derivative agrees with the original derivative on \(O\). Apply the set formula to indicators and then monotone convergence; separate positive and negative parts in the signed case. ◻

For a smooth scalar \(h\) at a regular point, a level interval of width \(d\tau\) corresponds to normal thickness \(d\tau/|\nabla h|\). This explains the Jacobian in the formula. For merely Lipschitz \(h\), individual levels need not be smooth hypersurfaces; the measure-theoretic identity remains valid.

The corresponding volume identity is precisely

\[\operatorname{Vol}(A\cap\{|\nabla h|>0\})=\int_{\mathbb R}\int_{A\cap h^{-1}(\tau)\cap\{|\nabla h|>0\}}\frac{1}{|\nabla h|}\,d\mathcal H^{n-1}\,d\tau.\]

It follows by applying weighted coarea to \(f=\mathbb1_A\mathbb1_{\{|\nabla h|>0\}}/|\nabla h|\). The critical set must be added separately when computing all of \(\operatorname{Vol}(A)\); for example, a constant \(h\) has zero gradient everywhere. The illustration depicts the regular-point thickness calculation.

A region A in n-dimensional space is sliced by a level set f^{-1}(y); the map f sends A onto the segment f(A) in m-dimensional space, and the slice lies over the point y. — Coarea formula.

Reading: Carmo (1992), Riemannian geometry.