The Helmholtz Decomposition
This appendix proves Theorem 11.137 of Real Analysis: every smooth compactly supported vector field on \(\R^{3}\) splits as in Equation (11.168) into a gradient and a curl, with the potentials given explicitly by Equation (11.169), and the two summands are unique among fields that vanish at infinity. It is the theorem that licenses the routine physical move of treating the irrotational and the solenoidal parts of a field separately — the longitudinal and transverse parts of the electromagnetic field, the P and S waves of a seismogram, the potential and vortical parts of a flow.
The paragraph following the proof link in Section 11.11.5 already gives the mechanism, and it is correct: each Cartesian component of \(\vect{w}\) is a Newtonian potential, so \(\nabla^{2}\vect{w}=\vect{u}\), and Equation (11.144) read backwards turns that Laplacian into the required sum. What the chapter does not supply, and what this section owes, is the analysis: that the singular integral Equation (11.169) converges at all, that it may be differentiated under the integral sign as often as one pleases, that the resulting field decays, and that the decay is enough to force uniqueness. That is the whole content of what follows.
One organisational point should be made at the outset. The argument uses two results of Partial Differential Equations — the fundamental-solution property of the Newtonian potential (Proposition 14.67) and the weak maximum principle (Theorem 14.75) — and Partial Differential Equations comes after Real Analysis. Nothing is circular: neither of those two results uses Theorem 11.137, or anything proved here. But the forward dependence is real and is why the derivation sits in the appendix rather than in the chapter, where it would have to be either postponed or duplicated. See Remark A45.9.
Throughout, \(\vect{u}\) is a smooth vector field on \(\R^{3}\) whose support is contained in the ball \(\abs{\vect{y}}\le R_{0}\), and we write
for the Newtonian kernel Equation (14.63). The section is pure vector analysis, but the SI bookkeeping is worth recording once because the physical parts read the decomposition off it: \(\Phi\) has the dimension of a reciprocal length and \(\dd^{3}z\) that of a volume, so \(\vect{w}\) carries the unit of \(\vect{u}\) multiplied by \(\mathrm{m}^{2}\); \(\phi\) and \(\vect{\Psi}\), being first derivatives of \(\vect{w}\), carry that of \(\vect{u}\) multiplied by \(\mathrm{m}\); and \(\nabla\phi\) and \(\nabla\times\vect{\Psi}\) carry the unit of \(\vect{u}\) itself, as Equation (11.168) requires.
What the derivation uses
From Real Analysis: the Leibniz integral rule (Theorem 11.109), both halves of the fundamental theorem of calculus (Theorems 11.42 and 11.43), uniform continuity (Theorem 11.25), the Cauchy criterion (Theorem 11.8), the second-order identities of the nabla calculus (Proposition 11.123), the class \(C^{1}\) (Definition 11.98) and the theorem that a \(C^{1}\) map is differentiable (Theorem 11.100), and the spherical volume element of Example 11.130 — which is itself an instance of Change of Variables in a Multiple Integral. From Partial Differential Equations: Proposition 14.67 and Theorem 14.75, both proved there.
The kernel is integrable, and uniformly so
For \(0\le\delta<\varepsilon\),
so that for every bounded \(g\) on \(\R^{3}\),
Rests on Example 11.130, Theorem 11.129 and Definition 11.125.
Derives Lemma A45.1. The shell \(\delta\le\abs{\vect{z}}\le\varepsilon\) is a compact region decomposing into finitely many simple regions (Definition 11.127), and \(\abs{\vect{z}}^{-1}\) is continuous on it, hence integrable. Passing to spherical coordinates, whose Jacobian determinant is \(r^{2}\sin\theta\) (Example 11.130, licensed by Theorem 11.129),
which is Equation (A45.2). Bounding the integrand of Equation (A45.3) by \(\left(4\pi\abs{\vect{z}}\right)^{-1}\sup\abs{g}\) and dropping \(\delta\) gives the estimate, and the bound does not involve \(\vect{x}\).
∎Let \(g\) be continuous on \(\R^{3}\) with support in \(\abs{\vect{y}}\le R_{0}\). For \(\varepsilon>0\) put
a proper integral over a compact region for each \(\vect{x}\). As \(\varepsilon\to0\) these converge uniformly on \(\R^{3}\) to a continuous limit \(\Phi*g\), and
the right-hand side being understood in the same excised sense. Rests on Lemma A45.1, Theorem 11.8 and Theorem 11.129.
Derives Proposition A45.2. Fix \(\vect{x}\) in a bounded set. The integrand of Equation (A45.5) vanishes unless \(\vect{x}-\vect{z}\) lies in the support of \(g\), i.e. unless \(\vect{z}\) lies in a fixed bounded set; so the integration is over a compact shell, on which the integrand is continuous, and the integral exists. For \(0<\varepsilon'<\varepsilon\) the difference is the integral over the shell \(\varepsilon'\le\abs{\vect{z}}\le\varepsilon\), bounded by \(\tfrac12\left(\sup\abs{g}\right)\varepsilon^{2}\) uniformly in \(\vect{x}\) by Equation (A45.3). The family is therefore uniformly Cauchy, so it converges pointwise (Theorem 11.8) and the convergence is uniform.
Each truncation is continuous in \(\vect{x}\): on a compact shell and for \(\vect{x}\) in a compact ball the integrand is uniformly continuous (Theorem 11.25), so the integral is continuous in the parameter — this is the last clause of Theorem 11.109, whose proof needs only that bound. A uniform limit of continuous functions is continuous: given \(\varepsilon>0\) choose \(\varepsilon_{0}\) with \(\abs{\left(\Phi*g\right)-\left(\Phi*g\right)_{\varepsilon_{0}}} <\varepsilon/3\) everywhere and then use the continuity of the truncation at the point in question, the three errors summing to \(\varepsilon\); this is the argument already made for series in Lemma 13.6. Finally Equation (A45.6) is the substitution \(\vect{y}=\vect{x}-\vect{z}\), a translation composed with a reflection, whose Jacobian determinant has modulus one (Theorem 11.129).
∎Differentiating under the integral sign
The singular factor in Equation (A45.5) does not depend on \(\vect{x}\). That is the whole trick: every derivative in \(\vect{x}\) falls on \(g\), where it is harmless, and the singularity is never differentiated.
Let \(O\subseteq\R^{n}\) be open and let \(f_{j}:O\to\R\) be of class \(C^{1}\), converging pointwise on \(O\) to \(f\), with each \(\pp_{i}f_{j}\) converging uniformly on \(O\) to a function \(h_{i}\). Then \(f\) is of class \(C^{1}\) on \(O\) and \(\pp_{i}f=h_{i}\). Rests on Theorem 11.42, Theorem 11.43 and Definition 11.98.
Derives Lemma A45.3. Each \(h_{i}\) is continuous, being a uniform limit of continuous functions (the \(\varepsilon/3\) argument recalled in the proof of Proposition A45.2). Fix \(\vect{x}\in O\) and \(\sigma>0\) with the segment \(\vect{x}+s\vect{e}_{i}\), \(\abs{s}\le\sigma\), inside \(O\). By the fundamental theorem of calculus (Theorem 11.43) applied to the one-variable \(C^{1}\) function \(s\mapsto f_{j}(\vect{x}+s\vect{e}_{i})\),
The left-hand side tends to \(f(\vect{x}+t\vect{e}_{i})-f(\vect{x})\); on the right the error is at most \(\abs{t}\sup_{O}\abs{\pp_{i}f_{j}-h_{i}}\), which tends to zero. Hence \(f(\vect{x}+t\vect{e}_{i})-f(\vect{x}) =\int_{0}^{t}h_{i}(\vect{x}+s\vect{e}_{i})\,\dd s\), and since \(h_{i}\) is continuous, differentiating at \(t=0\) (Theorem 11.42) gives \(\pp_{i}f(\vect{x})=h_{i}(\vect{x})\). All partial derivatives of \(f\) exist and are continuous, so \(f\) is of class \(C^{1}\) (Definition 11.98) and differentiable (Theorem 11.100).
∎Let \(g\) be of class \(C^{k}\), \(k\ge1\), with compact support. Then \(\Phi*g\) is of class \(C^{k}\) and, for every multi-index of order at most \(k\),
In particular \(\Phi*g\) is smooth when \(g\) is. Rests on Lemma A45.3, Theorem 11.109 and Lemma A45.1.
Derives Proposition A45.4. It suffices to prove the case \(m=1\) and to iterate, since \(\pp_{i}g\) is again \(C^{k-1}\) with compact support.
Fix a ball \(\abs{\vect{x}}<\rho\). For \(\vect{x}\) in it, the integrand of Equation (A45.5) vanishes unless \(\vect{z}\) lies in the compact shell \(D_{\varepsilon}=\set{\varepsilon\le\abs{\vect{z}}\le \rho+R_{0}}\), which decomposes into finitely many simple regions. On \(D_{\varepsilon}\times\left(-\rho,\rho\right)\) the function \((\vect{z},x^{i})\mapsto\Phi(\vect{z})g(\vect{x}-\vect{z})\) is continuous with continuous derivative \(-\Phi(\vect{z})\left(\pp_{i}g\right)(\vect{x}-\vect{z})\) in the parameter \(x^{i}\) — the kernel is bounded away from its singularity on \(D_{\varepsilon}\) — so the Leibniz integral rule (Theorem 11.109) applies in each variable \(x^{i}\) separately and gives
the sign of the chain rule being absorbed because the derivative is taken with respect to \(\vect{x}\) and not \(\vect{z}\): writing \(g(\vect{x}-\vect{z})\) and differentiating in \(x^{i}\) produces \(\left(\pp_{i}g\right)(\vect{x}-\vect{z})\) with no sign at all.
By Proposition A45.2 applied to \(g\) and to \(\pp_{i}g\), both sides of Equation (A45.9) converge uniformly as \(\varepsilon\to0\), the first to \(\Phi*g\) and the second to \(\Phi*\pp_{i}g\); each truncation is \(C^{1}\) on the ball by the same Leibniz rule. Lemma A45.3 therefore gives \(\pp_{i}\left(\Phi*g\right)=\Phi*\pp_{i}g\) there, and \(\rho\) was arbitrary.
∎Poisson's equation
Let \(\vect{u}\) be smooth with compact support and let \(\vect{w}=\Phi*\vect{u}\), meaning the convolution taken componentwise. Then \(\vect{w}\) is smooth and
Rests on Proposition A45.4, Proposition 14.67 and Definition 21.38.
Derives Proposition A45.5. Smoothness and \(\nabla^{2}w_{i}=\Phi*\left(\nabla^{2}u_{i}\right)\) are Proposition A45.4. Fix \(\vect{x}\) and put \(\varphi(\vect{z})=u_{i}(\vect{x}-\vect{z})\), a smooth function of compact support and hence a test function in the sense of Definition 21.38. Its Laplacian in \(\vect{z}\) is \(\left(\nabla^{2}u_{i}\right)(\vect{x}-\vect{z})\), the two sign changes of the chain rule cancelling in a second derivative. By Proposition 14.67, which says precisely that \(\int_{\R^{3}}\Phi\,\nabla^{2}\varphi\,\dd^{3}z=\varphi(\vect{0})\) for every test function,
which is Equation (A45.10) componentwise. Note that the integral on the left is the excised one of Proposition A45.2 and the one in Proposition 14.67 is the same limit, the kernel being absolutely integrable near the origin by Equation (A45.2).
∎Existence of the decomposition
Let \(\vect{u}\) be smooth on \(\R^{3}\) with compact support and let \(\vect{w}=\Phi*\vect{u}\). Then \(\phi=\nabla\cdot\vect{w}\) and \(\vect{\Psi}=-\nabla\times\vect{w}\) are smooth, satisfy Equation (11.169), and
Rests on Proposition A45.5, Proposition 11.123 and Equation (11.144).
Derives Theorem A45.6. \(\vect{w}\) is smooth by Proposition A45.4, so \(\phi\) and \(\vect{\Psi}\) are, and Equation (A45.6) identifies \(\vect{w}\) with the field displayed in Equation (11.169). The identity Equation (11.144) of Proposition 11.123, applied to the \(C^{2}\) field \(\vect{w}\) and rearranged, reads
and the left-hand side is \(\vect{u}\) by Equation (A45.10). Finally \(\nabla\cdot\vect{\Psi} =-\nabla\cdot\left(\nabla\times\vect{w}\right)=0\) by Equation (11.143), and \(\nabla\times\nabla\phi=\vect{0}\) by Equation (11.142), so the first summand is irrotational and the second solenoidal, as Theorem 11.137 asserts.
∎Decay at infinity
The uniqueness clause of Theorem 11.137 speaks of fields that tend to zero at infinity. The potentials just constructed do, and with a definite rate.
With \(\vect{u}\) supported in \(\abs{\vect{y}}\le R_{0}\) and \(\vect{w}=\Phi*\vect{u}\) as above, there is a constant \(C\), depending only on \(R_{0}\) and \(\sup\abs{\vect{u}}\), such that for \(\abs{\vect{x}}\ge2R_{0}\)
In particular \(\nabla\phi\) and \(\nabla\times\vect{\Psi}\) tend to zero at infinity, and so do \(\phi\) and \(\vect{\Psi}\) themselves. Rests on Proposition A45.2, Theorem 11.109 and Proposition A45.4.
Derives Proposition A45.7. Use the form Equation (A45.6), in which the integration runs over the fixed compact ball \(\abs{\vect{y}}\le R_{0}\) and the singular factor now depends on \(\vect{x}\). For \(\abs{\vect{x}}\ge2R_{0}\) and \(\vect{y}\) in that ball,
so the integrand is bounded and no singularity is present: the integral is proper, and the Leibniz rule (Theorem 11.109) may be applied directly in each \(x^{i}\), differentiating the kernel. Doing so,
and similarly the second derivatives of the kernel are bounded by \(3\abs{\vect{x}-\vect{y}}^{-3}\). Inserting Equation (A45.15) and bounding the integral by the volume \(\tfrac43\pi R_{0}^{3}\) of the ball containing the support times the supremum of the integrand gives all three estimates of Equation (A45.14) with the single constant \(C=8R_{0}^{3}\sup\abs{\vect{u}}\): the three bounds so obtained are \(\tfrac23R_{0}^{3}\sup\abs{\vect{u}}\), \(\tfrac43R_{0}^{3}\sup\abs{\vect{u}}\) and \(8R_{0}^{3}\sup\abs{\vect{u}}\) respectively, the factor \(\left(4\pi\right)^{-1}\) cancelling against \(\tfrac43\pi\) and the remaining powers of \(2\) coming from Equation (A45.15). The bound is not sharp and is not meant to be. Since \(\nabla\phi\) and \(\nabla\times\vect{\Psi}\) are built from second derivatives of \(\vect{w}\), they are \(O\left(\abs{\vect{x}}^{-3}\right)\); \(\phi\) and \(\vect{\Psi}\) are built from first derivatives and are \(O\left(\abs{\vect{x}}^{-2}\right)\).
∎Uniqueness
Let \(\phi_{1},\phi_{2}\) be \(C^{3}\) scalar fields and \(\vect{\Psi}_{1},\vect{\Psi}_{2}\) be \(C^{3}\) vector fields on \(\R^{3}\) with
and suppose each of the four fields \(\nabla\phi_{1},\nabla\times\vect{\Psi}_{1}, \nabla\phi_{2},\nabla\times\vect{\Psi}_{2}\) tends to zero as \(\abs{\vect{x}}\to\infty\). Then \(\nabla\phi_{1}=\nabla\phi_{2}\) and \(\nabla\times\vect{\Psi}_{1}=\nabla\times\vect{\Psi}_{2}\). Rests on Proposition 11.123, Theorem 14.75 and Equation (11.144).
Derives Theorem A45.8. Put \(\vect{h}=\nabla\phi_{1}-\nabla\phi_{2} =\nabla\times\vect{\Psi}_{2}-\nabla\times\vect{\Psi}_{1}\), a \(C^{2}\) field tending to zero at infinity. Read from the left it is a gradient, so \(\nabla\times\vect{h}=\vect{0}\) by Equation (11.142); read from the right it is a curl, so \(\nabla\cdot\vect{h}=0\) by Equation (11.143). Hence Equation (11.144) gives
so each Cartesian component \(h_{i}\) is harmonic on the whole of \(\R^{3}\) (Definition 14.66).
Fix \(\vect{x}_{0}\) and let \(R>\abs{\vect{x}_{0}}\). On the open ball \(B_{R}\) the function \(h_{i}\) is harmonic and continuous on the closure, so by the weak maximum principle (Theorem 14.75) it attains both its extremes on the sphere \(\abs{\vect{x}}=R\):
Both bounds tend to zero as \(R\to\infty\), because \(\vect{h}\to\vect{0}\) at infinity: given \(\varepsilon>0\) there is \(R_{1}\) with \(\abs{\vect{h}}<\varepsilon\) outside \(B_{R_{1}}\), so for \(R>R_{1}\) the two extremes lie in \(\left(-\varepsilon,\varepsilon\right)\). Hence \(\abs{h_{i}(\vect{x}_{0})}<\varepsilon\) for every \(\varepsilon>0\), and \(\vect{h}=\vect{0}\) identically.
∎Proof of Theorem 11.137. Derives Theorem 11.137. Existence, with the potentials Equation (11.169), is Theorem A45.6; the two summands are respectively curl-free and divergence-free by the same theorem, which is what Proposition 11.123 contributes; and the uniqueness clause is Theorem A45.8. That the potentials constructed here do satisfy the decay hypothesis of the uniqueness clause is Proposition A45.7.
∎Reading the result
Nothing from outside this treatise. Two results are taken from a later chapter, and that is worth stating plainly rather than letting a reader discover it: Proposition 14.67, which identifies Equation (A45.1) as the fundamental solution of the Laplacian, and Theorem 14.75, the weak maximum principle. Both are proved in Partial Differential Equations, both are proved there without any appeal to Theorem 11.137 or to anything in this section, and neither proof uses the Helmholtz decomposition anywhere in the book. So the forward reference is a matter of exposition, not of logic. It is also the reason the derivation belongs in an appendix: placed in Real Analysis it would have to import a chapter that has not yet been written, and placed in Partial Differential Equations it would be far from the vector calculus it completes.
The remaining ingredients are all from Real Analysis and are listed in What the derivation uses. Three things are deliberately not used: no Lebesgue convergence theorem — the passage to the limit \(\varepsilon\to0\) is uniform, by Lemma A45.1, and needs only the Cauchy criterion; no Fourier transform, although the decomposition has a one-line proof in Fourier variables that hides every question of convergence; and no theory of distributions beyond the single statement of Proposition 14.67, which is quoted as an identity between numbers, tested on one explicit test function.
Compact support is a convenience, not a necessity. The same argument runs whenever \(\vect{u}\) decays faster than \(\abs{\vect{x}}^{-2}\) with first derivatives decaying faster than \(\abs{\vect{x}}^{-3}\): the excision estimate Equation (A45.3) is untouched, and the only new work is to check that the integral over the far region converges, which those rates supply.
Some decay is indispensable, and the failure is not subtle. The constant field \(\vect{u}=\hat{\vect{z}}\) is a gradient, \(\vect{u}=\nabla z\); it is also a curl, \(\vect{u}=\nabla\times\left(x\,\hat{\vect{y}}\right)\), as one checks from Equation (11.126) by evaluating the three components of the curl of \(\left(0,x,0\right)\). So the split Equation (11.168) exists for it in two ways — everything in the gradient, or everything in the curl — and nothing in the local differential identities can choose between them. The decay condition in the uniqueness clause of Theorem 11.137 is what excludes such a field, and Theorem A45.8 shows it is exactly enough.
The regularity stated in Theorem A45.8 is a small strengthening of the chapter's phrasing, which leaves it implicit: the argument differentiates the potentials three times, so \(C^{3}\) potentials — equivalently \(C^{2}\) summands — are what Equation (A45.18) needs. For the potentials constructed here the point is moot, since they are smooth. Rests on Theorem A45.8, Theorem 11.137 and Equation (11.126).
The Helmholtz Decomposition discharges the derivation owed at Theorem 11.137 of Real Analysis. It also settles a debt recorded in Part III: Remark 34.46 of Continuum Mechanics and Elasticity states that the textbook route to the separation of P and S waves needs exactly this theorem — existence and uniqueness of the split, with the decay conditions that make it unique — and that Part II did not carry it, which is why the chapter re-routed the proof of Phenomenon 34.43 onto the acoustic tensor and plane waves instead. That re-routing stands as an independent derivation and is not superseded; what changes is that the alternative route is now available. The second site named there, Proposition 34.48, could not be re-routed: its displacement potentials are introduced as an ansatz sufficient for an existence claim, and the chapter says so. With Theorem A45.6 in hand that ansatz can be upgraded to a decomposition, and the Rayleigh wave shown to be the general plane-strain surface motion rather than one exhibited example.