Vector Calculus and the Topology of Domains in 3-Space, II Jason Cantarella, Dennis Deturck and Herman Gluck

Vector Calculus and the Topology of Domains in 3-Space, II Jason Cantarella, Dennis DeTurck and Herman Gluck

1. Introduction Suppose you have a vector field defined only on the boundary of a compact domain in 3-space. How can you tell whether your vector field can be extended throughout the domain so that it is divergence-free? Or curl-free? Or both? Or have specified divergence? Or specified curl? Or both? To answer these questions, you need to understand the relationship between the calculus of vector fields and the topology of their domains of definition. The Hodge Decomposition Theorem, which was discussed and proved in the first paper of this two-part series [12], provides the key by decomposing the space of vector fields on the domain into five mutually orthogonal subspaces which are topologically and an- alytically meaningful. Our goal in this paper is to show how this theorem can be used to derive the basic boundary-value theorems for vector fields, which in turn will answer the questions above. As we shall see, each of these questions can be restated as a problem in partial differential equations, and it will be essential for us to identify appropriate boundary conditions so that a solution will exist and be unique, or so that the set of solutions is parametrized in a meaningful way. In particular, we will consider the problem of finding an unknown vector field V on Ω whose curl and/or divergence are specified in advance. If just one of these is given, then the value of V on the boundary of Ω is also specified. If both the curl and divergence are given, then either the parallel or normal component of V along ∂Ω is specified. Appropriate integrability conditions must be imposed in each case. The resulting theorems provide us with a very natural follow-up to, and application of, the Hodge Decomposition Theorem.

1 2 cantarella, deturck, and gluck 2. The Hodge Decomposition Theorem To begin, we state the Hodge Decomposition Theorem, explain the conditions in the theorem, and give a brief outline of its proof. For further details, including examples of vector fields that belong to each of the five summands and a discussion of absolute and relative homology of domains in 3-space, see [12]. Let Ω be a compact domain in 3-space with smooth boundary ∂Ω. Let VF(Ω) be the infinite-dimensional vector space of all smooth vector fields in Ω, on which we 2 · use the L inner product V,W = Ω V W d(vol). In this paper, “smooth” always means “all partial derivatives of all orders exist and are continuous”. HODGE DECOMPOSITION THEOREM. The space VF(Ω) is the direct sum of five mutually orthogonal subspaces: VF(Ω) = FK ⊕ HK ⊕ CG ⊕ HG ⊕ GG with ker curl =HK⊕ CG ⊕ HG ⊕ GG im grad =CG⊕ HG ⊕ GG im curl =FK⊕ HK ⊕ CG ker div =FK⊕ HK ⊕ CG ⊕ HG, where FK=fluxless knots = {∇ · V =0,V· n =0, all interior fluxes = 0}, HK=harmonic knots = {∇ · V =0, ∇×V = 0,V· n =0}, CG=curly gradients = {V = ∇ϕ, ∇·V =0, all boundary fluxes = 0}, HG=harmonic gradients = {V = ∇ϕ, ∇·V =0,ϕlocally constant on ∂Ω}, GG=grounded gradients = {V = ∇ϕ, ϕ|∂Ω =0}, and furthermore,

∼ ∼ ∼ genus of ∂Ω HK = H1(Ω; R) = H2(Ω,∂Ω; R) = R ∼ ∼ ∼ (# components of ∂Ω) − (# components of Ω) HG = H2(Ω; R) = H1(Ω,∂Ω; R) = R .

Consider the subspace of VF(Ω) consisting of all divergence-free vector fields on Ω that are tangent to ∂Ω. These vector fields are used to represent incompressible fluid flows within fixed boundaries, and magnetic fields inside plasma containment devices; see [4], [6], [7], [23], [31],and[33]. When a problem in geometric knot theory is transformed to one in fluid dynamics, the knot is thickened to a tubular neighborhood of itself, and then represented by an incompressible fluid flow (sometimes called a fluid knot) in this neighborhood, much like blood pumping miraculously through an arterial loop; see [3], [26, 27, 29] and our paper [9]. With this last application in mind, we define

K = knots = {V ∈ VF(Ω) : ∇·V =0,V· n =0}, where n is the unit outward normal to the boundary of Ω, and where the condition ∇·V = 0 holds throughout Ω, while V · n = 0 holds only on its boundary. In general, vector calculus and the topology of domains in 3-space, ii 3 when we state conditions for our vector ﬁelds using n, they are understood to apply only on the boundary of Ω. At the same time, we deﬁne

G = gradients = {V ∈ VF(Ω) : V = ∇ϕ} for some smooth real-valued function ϕ deﬁned on Ω. Proposition 1. The space VF(Ω) is the direct sum of two orthogonal subspaces:

VF(Ω) = K ⊕ G .

The Hodge Decomposition Theorem provides a further decomposition of K into an orthogonal direct sum of two subspaces, and of G into an orthogonal direct sum of three subspaces. We start by decomposing the subspace K. Let Σ stand generically for any smooth orientable surface in Ω whose boundary ∂Σ lies in the boundary ∂Ω of the domain Ω. We call Σ a cross-sectional surface and write (Σ,∂Σ) ⊂ (Ω,∂Ω). We orient Σ by choosing one of its two unit normal vector fields n. Then, for any vector field V on Ω, we define the flux of V through Σtobe the value of the integral Φ= V · n d(area). Σ Assume that V is divergence-free and tangent to ∂Ω. Then the value of this flux depends only on the homology class of Σ in the relative homology group H2(Ω,∂Ω; R). For example, if Ω is an n-holed solid torus, this homology group is generated by disjoint oriented cross-sectional disks Σ1,Σ2,..., Σn, positioned so that cutting Ω along these disks will produce a simply-connected region, as in Figure 1. The fluxes fig 2 Φ1,Φ2,...,Φn of V through these disks determine the flux of V through any other from H1 cross-sectional surface. If the flux of V through every cross-sectional surface vanishes, then we say all interior fluxes = 0, and define

FK = ﬂuxless knots = {V ∈ VF(Ω) : ∇·V =0,V · n =0, all interior ﬂuxes = 0}.

Next we deﬁne

HK = harmonic knots = {V ∈ VF(Ω) : ∇·V =0, ∇×V = 0,V· n =0}.

We showed in [12] that HK is isomorphic to the homology group H1(Ω, R)and also, via Poincar´e duality, to the relative homology group H2(Ω,∂Ω; R). This is a ﬁnite-dimensional vector space, with dimension equal to the (total) genus of ∂Ω, which is obtained by adding the genera of the boundary components of the domain Ω. 4 cantarella, deturck, and gluck Proposition 2.ThesubspaceK is the direct sum of two orthogonal subspaces: K =FK⊕ HK .

Now we decompose the subspace G of gradient vector ﬁelds. Deﬁne DFG = divergence-free gradients = {V ∈ VF(Ω) : V = ∇ϕ, ∇·V =0}, which is the subspace of gradients of harmonic functions, and

GG = grounded gradients = {V ∈ VF(Ω) : V = ∇ϕ, ϕ|∂Ω =0}, which is the subspace of gradients of functions that vanish on the boundary of Ω. Proposition 3.ThesubspaceG is the direct sum of two orthogonal subspaces: G =DFG⊕ GG .

Now we decompose the subspace DFG of divergence-free gradient vector fields. If V is a vector field defined on Ω, we will say that all boundary fluxes of V are zero, if the flux of V through each component of ∂Ω is zero. Note that in general this is a stronger condition than saying that the flux of V through the boundary of each component of Ω is zero. This latter condition is used in Proposition 6. We define CG = curly gradients = {V ∈ VF(Ω) : V = ∇ϕ, ∇·V =0, all boundary fluxes = 0}. We call CG the subspace of curly gradients because these are the only gradient vector fields that lie in the image of curl. Finally, we define the subspace HG = harmonic gradients = {V ∈ VF(Ω) : V = ∇ϕ, ∇·V =0,ϕlocally constant on ∂Ω}, meaning that ϕ is constant on each component of ∂Ω.

HG is isomorphic to the homology group H2(Ω; R) and also, via Poincar´e duality, to the relative homology group H1(Ω,∂Ω; R). This is a ﬁnite-dimensional vector space, with one generator for each component of ∂Ω, and one relation for each component of Ω. Hence, its dimension is equal to the number of components of ∂Ωminus the number of components of Ω. Proposition 4.ThesubspaceDFG is the direct sum of two orthogonal subspaces: DFG = CG ⊕ HG .

With Propositions 1 through 4 in hand, the Hodge decomposition of VF(Ω), VF(Ω) = FK ⊕ HK ⊕ CG ⊕ HG ⊕ GG, vector calculus and the topology of domains in 3-space, ii 5 follows immediately. The related expressions for the kernels of curl and div,andthe images of grad and curl are also fairly direct consequences, as shown in [12]. We pause to record some observations from [12] that contrast harmonic knots with harmonic gradients. Harmonic knots are smooth vector fields in Ω which are divergence-free, curl-free, and tangent to the boundary of Ω. A harmonic knot is completely determined by the values of its circulation around a set of curves in Ω that generate H1(Ω; R), or alternatively, by its fluxes through a set of surfaces that generate H2(Ω,∂Ω; R). Harmonic gradients, on the other hand, are smooth vector fields in Ω which are divergence-free, curl-free, and orthogonal to the boundary of Ω. A harmonic gradient is completely determined by its fluxes through the components of the boundary of Ω (which are independent except for the condition that the total flux through the boundary of each component of Ω must be zero, since a harmonic knot is divergence- free). We will use these observations several times in what follows.

3. Four basic questions about vector fields. To prepare ourselves for the proofs that come later in this paper, we review the answers to four basic questions about vector fields that were given in [12]. Question 1. Given a vector field defined on a compact domain in 3-space, how do you know whether it is the gradient of a function? Let V be our vector field, defined on the compact domain Ω with smooth boundary; we want to know if there is a smooth real-valued function ϕ on Ω with V = ∇ϕ. If V is a gradient vector field, then its circulation around every loop is zero. Conversely, if the circulation of V around every loop is zero, then V = ∇ϕ with ϕ(x) defined by integrating V along any path to the point x from a chosen base point in the connected component of Ω that contains x. But this is not a practical test to apply. Answer to Question 1. The vector field V is a gradient if and only if ∇×V =0 and the integrals of V around the loops C1,C2,...,Ck are zero, where these loops form a basis for the one-dimensional homology H1(Ω). To prove that our answer is correct, first observe that the conditions are clearly necessary, so suppose that V satisfies them. By Stokes’ theorem, if V is curl-free, then its circulation around any loop C depends only on the homology class of C in H1(Ω). But any C is homologous to a linear combination of the basis loops C1,...,Ck, around each of which V has circulation zero, and so V must also have circulation zero around C. Then, as we pointed out above, V is certainly a gradient field. Although we didn’t use the Hodge Decomposition Theorem to reach this conclu- sion, we can use it to reflect on what we just said. The condition ∇×V = 0 tells us that V lies in the kernel of curl, namely HK ⊕ CG ⊕ HG ⊕ GG. The condition that V 6 cantarella, deturck, and gluck has zero circulation around the basis loops for H1(Ω) tells us that the HK-component of V is zero, so V lies in CG ⊕ HG ⊕ GG, which is just the subspace of gradient fields. Question 2. Given a vector field defined on a compact domain in 3-space, how do you know whether it is the curl of another vector field? Let V be our vector field defined on the compact domain Ω with smooth boundary; we want to know if there is a vector field U on Ω with ∇×U = V . If V is a curl, then its flux through every closed surface is zero, as an immediate consequence of Stokes’ theorem. Conversely, if the flux of V through every closed surface S in Ω is zero, then V is a curl. Again, this is not a practical test to apply. Answer to Question 2. The vector field V is the curl of another vector field if and only if ∇·V =0and the flux of V through each component of ∂Ω is zero. The confirmation that we have given the correct answer comes from the Hodge Decomposition Theorem. The condition ∇·V = 0 tells us that V lies in the kernel of div, namely FK ⊕ HK ⊕ CG ⊕ HG. The condition that the flux of V through each component of ∂Ω is zero, together with the observation at the end of section 2, tells us that the HG-component of V is zero, and hence that V lies in FK ⊕ HK ⊕ CG, which is just the image of curl. Question 3. Can you find a nonzero vector field on a given compact domain in 3-space which is divergence-free, curl-free and tangent to the boundary?

Answer to question 3. Such a vector field exists if and only if H1(Ω) =0. This is simply a part of the Hodge Decomposition Theorem. The vector fields we are looking for are the harmonic knots; they form the subspace HK of VF(Ω), and this is isomorphic to H1(Ω). To check that H1(Ω) is nonzero, we simply look at the boundary components of Ω and make sure that they are not all of genus zero. If this is true, we can refer to the proof of the Hodge Decomposition Theorem (see section 9 of [12]) to see how the harmonic knots are constructed from magnetic fields. Question 4. Can you find a nonzero vector field on a given compact domain in 3-space which is divergence-free, curl-free and orthogonal to the boundary?

Answer to question 4. Such a vector ﬁeld exists if and only if H2(Ω) =0. This is also just a part of the Hodge Decomposition Theorem. The vector ﬁelds we are looking for are the harmonic gradients; they form the subspace HG of VF(Ω), and this is isomorphic to H2(Ω). To check that H2(Ω) is nonzero, we simply look for a connected component of Ω that has more than one boundary component. If such a component is present, we can again refer to the proof of the Hodge Decomposition Theorem (see section 12 of [12]) to see how the harmonic gradients are constructed from solutions to the Dirichlet problem for the Laplace equation.

4. Divergence-free vector fields. We now turn to the fundamental theorems about vector fields alluded to in the Introduction. We consider existence problems for vector fields with preassigned diver- vector calculus and the topology of domains in 3-space, ii 7 gence or curl (or both). In each case, we identify the appropriate boundary conditions needed to guarantee both existence and uniqueness (at least up to a finite-dimensional space). Our approach in each case will be to start with “homogeneous” problems (search- ing for curl-free or divergence-free fields) in order to separate the issue of boundary conditions from the equations on the interior of the domain. In this section, we begin our study of vector fields with preassigned divergence by looking at the homogeneous case, namely, at the existence of fluid knots with preassigned boundary field. Proposition 5. Anysmoothvectorfielddefinedonandtangentto∂Ω can be extended to a smooth, divergence-free vector field on Ω. Proof.LetV be the unknown vector field, already defined on and tangent to ∂Ω. Then V defines a one-parameter family of diffeomorphisms (i.e.,aflow)on∂Ω. If the flow of V is area-decreasing (increasing) on some portion of ∂Ω, we must compensate by making the extension of V into the interior of Ω length-expanding (contracting) in the direction normal to the boundary, so that it can be volume-preserving on Ω, asshowninFigure2. from the To see this in concrete terms, suppose that we specify original V = v(y,z)  + w(y,z) k paper on the yz-plane. Then we can simply define ∂v ∂w V = − + xı + v(y,z)  + w(y,z) k ∂y ∂z to obtain a divergence-free extension over all of 3-space. If Ω is the unit ball in 3-space and the vector field V is defined on and tangent to its boundary, then a similar computation in spherical coordinates provides an extension of V to a divergence-free vector field defined on a neighborhood Ω of the boundary. Visualize Ω as the region between two concentric spheres. Then V is divergence- free and tangent to the outer boundary of Ω, which generates the two-dimensional homology of Ω. We would like to conclude that V is in the image of curl on Ω. By the Hodge Decomposition Theorem, the image of curl is FK ⊕ HK ⊕ CG. But we already know that V is in the kernel of div,whichisFK⊕ HK ⊕ CG ⊕ HG. Thus we need to show that V has no component in the direction of HG. But recall that the space of harmonic gradients is isomorphic to the homology group H2(Ω , R), which is generated by the outer boundary of Ω . Thus, we need to verify that the flux of V through the outer boundary of Ω vanishes. Since V is tangent to this outer boundary, we can conclude that V is in the image of curl on Ω, so write V = ∇×U on Ω. Now extend U from a slightly smaller neighborhood of the boundary of Ω to the entire ball Ω (by multiplying U by a scalar function that is identically equal to 1 on 8 cantarella, deturck, and gluck a small neighborhood of ∂Ω, but which decreases to be identically zero by the time it leaves Ω). To finish, we simply let V = ∇×U on all of Ω. The proof in general is modeled after this argument for the unit ball. We recall the formula for computing divergence in general orthogonal coordinates u, v, w on a region of 3-space. Since the coordinates are orthogonal, the vectorial element of arc length is given in these coordinates by the formula ds = a(u, v, w)du u + b(u, v, w)dv v + c(u, v, w)dw w , where u, v and w are unit vector fields in the coordinate directions. Let the vector field V be given by

V (u, v, w)=Vu(u, v, w)u + Vv(u, v, w)v + Vw(u, v, w)w . Then the divergence of V is 1 ∂ ∂ ∂ ∇·V = (bcVu)+ (acVv)+ (abVw) abc ∂u ∂v ∂w (see [19], p. 513). We will extend V from the boundary of Ω, where it is preassigned, to a divergence- free vector field defined on a neighborhood of the boundary, by solving a first-order ODE based on the above formula for divergence. First we construct and parametrize a neighborhood of ∂Ω inside Ω, as follows. At each point p ∈ ∂Ω, let n(p) denote the outward pointing unit normal vector. Define amaph:[0,ε] × ∂Ω → R3 by h(u, p)=p − un(p). Then, for sufficiently small ε,themaph will be a diffeomorphism of [0,ε] × ∂Ωonto a neighborhood of ∂ΩinΩ.Ifwefixu ∈ [0,ε] and define hu: ∂Ω → h(u × ∂Ω) by hu(p)=h(u, p), then hu is a diffeomorphism of ∂Ωontothesurfaceh(u × ∂Ω), which is “parallel” to ∂Ω and at a constant distance u from it. We use v and w as the local coordinates on ∂Ω. With respect to this coordinate system, the function a(u, v, w) in the vectorial element of arc length ds will be identically equal to 1, because n is a unit vector normal to h(u × ∂Ω) for all sufficiently small u. The divergence-free vector field V on a neighborhood of ∂Ω in Ω will be defined in two parts, V = V ⊥ + V , where V ⊥ is orthogonal to the surface h(u × ∂Ω) and V is parallel to it. We define V first: V (h(u, p)) = (hu)∗V (p).

In other words, the diffeomorphism hu transfers the vector field V from ∂Ωtothe vector field V on the surface h(u × ∂Ω) to give us the parallel component of V . vector calculus and the topology of domains in 3-space, ii 9 We get the normal component V ⊥ of V by solving a first-order ODE, as follows. Let u, v, w be local orthogonal coordinates in Ω on a neighborhood of the point p ∈ ∂Ω, with the u coordinate the same as defined above, and with the v and w coordinates along the surfaces hu(∂Ω). In these local coordinates, we write the desired vector field V as ⊥ V (u, v, w)=(Vuu)+(Vvv + Vww )=V + V . Since V has already been determined, we know the component functions Vv and Vw. We determine Vu by setting 1 ∂ ∂ ∂ ∇·V = (bcVu)+ (acVv)+ (abVw) =0, abc ∂u ∂v ∂w or equivalently, since a =1, ∂ ∂ ∂ (bcVu)=− (cVv)+ (bVw) . ∂u ∂v ∂w

This is a first-order ODE for the unknown function bcVu,inwhichwethinkofu as the independent variable and v and w as parameters. The value of this function is given at u = 0. Then there is a unique solution for 0 ≤ u ≤ ε(v, w), where both ε(v, w) and the solution depend smoothly on the parameters v and w.Since∂Ωis ∗ compact, the function bcVu is then defined on some neighborhood 0 ≤ u ≤ ε of ∂Ω. From this we compute the value of Vu, since the quantities b(u, v, w)andc(u, v, w) are known. This determines the orthogonal component ⊥ V = Vu(u, v, w)u, so that V = V ⊥ + V is divergence-free in some neighborhood of the point p ∈ ∂Ω. ∗ In this way we construct V over the neighborhood Ω = h([0,ε ] × ∂Ω) of ∂ΩinΩ. Starting with the smooth vector field V , defined on and tangent to ∂Ω, we have so far extended it to a smooth divergence-free vector field defined on the neighborhood Ω. Now we complete the argument just as we did for the unit ball. That is, V is divergence-free and tangent to the outer boundary, ∂Ω, of Ω.Since ∗ Ω is homeomorphic to [0,ε ] × ∂Ω, the components of this outer boundary generate the two-dimensional homology of Ω. And as before, a vector field that is divergence- free and has zero flux through each closed surface in a family that generates the 2-dimensional homology of the domain must be in the image of curl. So we write V = ∇×U on Ω.ThenweextendU from a slightly smaller neighborhood of ∂Ωto all of Ω, just as we did on the unit ball, and let V = ∇×U on Ω. This completes the proof of Proposition 5. Proposition 6. Asmoothvectorfielddefinedon∂Ω can be extended to a smooth, divergence-free vector field on Ω if and only if it has zero total flux through the boundary of each component of Ω. 10 cantarella, deturck, and gluck Proof. The zero-total-flux condition is clearly necessary, so we assume it is satisfied. Let ϕ be a smooth function on Ω which is a solution of the Laplace equation ∆ϕ = 0 with Neumann boundary condition ∂ϕ/∂n = V · n. As discussed in [12],the zero-total-flux condition through the boundary of each component of Ω is exactly the integrability condition needed for the existence of ϕ.

Let V1 = ∇ϕ,sothat∇·V1 =0andV1 · n = ∂ϕ/∂n = V · n.ThusV − V1 is tangent to ∂Ω.

Now let V2 be a divergence-free vector ﬁeld on Ω with V2 =(V − V1) , ∂Ω ∂Ω whose existence is guaranteed by Proposition 5.

Then V = V1 + V2 is divergence-free and extends the vector ﬁeld given on ∂Ω, completing the proof of Proposition 6.

5. Vector fields with preassigned divergence Proposition 7. Let f beasmoothfunctiondefinedonΩ,andV asmoothvector field defined on ∂Ω.ThenV can be extended to a smooth vector field V on Ω such that ∇·V = f if and only if f d(vol) = V · n d(area) Ωi ∂Ωi for each component Ωi of Ω. Proof. We copy the proof of Proposition 6, except that here the smooth function ϕ is a solution of the Poisson equation ∆ϕ = f with Neumann boundary data ∂ϕ/∂n = V · n, which exists thanks to the integrability condition given above (see [12] for details).

Then let V1 = ∇ϕ,sothat∇·V1 = f and V1 · n = ∂ϕ/∂n = V · n.ThusV − V1 is tangent to ∂Ω, and we ﬁnish the argument exactly as for Proposition 6.

6. Curl-free vector fields Here again is a simple way to begin. Proposition 8. Any smooth vector field defined on and orthogonal to ∂Ω can be extended to a smooth, curl-free vector field on Ω. Proof. We begin by noting that a curl-free vector field on Ω that is orthogonal to ∂Ω must be a gradient vector field, V = ∇ϕ for some function ϕ on Ω, and that ϕ must be constant on each component of ∂Ω. To see this, first recall from the answer to Question 1 in section 3 that a curl-free vector field will be a gradient vector field if and only if it has zero integral around a family of loops that generate H1(Ω). Since vector calculus and the topology of domains in 3-space, ii 11 the map H1(∂Ω) → H1(Ω) is onto (see section 6 of [12]), these loops may be chosen on the boundary of Ω, where the integral is certain to be zero because V is orthogonal to the boundary. The fact that V = ∇ϕ is orthogonal to the boundary also tells us that ϕ is constant on each component of ∂Ω. Hence, V must lie within HG ⊕ GG. In fact, we will construct ϕ to be zero on ∂Ω, so that V will lie in GG. First we construct ϕ on a neighborhood of ∂Ω in Ω, and for this purpose we use the neighborhood and parametrization h:[0,ε] × ∂Ω → Ω that was introduced during the proof of Proposition 5. On this neighborhood, we continue to use the coordinate function u,withu =0on∂Ω. Let n(p) be the outward pointing unit normal vector field along ∂Ω, and write V = a(p)n(p)along∂Ω. Then define ϕ on the neighborhood h([0,ε] × ∂Ω) by ϕ(h(u, p)) = −a(p)u. Because ∂Ωisalevelsurfaceofϕ, we know that the gradient of ϕ is orthogonal to ∂Ω. Thus we can express (∇ϕ)(p)=a(p)n(p)=V (p)along∂Ω, as desired. Finally, we extend ϕ from a slightly smaller neighborhood of ∂Ωtoasmooth function defined on all of Ω in any way we please. The resulting gradient vector field V = ∇ϕ on Ω is curl-free and agrees with the given vector field V on ∂Ω, completing the proof of Proposition 8. Proposition 9. AsmoothvectorfieldV defined on and tangent to ∂Ω can be extended to a smooth, curl-free vector field on Ω if and only if

(1) (∇×V )⊥ =0on ∂Ω,and (2) V · ds = 0 for all closed curves C on ∂Ω that bound in Ω. C

Before beginning the proof, we need to explain condition (1) and comment on condition (2). Recall from the formula for curl in rectangular coordinates that the z-component of ∇×V depends only on the x and y components of V . In general, if the vector field V is known along a surface S in 3-space, then the component of its curl normal to S is also known along S. This follows from the curl theorem, since 1 (∇×V ) · n(p) = lim V · ds, area Σ ∂Σ where Σ is a small patch of the surface S around the point p, and the limit is taken as Σ shrinks down to p. Notice that only the component of V tangent to S is needed to compute the above integral, so that (∇×V )⊥ along S depends only on V along S. As a consequence, if we preassign V along ∂Ω and hope to extend it to a curl-free vector field on Ω, then we must assume that (∇×V )⊥ = 0 along ∂Ω. In the present case, V is tangent to ∂Ω, so V = V . This explains condition (1). 12 cantarella, deturck, and gluck Condition (2) is certainly necessary for the existence of a curl-free extension of V over the domain Ω. Note that condition (2), as stated, implies condition (1). But in the presence of condition (1), condition (2) need only be verified for a family of closed curves C spanning the kernel of the surjective map H1(∂Ω) → H1(Ω). This follows from Stokes’ theorem, since a curve in the kernel of the map bounds in Ω, and so the integral of a curl-free vector field around such a curve must be zero. Now we prove Proposition 9. Proof. We start with the smooth vector field V defined on and tangent to ∂Ω, and satisfying conditions (1) and (2).

Let V1 be a harmonic knot deﬁned on Ω and having the same periods on ∂Ωas V . Then the vector ﬁeld V − V1 on ∂Ω is tangent to ∂Ω and integrates to zero around every closed curve on this surface. Hence there is a function ϕ on ∂Ωwith ∇ ϕ = V − V1,where∇ denotes the gradient operator along the surface ∂Ω. Next we extend ϕ to a function on a neighborhood of ∂Ωsothat∂ϕ/∂n = 0 along ∂Ω. This is easily done using our now familiar neighborhood h([0,ε]×∂Ω); we simply set ϕ(h(u, p)) = ϕ(p).

As a consequence, ∇ϕ = V − V1 along ∂Ω. Now we simply extend ϕ from a slightly smaller neighborhood of ∂ΩtoallofΩ, put V2 = ∇ϕ, and ﬁnally let V = V1 + V2.ThenV is curl-free because both V1 and V2 are, and V assumes the given values on ∂Ω because V2|∂Ω = V − V1|∂Ω. This completes the proof of Proposition 9. We combine Propositions 8 and 9 as follows.

Proposition 10. AsmoothvectorfieldV0 defined on ∂Ω can be extended to a smooth, curl-free vector field on Ω if and only if

∇× ⊥ (1) ( V0 ) =0on ∂Ω,and

(2) V0 · ds = 0 for all closed curves C on ∂Ω that bound in Ω. C

⊥ Proof. The proof is immediate: we simply split V0 into V0 and V0 along ∂Ω, and ⊥ use Propositions 8 and 9 to extend V0 and V0 , respectively, to curl-free vector fields on Ω. Finally, observe that, as in Proposition 9, in the presence of condition (1), condition (2) need only be verified for a finite set of curves on ∂Ω.

7. Vector fields with preassigned curl Proposition 11. Let V beasmoothvectorfielddefinedon∂Ω and W asmooth vector field defined on Ω.ThenV can be extended to a smooth vector field V on Ω such that ∇×V = W vector calculus and the topology of domains in 3-space, ii 13 if and only if

(1) ∇·W =0, (2) (∇×V )⊥ = W ⊥ on ∂Ω,and (3) V · ds = W · n d(area) for all surfaces Σ in Ω with ∂Σ ⊂ ∂Ω,wheren is ∂Σ Σ the unit normal vector ﬁeld to Σ, consistent with the orientation of ∂Σ.

Proof. Conditions (1), (2) and (3) are clearly necessary. Moreover, if conditions (1) and (2) are satisfied, it is easy to see (as in Proposition 9) that condition (3) need only be checked for a set of surfaces that span H2(Ω,∂Ω; R). So assume that the vector fields V on ∂ΩandW on Ω are given, and satisfy conditions (1), (2) and (3). We assert that condition (2) implies that W has zero flux through each component of ∂Ω. To see this, take the vector field V on ∂Ω and extend it to any smooth vector field on Ω (or even on just a neighborhood of ∂Ω), without trying to satisfy the equation ∇×V = W . We’ll still have (∇×V )⊥ = W ⊥ on ∂Ω by condition (2). And since the curl of a vector field always has zero flux through a closed surface, the vector field W must have zero flux through each component of ∂Ω, as asserted. When we combine this with the fact that W is divergence-free, according to condition (1), we conclude as in the proof of Proposition 5 that W is in the image of curl.

So let V1 be any smooth vector ﬁeld on Ω such that ∇×V1 = W .

Now let V0 = V −V1 on ∂Ω. Then conditions (2) and (3) imply that V0 satisﬁes conditions (1) and (2) of Proposition 10. That proposition then guarantees the existence of a smooth vector ﬁeld V0 on all of Ω such that ∇×V0 =0andV0|∂Ω = V − V1|∂Ω.

Hence the vector ﬁeld V = V0 + V1 satisﬁes ∇×V = W and extends the given V from ∂Ω to Ω, completing the proof of Proposition 11.

8. Vector fields with preassigned divergence and curl Proposition 12. Let V beasmoothvectorfielddefinedonandtangentto∂Ω,let W beasmoothvectorfielddefinedonΩ,andletf be a smooth function defined on Ω. Then there exists a smooth vector field V defined on Ω such that

∇×V = W and ∇·V = f, with parallel component V as given on ∂Ω (but with V ⊥ unspeciﬁed on ∂Ω), if and only if

(1) ∇·W =0, (2) (∇×V )⊥ = W ⊥ on ∂Ω,and 14 cantarella, deturck, and gluck (3) V · ds = W · n d(area), for all surfaces Σ in Ω with ∂Σ ⊂ ∂Ω,wheren ∂Σ Σ is the unit normal vector ﬁeld to Σ, consistent with the orientation of ∂Σ.

Furthermore, the solution V is unique up to adding an element of the ﬁnite- dimensional space HG. Proof. Just as in the previous proposition, conditions (1), (2) and (3) are necessary, and if conditions (1) and (2) are satisﬁed, then condition (3) need only be checked for a set of surfaces spanning H2(Ω,∂Ω; R). Furthermore, using the observations at the end of section 2 just as before, conditions (1) and (2) guarantee that W lies in the image of curl.

So let V1 be any smooth vector ﬁeld on Ω such that ∇×V1 = W .

Subtracting V1 from the unknown vector ﬁeld V , subtracting ∇·V1 from f,and subtracting V1 from the preassigned V along ∂Ω, we are in eﬀect reduced to proving Proposition 12 in the special case that W = 0. For simplicity of notation, we take this to be our task. Thus our conditions now read:

(2) (∇×V )⊥ =0on∂Ω, and (3) V · ds = 0, for all closed curves C on ∂Ω that bound in Ω. C

Now, as in the proof of Proposition 9, we let V2 be a harmonic knot deﬁned on Ω and having the same periods on ∂ΩasV . Subtracting V2 from the unknown vector ﬁeld V , and subtracting V2 from the preassigned V along ∂Ω, we have further reduced our task to that of proving Proposition 12 in the special case that W =0 and all the periods of V on ∂Ω are zero. Again, for simplicity of notation, we take this to be our task. Since all the periods of V on ∂Ω are zero, there is a function ϕ0 on ∂Ωwith ∇ ϕ0 = V .

Now we can solve the Poisson equation ∆ϕ = f −∇·V1 with Dirichlet boundary condition ϕ|∂Ω = ϕ0, and put V3 = ∇ϕ. This gives us

∇×V3 = ∇×(∇ϕ)=0 and ∇·V3 = ∇·(∇ϕ)=∆ϕ = f −∇·V1, as well as ∇ ∇ ∇ V3 =( ϕ) = ϕ = ϕ0 = preassigned value of V along ∂Ω.

Hence the vector field V3 has zero curl, divergence equal to f −∇·V1, and preassigned value of V along ∂Ω. It thus satisfies all the requirements imposed on it. vector calculus and the topology of domains in 3-space, ii 15 The original vector field V , whose existence is asserted in Proposition 12, is then the sum of three fields, V = V1 + V2 + V3,whereV1 satisfies ∇×V1 = W ,whereV2 is a harmonic knot with preassigned periods, and where V3 is a gradient vector field with preassigned divergence and preassigned parallel components along ∂Ω. The difference between any two vector fields satisfying all the conditions of Propo- sition 12 has zero curl, zero divergence, and zero parallel components along the boundary of Ω, and is therefore a harmonic gradient. This completes the proof of Proposition 12. Proposition 13. Let V ⊥ be a smooth vector field defined on and normal to ∂Ω,let W beasmoothvectorfielddefinedonΩ,andletf be a smooth function defined on Ω. Then there exists a smooth vector field V defined on Ω such that ∇×V = W and ∇·V = f, with normal component V ⊥ as given on ∂Ω (but with V unspecified on ∂Ω), if and only if

(1) ∇·W =0, (2) The ﬂux of W through each component of ∂Ω is zero, and ⊥ · (3) Ωi f d(vol) = ∂Ωi V n d(area) for each component Ωi of Ω.

Furthermore, the solution V is unique up to adding an element of the ﬁnite- dimensional space HK. Proof. Conditions (1), (2) and (3) are clearly necessary. Assume that these conditions hold, and note that conditions (1) and (2) guarantee that W lies in the image of curl, by the answer to Question 2 in Section 3.

So let V1 be any smooth vector ﬁeld on Ω such that ∇×V1 = W .

Subtracting V1 from the unknown vector field V , subtracting ∇·V1 from f,and ⊥ ⊥ subtracting V1 from the preassigned V along ∂Ω, we are in effect reduced to proving Proposition 13 in the special case that W = 0. For simplicity of notation, we take this to be our task. So now we must find a smooth vector field V on Ω such that ∇×V =0 and ∇·V = f, with normal component V ⊥ as given on ∂Ω, aided by the condition ⊥ (3) f d(vol) = V · n d(area) for each component Ωi of Ω. Ωi ∂Ωi

Since any such vector field V can be modified by a harmonic knot without changing its curl, its divergence or its normal component along ∂Ω, by the Hodge Decomposi- tion Theorem, we can search for V among the gradient vector fields. 16 cantarella, deturck, and gluck Let ϕ be a solution of the Poisson equation ∆ϕ = f with Neumann boundary condition ∂ϕ/∂n = V ⊥ ·n along ∂Ω. The function ϕ exists because (3) above provides the required integrability condition. Then the vector field V = ∇ϕ satisfies

∇×V = ∇×(∇ϕ)=0 and ∇·V = ∇·(∇ϕ)=∆ϕ = f, as well as ∂ϕ V ⊥ =(∇ϕ)⊥ = n =(V ⊥ · n) n = preassigned value of V ⊥ along ∂Ω. ∂n

Hence the current vector ﬁeld V has zero curl, divergence equal to f, and preassigned value of V ⊥ along ∂Ω, thus satisfying all the requirements imposed on it.

The original vector field V is the sum of two fields, V = V1 + V2,whereV1 satisfies ∇×V1 = W ,andwhereV2 is a gradient vector field with preassigned divergence and preassigned normal components along ∂Ω. The difference between any two vector fields satisfying all the conditions of Proposi- tion 13 has zero curl, zero divergence, and zero normal component along the boundary of Ω, and is therefore a harmonic knot. This completes the proof of Proposition 13.

9. Extensions and restrictions The questions posed at the beginning of this paper have been answered by Propo- sitions 6, 7, 10, 11, 12 and 13. It is interesting to note the pattern of these propositions. In seeking an unknown vector field V on the domain Ω, whose curl or divergence is specified throughout Ω, we are able to specify the value of V along the boundary of Ω, subject to appropriate integrability conditions. But if both the curl and divergence of V on Ω are specified in advance, then we are able to specify either the parallel or the normal component of V along the boundary of Ω, subject to appropriate integrability conditions, but not both. To understand why one cannot in general prescribe both the divergence and the curl of V on Ω together with both the tangential and normal components of V on ∂Ω, we will attempt to create such a proposition by simply combining the hypotheses of Propositions 12 and 13. That is, we let V be a smooth vector field defined on ∂Ω, and W a smooth vector field defined on Ω, and f a smooth function defined on Ω. We seek to extend V from ∂ΩtoallofΩsothat

∇×V = W and ∇·V = f.

At the very least, we must satisfy the integrability conditions

(1) ∇·W =0 vector calculus and the topology of domains in 3-space, ii 17 (2) (∇×V )⊥ = W ⊥ on ∂Ω (3) V · ds = W · n d(area), for all surfaces Σ in Ω with ∂Σ ⊂ ∂Ω, where n is ∂Σ Σ the unit normal vector field to Σ, consistent with the orientation of ∂Σ. ⊥ (4) f d(vol) = V · n d(area) for each component Ωi of Ω. Ωi ∂Ωi It is not necessary to explicitly include condition (2) from Proposition 13, that the flux of W through each component of ∂Ω is zero, because conditions (1) and (2) above already imply that W is in the image of curl, as we noted in the proof of Proposition 12. If we end the statement of our trial proposition here and hope to obtain the desired extension of V throughout Ω, then we will find that our problem is overdetermined. To make this clear, let us seek a vector field on our domain Ω which is divergence- free and curl-free (and hence lies in HK ⊕ CG ⊕ HG), and which is preassigned along ∂Ω. In other words, we are setting W =0andf = 0, and we will try to specify appropriate conditions on the boundary values of V . Consider the four integrability conditions stated above. The first is vacuously satisfied. The second, that (∇×V )⊥ = 0, amounts to requiring that the line integral of V along small loops on ∂Ω must vanish. The third condition requires that the line integral of V along loops on ∂Ω which bound inside Ω must also vanish. The fourth requires that the flux of V through the boundary of each component of Ω must vanish. To focus our thinking, let us kill the HK component of V by requiring that the periods of V along the generators of H1(Ω), which we know from [12] can be chosen on ∂Ω, all vanish. Then it follows that the line integrals of V along all loops on ∂Ω must be zero. Thus V ⊥ is the surface gradient of a function α: ∂Ω → R,thatis V = ∇α. Naturally, α is only determined up to (various) constants on the boundary components of Ω. Since we have killed the HK component of V , we know that V = ∇ϕ, and since ∇·V =0,wehave∆ϕ = 0. Thus we have transformed our problem into finding a harmonic function ϕ which simultaneously satisfies the Dirichlet condition ϕ|∂Ω = α and the Neumann condition ∂ϕ/∂n = β,whereV ⊥ = βn,andwhereβ is constrained to have zero integral over the boundary of each component of Ω. But this boundary-value problem is clearly overdetermined, since the Dirichlet condition alone already determines ϕ, and the preassigned Neumann condition may or may not be satisfied for this uniquely-determined ϕ. The operator that acts on functions on ∂Ω, mapping Dirichlet data (α) to its corresponding Neumann data (β) is a (highly non-local) pseudo-differential operator of order 1, called the Dirichlet- to-Neumann operator, or sometimes simply the Neumann operator. It has received a great deal of attention from mathematicians interested in scattering theory, the ∂-Neumann problem, and electrical conductivity problems (see [32], section 9.7 and Appendix C to Chapter 12). 18 cantarella, deturck, and gluck One extension of our results along the lines of Propositions 12 and 13 would be to prescribe both the curl and divergence of V on Ω, and a “mixed”, or “Robin”- type boundary condition, in which the value of V would be prescribed on some components of ∂Ω, and V ⊥ along the other components of ∂Ω. Let us assume that

(1) ∇·W =0 (2) the ﬂux of W through each component of ∂Ω is zero (3) (∇×V )⊥ = W ⊥ on the components of ∂ΩwhereV is prescribed (4) V · ds= W · n d(area) for all surfaces Σ ⊂ Ω such that ∂Σiscontained ∂Σ Σ in the components of ∂ΩwhereV is prescribed ⊥ · ⊥ (5) Ωi f d(vol) = ∂Ωi V n d(area) for any component Ωi of Ω such that V is prescribed over all of ∂Ωi.

Then there will exist a vector field V defined on Ω such that ∇×V = W and ∇·V = f, with the specified tangential and normal components on the respective parts of the boundary. The proof of this depends on the fact that the Robin problem for the Poisson equation (in which one prescribes Dirichlet data on part of ∂Ωand Neumann data on the rest of ∂Ω) is well-posed (see [18]).

10. Some remarks about the literature As mentioned in section 2, the Hodge Decomposition Theorem, as well as the boundary-value problems for vector fields we have proved in this paper, provide a framework for vector analysis that is used in a number of studies in fluid dynamics and plasma physics: mathematical theory of viscous incompressible flow [21];sta- tionary solutions of the Navier-Stokes equation [17]; spectral problems for self-adjoint realizations of the curl operator [10, 11, 28, 35]; Beltrami fields and force-free magnetic fields [8, 9, 22, 24, 25]; and discrete eigenstates of plasmas [33, 34]. In addition, we recommend the book [30], where the reader can find both the Hodge Decomposition Theorem and the boundary-value theorems for vector fields presented in the language of differential forms and in the generality of n dimensions. For foundational reading, and other viewpoints on and extensions of the material covered here, we recommend [1, 2, 5, 13, 14, 15, 16, 20].

Acknowledgements We are especially grateful to our students Ravi Goyal, Ariel Amdur and Nachi Gupta at the University of Pennsylvania for reading various drafts of this paper, for presenting it in their own words, and for helping us to say things better and more clearly. vector calculus and the topology of domains in 3-space, ii 19 REFERENCES

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Picard, On a selfadjoint realization of curl and some of its applications, Ricerche Mat. 47(1) (1998) 153–180. 29. R.L. Ricca and H.K. Moffatt, The helicity of a knotted vortex filament, Topological Aspects of Dynamics of Fluids and Plasmas (H.K. Moffatt, ed., 1992), 225–236. 30. G. Schwarz, Hodge Decomposition — A Method for Solving Boundary Value Prob- lems, Lecture Notes in Mathematics 1607, Springer-Verlag, New York (1995). 31. J.B. Taylor, Relaxation and magnetic reconnection in plasmas ,Rev.Mod.Phys., 58(3) (1986) 741–763. 32. M.E. Taylor, Partial Differential Equations II: Qualitative Studies of Linear Equa- tions, Applied Mathematical Sciences 116, Springer-Verlag, New York (1996). vector calculus and the topology of domains in 3-space, ii 21 33. Z. Yoshida, Discrete eigenstates of plasmas described by the Chandrasekhar- Kendall functions, Progress of Theoretical Phys. 86 (1991) 45–55. 34. Z. Yoshida, Eigenfunction expansions associated with curl derivatives in cylindri- cal geometries; completeness of Chandrasekhar-Kendall eigenfunctions,J.Math. Phys. 33(4) (1992) 1252–1256. 35. Z. Yoshida and Y. Giga, Remarks on spectra of operator rot,Math.Z.204(1990) 235–245. 22 cantarella, deturck, and gluck JASON CANTARELLA is an assistant professor of mathematics at the Uni- versity of Georgia. His research blends geometry, topology and analysis. He is a National Science Foundation postdoctoral fellow and a Project NExT fellow. Dept. of Mathematics, University of Georgia, Athens, GA, 30602. [email protected]

DENNIS DETURCK will have been chairman of the Mathematics Department at Penn. His research usually centers on problems in partial diﬀerential equations and diﬀerential geometry. He served as Associate Editor of the Monthly from 1986 to 1996 and recently won the Distinguished Teaching Award from the MAA’s Eastern Penn- sylvania and Delaware Section. Dept. of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395 [email protected]

HERMAN GLUCK is a professor of mathematics at Penn, with research inter- ests in geometry and topology and their applications to molecular biology and plasma physics. Dept. of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395 [email protected]

AMS CLASSIFICATION CODES: 14F40, 26-01, 26B12, 31-01, 31-02, 31B05, 31B20, 35-01, 35-02, 35J05, 53-01, 53-02, 57R25, 58-01, 58-02, 58A14, 58J32. KEYWORDS: boundary-value problem, curl, divergence, Dirichlet problem, gradient, harmonic forms, Helmholtz theorem, Hodge Decomposition Theorem, Hodge theory, Laplace equation, magnetic ﬁeld, Neu- mann problem, orthogonal decomposition, Poisson equation, 3-space, topology, vector calculus, vector ﬁelds