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

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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 dual- ity, 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 com- ponent 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.
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