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Concepcion: Scalar and Vector Potentials, Helmholtz

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Concepcion: Scalar and Vector Potentials, Helmholtz Introduction Scalar and vector potentials in the whole space Scalar and vector potentials in a bounded domain Helmholtz decomposition Homology and de Rham cohomology Scalar and vector potentials, Helmholtz decomposition, and de Rham cohomology Alberto Valli Department of Mathematics, University of Trento, Italy A. Valli Potentials, Helmholtz decomposition, de Rham cohomology Introduction Scalar and vector potentials in the whole space Scalar and vector potentials in a bounded domain Helmholtz decomposition Homology and de Rham cohomology Outline 1 Introduction 2 Scalar and vector potentials in the whole space 3 Scalar and vector potentials in a bounded domain 4 Helmholtz decomposition 5 Homology and de Rham cohomology A. Valli Potentials, Helmholtz decomposition, de Rham cohomology Introduction Scalar and vector potentials in the whole space Scalar and vector potentials in a bounded domain Helmholtz decomposition Homology and de Rham cohomology The objects Beyond a doubt, among the \stars" of vector calculus we have the operators grad div curl Aim of this talk is to understand better their properties and their connections with some topological concepts. A. Valli Potentials, Helmholtz decomposition, de Rham cohomology Introduction Scalar and vector potentials in the whole space Scalar and vector potentials in a bounded domain Helmholtz decomposition Homology and de Rham cohomology First results First well-known results are (just compute...): curl grad = 0 for each scalar function div curl H = 0 for each vector field H. We can thus write Theorem (1) If H = grad , then curl H = 0 (namely, H is curl-free). Theorem (2) If B = curl A, then div B = 0 (namely, B is divergence-free). A. Valli Potentials, Helmholtz decomposition, de Rham cohomology Introduction Scalar and vector potentials in the whole space Scalar and vector potentials in a bounded domain Helmholtz decomposition Homology and de Rham cohomology First results (cont'd) The natural question is: are these conditions sufficient? We will see that the answer depends on the geometry of the region Ω where we are working. A. Valli Potentials, Helmholtz decomposition, de Rham cohomology Introduction Scalar and vector potentials in the whole space Scalar and vector potentials in a bounded domain Helmholtz decomposition Homology and de Rham cohomology In the whole space... 3 Let us start from Ω = R . We need some tools. First of all we know [just compute...] that the function 1 1 K(x; y) = (1) 4π jx − yj satisfies −∆x K(x; y) = 0 for x 6= y R @B grad x K(x; 0) · n(x) dSx = −1 ; where B is the ball of center 0 and radius 1, and n the unit outward normal on @B. A. Valli Potentials, Helmholtz decomposition, de Rham cohomology Introduction Scalar and vector potentials in the whole space Scalar and vector potentials in a bounded domain Helmholtz decomposition Homology and de Rham cohomology Dirac δ0 distribution [Indeed, in a more advanced mathematical language, the function K(x; y) is the fundamental solution of the −∆ operator, namely, it satisfies −∆x K(x; y) = δ0(x − y) in the distributional sense, δ0 being the Dirac delta distribution centered at 0. Roughly speaking, for each (suitable...) function f the Dirac delta distribution satisfiesZ δ0(x − y)f (y) dy = f (x) : R3 We also know that the function Z u(x) = K(x; y)f (y) dy R3 3 satisfies −∆u = f in R . In fact (formally...) R R −∆u(x) = −∆x [ 3 K(x; y)f (y) dy] = 3 [−∆x K(x; y)]f (y) dy R R = R δ (x − y)f (y) dy = f (x) :] R3 0 A. Valli Potentials, Helmholtz decomposition, de Rham cohomology Introduction Scalar and vector potentials in the whole space Scalar and vector potentials in a bounded domain Helmholtz decomposition Homology and de Rham cohomology Scalar and vector potentials Let us come to the determination of a scalar potential for a curl-free vector field H (namely, a scalar function such that grad = H) and of a vector potential A for a divergence-free vector field B (namely, a vector field A such that curl A = B). 3 Consider a vector field H and define in R the function Z (x) = − K(x; y)div H(y) dy : (2) R3 3 Consider a vector field B and define in R the vector field Z A(x) = K(x; y)curl B(y) dy : (3) R3 A. Valli Potentials, Helmholtz decomposition, de Rham cohomology Introduction Scalar and vector potentials in the whole space Scalar and vector potentials in a bounded domain Helmholtz decomposition Homology and de Rham cohomology Theorems Theorem (3) Assume that H decays sufficiently fast at infinity and satisfies 3 3 curl H = 0 in R . The function satisfies grad = H in R . Proof. It is easily shown that 1 x − y D K(x; y) = − i i = −D K(x; y) ; xi 4π jx − yj3 yi hence (formally, and using that Di Hj = Dj Hi ...) R R D (x) = − 3 Dx K(x; y)div H(y) dy = 3 Dy K(x; y)div H(y) dy i R i R i P R P R = − 3 Dy Dy K(x; y)H (y) dy = 3 Dy K(x; y)D H (y) dy j R j i j j R j i j P R R = 3 Dy K(x; y)D H (y) dy = − 3 ∆y K(x; y)H (y) dy j R j j i R i = R δ (x − y)H (y) dy = H (x) : R3 0 i i A. Valli Potentials, Helmholtz decomposition, de Rham cohomology Introduction Scalar and vector potentials in the whole space Scalar and vector potentials in a bounded domain Helmholtz decomposition Homology and de Rham cohomology Theorems (cont'd) Theorem (4) Assume that B decays sufficiently fast at infinity and satisfies 3 div B = 0 in R . The vector field A satisfies curl A = B (and 3 div A = 0) in R . Proof. We have R D1A2(x) = 3 Dx K(x; y)(D3B1 − D1B3)(y) dy RR 1 = − 3 Dy K(x; y)(D3B1 − D1B3)(y) dy R R 1 = 3 [−Dy Dy K(x; y)B3(y) + Dy Dy K(x; y)B1(y)] dy RR 1 1 3 1 = 3 [−Dy Dy K(x; y)B3(y) − Dy K(x; y)D1B1(y)] dy : R 1 1 3 Similarly, Z D2A1(x) = [Dy2 Dy2 K(x; y)B3(y) + Dy3 K(x; y)D2B2(y)] dy : R3 A. Valli Potentials, Helmholtz decomposition, de Rham cohomology Introduction Scalar and vector potentials in the whole space Scalar and vector potentials in a bounded domain Helmholtz decomposition Homology and de Rham cohomology Theorems (cont'd) Since D1B1 + D2B2 = −D3B3, we find R − 3 Dy K(x; y)[D1B1(y) + D2B2(y)] dy R R 3 R = 3 Dy K(x; y)D3B3(y) dy = − 3 Dy Dy K(x; y)B3(y) dy ; R 3 R 3 3 hence R D1A2(x) − D2A1(x) = − 3 ∆y K(x; y)B3(y) dy R R = 3 δ (x − y)B3(y) dy = B3(x) : R 0 Repeating the same computations for the other components, the first part of the thesis follows. On the other hand R D1A1(x) = − 3 Dy K(x; y)(D2B3 − D3B2)(y) dy RR 1 = − 3 [Dy K(x; y)D2B3(y) − Dy K(x; y)D1B2(y)] dy ; R 1 3 and, proceding similary for D2A2 and D3A3, the second part of the thesis is easily verified. A. Valli Potentials, Helmholtz decomposition, de Rham cohomology Introduction Scalar and vector potentials in the whole space Scalar and vector potentials in a bounded domain Helmholtz decomposition Homology and de Rham cohomology Leading idea What has been the idea? If satisfies grad = H, then −div H = −div grad = −∆ ; hence we can use the (scalar) integral representation formula in terms of the fundamental solution K; if A satisfies curl A = B (and div A = 0), then curl B = curl curl A = curl curl A − grad div A = −∆A ; hence we can use the (vector) integral representation formula in terms of the fundamental solution K. A. Valli Potentials, Helmholtz decomposition, de Rham cohomology Introduction Scalar and vector potentials in the whole space Scalar and vector potentials in a bounded domain Helmholtz decomposition Homology and de Rham cohomology Biot{Savart formulas An alternative (and essentially equivalent) point of view is the one leading to the Biot{Savart formulas: Scalar potential: look for = div grad '. If satisfies grad = H, then H = grad div grad ' = ∆ grad '; hence Z grad ' = − K(x; y)H(y) dy R3 and 1 Z x − y (x) = 3 · H(y) dy : (4) 4π R3 jx − yj A. Valli Potentials, Helmholtz decomposition, de Rham cohomology Introduction Scalar and vector potentials in the whole space Scalar and vector potentials in a bounded domain Helmholtz decomposition Homology and de Rham cohomology Biot{Savart formulas (cont'd) Vector potential: look for A = curl Q (with div Q = 0). If A satisfies curl A = B, then B = curl curl Q = −∆Q ; hence Z Q(x) = K(x; y)B(y) dy R3 and 1 Z y − x A(x) = 3 × B(y) dy : (5) 4π R3 jx − yj A. Valli Potentials, Helmholtz decomposition, de Rham cohomology Introduction Scalar and vector potentials in the whole space Scalar and vector potentials in a bounded domain Helmholtz decomposition Homology and de Rham cohomology In a bounded domain... The problem is more complicated in a bounded domain Ω. However, some well-known results are usually presented in any calculus course. Theorem (5) Assume that H satisfies curl H = 0 in Ω and that any closed curve in Ω is the boundary of a suitable surface S ⊂ Ω.
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