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Vector, Dyadic, and Integral Relations

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Vector, Dyadic, and Integral Relations A Vector, Dyadic, and Integral Relations A.1 Vector Identities A·B=B·A A x B = -B x A A· (B x C) = B· (C x A) = C· (A x B) A x (B x C) = B (A· C) - C (A· B) V (qn/J) = ¢ V1jJ + 'ljJ\l ¢ V· (1jJA) = A· V1fJ +~'V . A Vx (1jJA) = V1jJ x A+'ljJ\l x A V· (A x B) = B . V x A - A· V x B Vx (A x B) = (B· V) A - (A· V) B + A V . B - BV . A V (A· B) = (A· V) B + (B· V) A + A x V x B + B x V x A V x V x A = VV . A - v2 A V . V'ljJ = V21jJ V x V?/J = 0 V· V x A =0 605 606 A. Vector, Dyadic, and Integral Relations A.2 Dyadic Identities (AB) . (CD) = A (B . C) D (AB) x C = A (B x C) A· (C· B) = (A .C). B = A· C· B (A. C) x B = A· (C x B) = A· C x B (A x C) . B = A x (C· B) (A x C) x B = Ax (C x B) = AxC x B (A· C) . D = A· (C . D) = A· C . D (C· D) . A = C· (D . A) = C . D· A (C . D) x A = C . (D x A) = C . D x A (A x C)· D = Ax (C· D) = A x C· D (C x A) . D = C . (A x D) A- (B x C) = -B· (A x C) = (A x B) . C (C x A)· B = C· A x B = - (C x B)· A A x (B x C) = B (A· C) - (A· B) C A· C = (C) T . A = (A· C) T (cT) T= C T Ax C = - [(C) T X A] A·1=1·A AxI=IxA (I x A) . B = A· (I x B) = A x B (I x A) . C = A x C (A x I) . B = A x B (A B) . C = A (B· C) \1 (1/JA) = 1/J\1 A + (\11/J) A \1. (1/JC) = (\11/J) . C + 1/J\1. C \1x (1/JC) = \11/J x C+1/J\1 x C \1 x \1xC = \1 (\1. C) - \1 2 C \1. (1/JI) = \11/J \1. (I x A) = \1 x A \1x (1/JI) = \11/J x I \1 x \1x (1/JI) = \1\11/J - 1\1 2 1/J \1 . \1 A = \1 2 A \1·\1xC=O A.3 Dyadic Analysis 607 A.3 Dyadic Analysis Letting Xi denote the Cartesian variables (x, y, z), we represent a dyadic B as 333 B = LLBiJXiX] = LBjXj = B 1x+B2y+B3z i=1 j=1 j=1 and 333 BT = LLBij Xj Xi = LxjBj = XB1 + yB2 + ZB3. ;=1 j=1 j=1 Then we have the following. :> 3 3 A· B = L L AiBij Xj = L (A· B j ) Xj i=1 j=1 j=1 3 3 B·A= ~~ AB· X - LL.1 1.1 1 ;=1 j=1 3 = L B j (Xj . A) = Bdx· A) + (y. A) Y+ (z· A) Z j=1 :> A x B = L (A x B j ) Xj = (A x Bdx+ (A x B2)y + (A x B3)Z )=1 3 B x A = L B j (Xj x A) = Bl (x x A) + (y x A) Y + (z x A) Z j=1 33, V.B = ~~ oBij X. - LL OX; ] ;=1 j=1 3 = L (V . B j ) Xj = (V· B 1 ) X + (V . B 2 ) y + (V . B 3 ) Z j=1 333 VxB = LL (VBij x Xi)Xj = L (V x B j ) Xj ;=1 j=1 )=1 608 A. Vector, Dyadic, and Integral Relations 3 3 3 \7B = L L ~~J X;X-j = L (\7 Bj ) Xj = (\7 Bd X + (\7 B 2 ) Y+ (\7 B 3 ) Z i=lj=1 I j=1 In Cartesian coordinates the identity dyadic is I = xx + yy + ZZ, and \7. (~I) = \7~, \7x (~I) = \7~ x I. A.4 Integral Identities In the following V is a volume (<;;;; R 3 ) bounded by closed surface S, and n is a unit normal vector that points outward from the surface S, with n dS = dS. In two dimensions S is an open surface and 1 denotes the unit tangential vector to the edge (rim) of the open surface, with m a unit normal vector that points outward from the open surface S with m following the right screw rule by turning 1. For an open surface S the unit vector n is perpendicular to the rim of the surface, locally tangent to the open surface, with n = 1 x m. Note that we use bold for n, 1, m, but other unit vectors are indicated with a caret, e.g., x, y, Z. The dimensionality of the following integral theorems should be apparent from the given form. Various mathematical conditions are placed on the functions appearing in the following theorems. For instance, the divergence theorem fv \7. A dV = is n . A dS is valid for any regular region 1 V with the vector function A continuous and piecewise continuously differentiable in V. The condition requiring that the function be piecewise continuously differentiable on the boundary surface S may be removed as long as the left-side integral in the theorem converges. Similar conditions can be stated for the various Stokes' and gradient theo­ rems, which in turn are used to derive the Green's theorems. While in many practical cases these conditions can be considerably weakened, the discus­ sion of the dyadic Green's theorem in Section 1.3.5 illustrates that this is not always the case, depending on the order of the singularity involved. In general, it is assumed that the regions in question are regular and that the involved functions are continuous and sufficiently continuously differentiable for the theorems to hold. The derivation of the standard theorems may be found in most calculus books, and the elevation to vector and vector-dyadic form is described in reference [3] of Chapter 1. 1 A regular region is a bounded closed region whose boundary is a closed, sufficiently smooth surface. In this definition the bo~dary is part of the region. Alternatively, often one considers open regions D, where D is used to include the boundary. A.4 Integral Identities 609 Vector Gauss or Divergence Theorem Iv V, A dV = is n ' A dS /' V ' A dS = in 'A dl .J s I Dyadic Gauss or Divergence Theorem /' V, A dV = 1 n' A dS .!v Is Stokes' Theorem is n ' V x A dS = iI, A dl Dyadic Stokes' Theorem /' n' Vx AdS= iI, Adl .J s ' I Vector Stokes' Theorem Iv V x A dV = is n x A dS Scalar Stokes' Theorem 1. n x V1/; dS= .i I1/; dl Vector Gradient Theorem j' \h/) dV = 1 n1/; dS v Is Dyadic Gradient Theorem Iv V A dV = is n A dS Green's First Theorem Iv [if!1 V2~i2 + V1/;l ' V1/;2J dV = is 1/;1 V1/;2 ,dS is [1fJ1 V21/;2 + VUi1 ' V1/;2] dV = .i 1/;1 V1/;2 ' dl 610 A. Vector, Dyadic, and Integral Relations Green's Second Theorem i [7f!1 \727f!2 - 7f!2 \727f!1] dV = Is (7f!1 \7 7f!2 - 7f!2 \7 7f!d . dS = 1 (7f!1 a7f!2 _ 7f!2 a7f!l) dB Js an an Is [7f!1 \727f!2 - 7f!2 \727f!1] dB = i (7f!1 \7 7f!2 - 7f!2 \7 7f!d . n dl = (7f!1 a7f!2 - 7f!2 a7f!l ) dl iI an an Vector Green's First Theorem i [(\7 x A)· (\7 x B) -A· \7 x \7 x B] dV = Is n· (A x \7 x B) dB Vector Green's Second Theorem i [\7 x \7 x A . B - A . \7 x \7 x B] dV = Is [Ax (\7 x B) + (\7 x A) xB]· dS Vector-Dyadic Green's First Theorem i [\7 x A . \7 x B - A . \7 x \7 x B] dV = Is n· (A x \7 x B) dB Vector-Dyadic Green's Second Theorem i (\7 x \7 x A) ·B-A· (\7 x \7xB) dV = Is n· [A x \7xB + (\7 x A) x B] dB = Is ((n x A) . (\7xB) + (n x \7 x A) . B) dB which can be converted, using vector and dyadic identities, to i [(\7 2 A) . B-A-\72 B] dV = - f {(n x A)· (\7xB) + (n x \7 x A)· B + [n . A \7 . B - n . B \7 . A]} dB. A.4 Useful Formulas: Position Vector and Scalar Green's Functions 611 o [-[ Figure A.I: Geometry of vectors I' and 1". Note that Green's theorems can be considered as extensions of integra­ tion by parts to higher dimensions. Indeed, in one dimension we get A.5 Useful Formulas Involving the Position Vector and Scalar Green's Functions ,\,N ~ I' = 6;=1 xiXi R(r, 1") = I' - 1", R(r', 1') = 1"-1' R(r, 1") = IR(r, 1")1 = II' - 1"1 = (~:'1 (Xi - x;)2f/2= R(r', 1') ~( ') - R(r,r') - R~(' ) R 1',1' - R(r,r') -- I' ,I' \1R(r, 1") = R(r, 1") = \1'R(r',r) = -\1'R(r,r') = -\1R(r', 1') \1 1 - \7R(ror') - \1' 1 - \1' 1 - \1 1 R(r,r') -- R2(rorl) - R(r'or) -- R(r,r' ) - - R(r'or) ",a.
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