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Nabla in Curvilinear Coordinates

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Nabla in Curvilinear Coordinates Nabla in Curvilinear Coordinates Reference: M. R. Spiegel, Schaum's Outline of ::: Vector Analysis ::: , Chapter 7 (and part of Chap. 8). (Page references are to that book.) ~ ^ ^ Suppose that A = Arr^ + Aθθ + Aφφ with respect to the usual basis of unit vectors in spherical coordinates. What is A~?Itisnot @Ar + ∇· @r @Aθ @Aφ ∂θ + ∂φ . The reason is that the spherical unit vectors themselves are functions of position, and their own derivatives must be taken into account. The correct formula is [p. 165, solution 63(b)] 1 @ 1 @ 1 @A A~= (r2A )+ (sin θA )+ φ : ∇· r2 @r r rsin θ ∂θ θ rsin θ ∂φ Where did this come from? To set the stage, let's recall that the most elementary local basis asso- ciated with a coordinate system is the tangent vectors @~r @~r @~r @~r in the spherical case, ; ; : @uj @r ∂θ ∂φ The metric tensor is defined as the matrix of inner products @~r @~r gij = : @ui · @uj This means that the formula for arc length is 2 3 2 2 2 2 2 ds = gij dui duj = dr + r dθ + r sin θdφ in sphericals : Xij (In this context dv du =+du dv, not du dv as in surface integrals.) − 1 Orthogonal coordinates are the very important special cases where g is a diagonal matrix: 2 h1 00 2 g = 0h2 0(in three-dimensional cases): 00h2 3 That is, by definition @~r h = ; j @u j in the spherical case, hr =1;hθ=r; hφ = r sin θ. In orthogonal coordinates we can define the orthonormal basis of unit vec- tors, 1 @~r u^j = : hj @uj The other standard basis vectors, the normal vectors u ,satisfy ∇ j u^ =h u : j j∇ j Now we turn to the gradient. Note that there is some ambiguity in the term \components of the gradient in spherical coordinates", because we have three natural spherical bases to choose among. The most useful choice for \everyday" physics is the orthonormal basis, which is obtained by a rotation from the Cartesian basis ^{; |;^ k^ . By the chain rule one sees, for instance, that { } @f @f @x @~r = j = f = h θ^ f: ∂θ @x ∂θ ∇ · ∂θ θ ·∇ Xj j But since this basis is ON, this is equivalent to saying that the θ^ component 1 @f of f is h − . In general [p. 137], ∇ θ ∂θ 1 @f f = u^j ; ∇ hj @uj Xj 2 which in the spherical case is [p. 165, solution 63(a)] @f 1 @f 1 @f f = r^+ θ^+ φ.^ ∇ @r r ∂θ r sin θ ∂φ To treat the divergence and curl, first note that u^ =^u u^ =hh( u) ( u) 1 2× 3 2 3 ∇ 2 × ∇ 3 (and the two obvious cyclic permutations of this formula). Now consider one term of the divergence: (A u^ )= (A h h u u ) ∇· 1 1 ∇· 1 2 3∇ 2 ×∇ 3 = (A h h ) ( u u )+A h h ( u u ): ∇ 1 2 3 · ∇ 2 ×∇ 3 1 2 3∇· ∇ 2 ×∇ 3 But the last term is ~0, because (B~ C~)=C~ ( B~) B~( C~)and u =0.So ∇· × · ∇× − ∇× ∇×∇ j (A u^ )= (A h h ) ( u u ) ∇· 1 1 ∇ 1 2 3 · ∇ 2×∇ 3 1 = (^u2 u^3) (A1h1h2) h2h3 × ·∇ u^1 @ 1 = (A1h2h3) u^1 +(^u2 andu ^3 terms) ; h1h3 ·@u1 · h1 where the previous formula for the gradient has been used. Only theu ^1 term survives the dot product, so 1 @ (A1u^1)= (A1h2h3): ∇· h1h2h3 @u1 Adding the two cyclic analogues of this term one gets [p. 150] ~ 1 @ @ @ A= (A1h2h3)+ (A2h3h1)+ (A3h1h2) : ∇· h1h2h3 @u1 @u2 @u3 In spherical coordinates this reduces to the formula given at the beginning. Incidentally, the denominator in the divergence formula, h1h2h3 = √det g, is equal to the Jacobian determinant that occurs in the curvilinear 3 volume formula. This fact is a consequence of Part 2 of the Volume Theo- rem on p. 340 of Linearity, where the metric tensor was introduced without aname. Similarly [p. 150; p. 154 for spherical case], the curl is h1u^1 h2u^2 h3u^3 1 A~= @ @ @ : @u1 @u2 @u3 ∇× h1h2h3 h A h A h A 1 1 2 2 3 3 An alternative way of deriving the divergence and curl formulas [pp. 151{152] is based on the integral definitions of divergence and curl [p. 365 of Linearity]. The factors hj show up in the relationships between coordinate increments (∆uj ) and physical lengths, areas, and volumes. 4.
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