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old Chrstoffel div B

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Section of Phil's tensor documentation on curvilinear coordinates, saved after being removed from Section 9 on the divergence because the material now appears in Section 15. It derives div B as the contraction of the covariant derivative B^a;a, evaluates the contracted Christoffel symbol using the metric determinant, and obtains div B = (1/√g) ∂n(√g B^n). It notes this agrees with the geometric derivation.

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This section was removed from Section 9 on the divergence, since it now appears in Section 15. (e) The Christoffel derivation of div B This derivation is done in Picture C where the curvilinear coordinates are x so that ∂a means ∂/∂xa and g ≠ 1 is the curvilinear metric tensor, Mention of x(0)-space is completely avoided. In fact, even the x coordinate symbol rarely appears and could also have been removed. Function div B is in x-space, and once seen as a tensorial scalar, can be regarded as also being in x(0) space, [div B](0)(x(0)) = [div B] (x) // = Ba;a We start with the covariant derivative of a contravariant vector field component, which object is known to be a mixed rank-2 tensor (see Appendix G (e) form 2, and also (g)), Bb;a ≡ ∂aBb + ΓbanBn where Γcab = ½ gcd( ∂agbd + ∂bgad – ∂dgab ) . Then the tensorial scalar object div B is defined by index contraction to be div B ≡ Ba;a ≡ ∂aBa + Γaan Bn Evaluation of Γaan gives (see Appendix G (g) ) Γaan = ½ gad( ∂agnd + ∂ngad – ∂dgan ) = ½ gad ∂ngad = ½ (1/g)∂ng = (1/) ∂n() Therefore div B = Ba;a ≡ ∂aBa + Γaan Bn = ∂nBn + (1/) ∂n()Bn = [1/] ∂n [ Bn] in agreement with the geometric derivation. Thus the entire distinction from ∂nBn is the second term above which arises from the affine connection Γban part of the covariant derivative.