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Extra resources for A Course of Differential Geometry

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Let’s derive the rule for covariant differentiation of tensor fields in a curvilinear coordinate system. We consider a vectorial field A to begin with. This is a field whose components are specified by one upper index: Ai (u1 , u2 , u3 ). In order to calculate the components of the field B = ∇A we choose some auxiliary Cartesian coordinate system x ˜1 , x ˜2 , x ˜3 . 1) from Chapter II, and finally, we transform the components of ∇A from Cartesian coordinates back to the original curvilinear coordinates.

Js =ξ q=1 ir X q ∇q Aij11... js . This relationship is equivalent to the property (3) in the statement of the theorem. In order to prove the fourth property in the theorem one should carry out the following calculations with the components of A, B and X: 3 3 i ... i ir r+1 r+m = X q ∂/∂xq Aij11... js Bjs+1 ... js+n Xq q=1 q=1 i 3 i ... i ir r+m × Bjr+1 + Aij11... js s+1 ... js+n Xq ir ∂Aij11... js ∂xq ... i r+m ∂Bjr+1 s+1 ... js+n ∂xq q=1 × . And finally, the following series of calculations 3 Xq q=1 3 ∂ ∂xq i ...

It is denoted ρ = rot F. 5) the rotor or a vector field F is the contraction of the tensor field ˆ⊗ω⊗g ˆ ⊗ ∇F with respect to four pairs of indices: rot F = C(ˆ ˆ ⊗ ∇F). g g⊗ω⊗g Remark. 11), then the rotor of a vector field should be understood as a pseudovectorial field. Suppose that O, e1 , e2 , e3 is a rectangular Cartesian coordinate system in E with orthonormal right-oriented basis e1 , e2 , e3 . The Gram matrix of the basis e1 , e2 , e3 is the unit matrix. Therefore, we have gij = g ij = δji = 1 for i = j, 0 for i = j.

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