Mathematical Methods for Physics and Engineering - Matematica.NET

26.20 VECTOR OPERATORS IN TENSOR FORMwhere the expression in parentheses is the required covariant derivativeT ij ij∂T; k= +Γ i∂ulkT lj +Γ j k lk T il . (26.89)Using (26.89), the derivative of the tensor T with respect to u k cannowbewrittenintermsof its contravariant components as∂T∂u = T ij k ; k e i ⊗ e j . ◭Results similar to (26.89) may be obtained for the the covariant derivatives ofthe mixed and covariant components of a second-order tensor. Collecting theseresults together, we haveT ij ; k = T ij ,k +Γi lkT lj +Γ j lk T il ,T i j; k = T i j,k +Γ i lkT l j − Γ l jkT i l,T ij; k = T ij, k − Γ l ikT lj − Γ l jkT il ,where we have used the comma notation for partial derivatives. The position ofthe indices in these expressions is very systematic: for each contravariant index(superscript) on the LHS we add a term on the RHS containing a Christoffelsymbol with a plus sign, and for every covariant index (subscript) we add acorresponding term with a minus sign. This is extended straightforwardly totensors with an arbitrary number of contravariant and covariant indices.We note that the quantities T ij ; k , T i j; k and T ij; k are the components of thesame third-order tensor ∇T with respect to different tensor bases, i.e.∇T = T ij ; k e i ⊗ e j ⊗ e k = T i j; ke i ⊗ e j ⊗ e k = T ij; k e i ⊗ e j ⊗ e k .We conclude this section by considering briefly the covariant derivative of ascalar. The covariant derivative differs from the simple partial derivative withrespect to the coordinates only because the basis vectors of the coordinatesystem change with position in space (hence for Cartesian coordinates there is nodifference). However, a scalar φ does not depend on the basis vectors at all andso its covariant derivative must be the same as its partial derivative, i.e.φ ; j = ∂φ∂u j = φ ,j. (26.90)26.20 Vector operators in tensor formIn section 10.10 we used vector calculus methods to find expressions for vectordifferential operators, such as grad, div, curl and the Laplacian, in general orthogonalcurvilinear coordinates, taking cylindrical and spherical polars as particularexamples. In this section we use the framework of general tensors that we havedeveloped to obtain, in tensor form, expressions for these operators that are validin all coordinate systems, whether orthogonal or not.971