Mathematical Methods for Physics and Engineering - Matematica.NET

26.20 VECTOR OPERATORS IN TENSOR FORM◮Suppose A =[a ij ], B =[b ij ] and that B = A −1 . By considering the determinant a = |A|,show that∂a∂u = ab ji ∂a ijk ∂u . kIf we denote the cofactor of the element a ij by ∆ ij then the elements of the inverse matrixare given by (see chapter 8)However, the determinant of A is given byb ij = 1 a ∆ji . (26.93)a = ∑ ja ij ∆ ij ,in which we have fixed i and written the sum over j explicitly, for clarity. Partiallydifferentiating both sides with respect to a ij , we then obtain∂a=∆ ij , (26.94)∂a ijsince a ij does not occur in any of the cofactors ∆ ij .Now, if the a ij depend on the coordinates then so will the determinant a and, by thechain rule, we have∂a∂u = ∂a ∂a ij ∂ak ∂a ij ∂u k =∆ij ij∂u = ab ji ∂a ijk ∂u , (26.95)kin which we have used (26.93) and (26.94). ◭Applying the result (26.95) to the determinant g of the metric tensor, andremembering both that g ik g kj = δ i j and that gij is symmetric, we obtain∂g∂u k = ∂g ggij ij∂u k . (26.96)Substituting (26.96) into (26.92) we find that the expression for the Christoffelsymbol can be much simplified to giveΓ i ki = 1 ∂g2g ∂u k = √ 1 ∂ √ gg ∂u k .Thus finally we obtain the expression for the divergence of a vector field in ageneral coordinate system as∇ · v = v i ; i = √ 1 ∂g ∂u j (√ gv j ). (26.97)LaplacianIf we replace v by ∇φ in ∇ · v then we obtain the Laplacian ∇ 2 φ. From (26.91),we havev i e i = v = ∇φ = ∂φ∂u i ei ,973