Mathematical Methods for Physics and Engineering - Matematica.NET

TENSORSIn order to compare the results obtained here with those given in section10.10 for orthogonal coordinates, it is necessary to remember that here we areworking with the (in general) non-unit basis vectors e i = ∂r/∂u i or e i = ∇u i .Thus the components of a vector v = v i e i are not the same as the components ˆv iappropriate to the corresponding unit basis ê i . In fact, if the scale factors of thecoordinate system are h i , i =1, 2, 3, then v i = ˆv i /h i (no summation over i).As mentioned in section 26.15, for an orthogonal coordinate system with scalefactors h i we have{h2g ij = i if i = j,0 otherwise{1/hand g ij 2= i if i = j,0 otherwise,and so the determinant g of the matrix [g ij ] is given by g = h 2 1 h2 2 h2 3 .GradientThe gradient of a scalar φ is given by∇φ = φ ; i e i = ∂φ∂u i ei , (26.91)since the covariant derivative of a scalar is the same as its partial derivative.DivergenceReplacing the partial derivatives that occur in Cartesian coordinates with covariantderivatives, the divergence of a vector field v in a general coordinate systemis given by∇ · v = v i ; i = ∂vi∂u i +Γ i kiv k .Using the expression (26.82) for the Christoffel symbol in terms of the metrictensor, we find(Γ i ki = 1 ∂gil2 gil ∂u k + ∂g kl∂u i − ∂g )ki∂u l = 1 ∂g 2 gil il∂u k . (26.92)The last two terms have cancelled becauseg il ∂g kl∂u i = g li ∂g ki∂u l = g il ∂g ki∂u l ,where in the first equality we have interchanged the dummy indices i and l, andin the second equality have used the symmetry of the metric tensor.We may simplify (26.92) still further by using a result concerning the derivativeof the determinant of a matrix whose elements are functions of the coordinates.972