Advanced Calculus fi..

206 Advanced Calculus, Fifth EditionContravariant derivatives are found in similar fashion. In fact one has the followingsimple rule: For every tensor A,To justify this, we observe that the right-hand side is a tensor of the type desiredand that in standard coordinates (ti) it reduces to the set of partial derivatives of thecomponents of A with respect to xk. Equation (3.98) expresses contravariant derivativesin terms of covariant derivatives and shows that AkA and A ~ are A associatedtensors.For an invariant f, one findsThus the covariant derivative of an invariant f is simply the gradient of f:af3Ak f = grad f = - (3.100)axk 'The contravariant derivative of f is the associated contravariant vector.We can now define the divergence of a vector field. We want this to reduce tothe usual divergence in standard coordinates. If u has contravariant components u',then we definesince this reduces to the expected expressiondiv u = div u' = A,u", (3.101)I.din standard coordinates. Since A,uU is obtained by contraction of the tensor A ~u~,it is a tensor of order 0, an invariant. We can obtain the same invariant from thecovariant components of u:div u = div ui = Aau,, (3.103)for in standard coordinates the components become Ui = u', and again (3.102) isobtained.One can also obtain (Problem 13) the following useful expression for the divergence:1 adiv u = - - (figaAu,).42 axhFrom the divergence and gradient we obtain the Laplacian of an invariant:dv2 f = div grad f = AaA, f.(3.105)We can also obtain this from the contravariant components of grad f:v2 f = AuAff. (3.106)