Mathematical Methods for Physics and Engineering - Matematica.NET

26.18 DERIVATIVES OF BASIS VECTORS AND CHRISTOFFEL SYMBOLSthe outer product of the two tensors, or any contraction of them, is a relativetensor of weight w 1 + w 2 . As a special case, we may use ɛ ijk and ɛ ijk to constructpseudovectors from antisymmetric tensors and vice versa, in an analogous wayto that discussed in section 26.11.For example, if the A ij are the contravariant components of an antisymmetrictensor (w =0)thenp i = 1 2 ɛ ijkA jkare the covariant components of a pseudovector (w = −1), since ɛ ijk has weightw = −1. Similarly, we may show thatA ij = ɛ ijk p k .26.18 Derivatives of basis vectors and Christoffel symbolsIn Cartesian coordinates, the basis vectors e i are constant and so their derivativeswith respect to the coordinates vanish. In a general coordinate system, however,the basis vectors e i and e i are functions of the coordinates. Therefore, in orderthat we may differentiate general tensors we must consider the derivatives of thebasis vectors.First consider the derivative ∂e i /∂u j . Since this is itself a vector, it can bewritten as a linear combination of the basis vectors e k , k =1, 2, 3. If we introducethe symbol Γ k ijto denote the coefficients in this combination, we have∂e i∂u j =Γk ije k . (26.75)The coefficient Γ k ij is the kth component of the vector ∂e i/∂u j .Usingthereciprocityrelation e i · e j = δj i , these 27 numbers are given (at each point in space)byΓ k ij = e k · ∂e i∂u j . (26.76)Furthermore, by differentiating the reciprocity relation e i · e j = δj i with respectto the coordinates, and using (26.76), it is straightforward to show that thederivatives of the contravariant basis vectors are given by∂e i∂u j = −Γi kje k . (26.77)The symbol Γ k ijis called a Christoffel symbol (of the second kind), but, despiteappearances to the contrary, these quantities do not form the components of athird-order tensor. It is clear from (26.76) that in Cartesian coordinates Γ k ij =0for all values of the indices i, j and k.965