Tham khảo tài liệu 'mathematical method in science and engineering episode 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | OPERATIONS WITH GENERAL TENSORS 193 Similarly the covariant derivative of a contravariant vector is defined as dxi The covariant derivative is also shown as di that is djUi Ui-j. The covariant derivative of a higher-rank tensor is obtained by treating each index at a time as rrtiii . __ 3l32 - I J I rTili2. I I 2 1 rpiiỉ. I 1 3132---- k Qxk I J l f 1 3 32-- I kl I 1 3i 32-- r _ ím ỊTili2. _im i khỉ kj2ị Covariant derivatives distribute like ordinary derivatives that is AB .ị A-iB AB-i and aA bB .ị aA-i bB-i where A and B are tensors of arbitrary rank and a and b are scalars. Some Covariant Derivatives In the following we also show equivalent ways of writing these operations commonly encountered in the literature. 1. Using definition Equation we can write the covariant derivative of a scalar function Ị as an ordinary derivative V 4 d3V This is also the covariant component of the gradient Vế . 2. Using the symmetry of Christoffel symbols the curl of a vector field V can be defined as the second-rank tensor djVi djVj Vi-j Vj-i ij ch ị dVj dxJ dxi 194 COORDINATES AND TENSORS Note that because we have used the symmetry of the Christoffel symbols the curl operation can only be performed on the covariant components of a vector. 3. The covariant derivative of the metric tensor is zero dkOij 9ij-k 0 with Equation and the definition of Christoffel symbols the proof is straightforward. 4. A frequently used property of the Christoffel symbol of the second kind is PL ỂíMẩì. 2 dxk dxk In the derivation we use the result aí ưĩẵ dxK OXK from the theory of matrices where g det gij. 5. We can now define covariant divergence as V-Ũ dịé .ip 10-220 pẰk 24 1O221 If V1 is a tensor density of weight 1 divergence becomes v vii dịV which is again a scalar density of weight 1. 6. Using Equation we write the contravariant component of the gradient
