Tham khảo tài liệu 'intro to differential geometry and general relativity - s. warner episode 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | dX dxq i pq i Xp dxq dt The quantity in brackets converts the vector dxq dt into the vector DX dt. Moreover since every contravariant vector has the form dxq dt recall the definition of tangent vectors in terms of paths it follows that the quantity in brackets looks like a tensor of type 1 1 and we call it the qth covariant partial derivative of X Definition The covariant partial derivative of the contravariant field Xp is the type 1 1 tensor given by Covariant Partial Derivative of X Ý - i pq J Xp Some texts use VqXi. Do you see now why it is called the covariant derivative Similarly we can obtain the type 0 2 tensor check that it transforms corectly Covariant Partial Derivative of Yp dYn Y p piq dxq i pq i Yi Notes 1. All these forms of derivatives satisfy the expected rules for sums and also products. See the exercises. 2. If C is a path on M then we obtain the following analogue of the chain rule dX p dxk dt X M . See the definitions. Exercise Set 8 1. a Show that i jk i kj b If Tjk are functions that transform in the same way as Christoffel symbols of the second kind called a connection show that Yjk - rklj is always a type 1 2 tensor called the associated torsion tensor . c If ajj and gjj are any two symmetric non-degenerate type 0 2 tensor fields with associated Christoffel symbols r i 1 i u L and Wg respectively. Show that i jk a i jk g 61 is a type 1 2 tensor. 2. Covariant Differential of a Covariant Vector Field Show that if Yị is a covariant vector then__ DYp dYp - 1 L pq Yi d-vq. are the components of a covariant vector field. That is check that it transforms correctly. 3. Covariant Differential of a Tensor Field Show that it we define DTh d h p p h rq. T d - l pq. Th dxq. l then the coordinates transform like a 1 1 tensor. 4. Obtain the transformation equations for Chritstoffel symbols of the first and second kind. You might wish to consult an earlier printing of these notes or the Internet site. 5. Show directly that the .
