Tham khảo tài liệu 'intro to differential geometry and general relativity - s. warner episode 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | these coordinates are kk k and r r r det Er Er . Er respectively then at the point m h .kn det Dk1 Dk2 Dkn x - ởxÌ2 x - l1l2-ln 3x 1 dxk2 dxkn by definition of the determinant since i l i i is just the sign of the permutation _ dxi1 rix1- dxln dyr1 dyr2 dyr l lx--l i dyri dyr2 dyrn dX dX 2 dxX _ _ dyr1 dyr2 dyrn 1 x dỹk1 dxk2 dxk showing that the tensor transforms correctly. Finally we assert that det Dk Dk Dk is a smooth function of the point m. This depends on the change-of-coordinate matrices to the inertial coordinates. But we saw that we could construct inertial frames by setting dx_ kv J V j where the V j were an orthogonal base of the tangent space at m. Since we can vary the coordinates of this base smoothly the smoothness follows. Example In E3 the Levi-Civita tensor coincides with the totally antisymmetric third-order tensor ljk in Exercise Set 5. In the Exercises we see how to use it to generalize the crossproduct. Exercise Set 9 1. Recall that we can define the arc length of a smooth non-null curve by t dxdx y g du a Assuming that this function is invertible so that we can express xl as a function of s show that f2 1. 2. Derive the equations for a geodesic with respect to the parameter t. 3. Obtain an analogue of Corollary for the covariant partial derivatives of type 2 0 tensors. 71 4. Use inertial frames argument to prove that gablc gab c 0. Also see Exercise Set 4 1. 5. Show that if the columns of a matrix D are orthonormal then det D 1. 6. Prove that if E is the Levi-Civita tensor then in any frame Ei i i 0 whenever two of the indices are equal. Thus the only non-zero coordinates occur when all the indices differ. 7. Use the Levi-Civita tensor to show that if x is any inertial frame at m and if X 1 . . . X n are any n contravariant vectors at m then det X 1 . . . X n is a scalar. 8. The Volume 1-Form A Generalization of the Cross Product If we are given n-1 vector fields X 2 X 3 . . . X n on the n-manifold M define a covariant vector field by X
