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Intro to Differential Geometry and General Relativity - S. Warner Episode 3

Tham khảo tài liệu 'intro to differential geometry and general relativity - s. warner episode 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 1 2 1 d 2 d n d by the formula G v v . . . v v y v 2 . . . v y i d v T7 . dx Then we can verify that F and G are inverses as follows 12 n i d F G v v . . . v F v - local coordinates of the vector v -T v2 2 . . . vn . 12 n ox ox ox But in view of the simple local coordinate structure of the vectors 777 the i th coordinate dx of this field is 1 i-1 i i 1 n i v 0 . . . v 0 v 1 v 0 . . . v 0 v. In other words i th coordinate of F G v F G v v so that F G v v. Conversely G F w w1 WA . . . w t ox ox ox where w are the local coordinates of the vector w. Is this the same vector as w Well let us look at the ambient coordinates since if two vectors have the same ambient coordinates they are certainly the same vector But we know how to find the ambient coordinates of each term in the sum. So the j th ambient coordinate of G F w is 21 G F w j w1dy- w2 ị . . . wn ịn dx dx dx using the formula for the ambient coordinates of the d dxl Wj using the conversion formulas Therefore G F w w and we are done. o That is why we use local coordinates there is no need to specify a path every time we want a tangent vector Note Under the one-to-one correspondence in the proposition the standard basis vectors in n n_.1 n n_.2 n n_.n n correspond to the tangent vectors d dx d dx . . . d dx . Therefore the latter vectors are a basis of the tangent space Tm. 1. Suppose that v is a tangent vector at m é M with the property that there exists a local coordinate system x at m with v 0 for every i. Show that v has zero coordinates in every coefficient system and that in fact v 0. 2. a Calculate the ambient coordinates of the vectors d d0 and d d0 at a general point on S2 where 0 and 0 are spherical polar coordinates 0 x1 0 x2 . b Sketch these vectors at some point on the sphere. 3. Prove that - f . dx dx ax 4. Consider the torus T2 with the chart x given by yi a b cos x1 cos x2 y2 a b cos x1 sin x2 y3 b sin x1 0 x 2n. Find the ambeint coordinates of the two orthogonal tangent vectors at a general point