Tham khảo tài liệu 'intro to differential geometry and general relativity - s. warner episode 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | b Let 0 be a scalar field. Its ambient gradient grad 0 is given by grad 0 . . . I0 ỔJ1 dJs that is the garden-variety gradient you learned about in calculus. This gradient is in general neither covariant or contravariant. However we can use it to obtain a 1-form as follows If V is any contravariant vector field then the rate of change of 0 along V is given by 0. If V happens to be a unit vector at some point then this is the directional derivative at that point. In other words dotting with grad 0 assigns to each contravariant vector field the scalar field F v 0 which tells it how fast 0 is changing along V. We also get the 1-form identities F V W F V F W F aV aF V . The coordinates of the corresponding covariant vector field are F d dx a dxl .grad 0 dy1 dy2 dy d0 d0 ax dxi . dxi . dyi . . . dys d0 s dxl which is the example that first motivated the definition. c Generalizing b let be any smooth vector field in Es defined on M. Then the operation of dotting with is a linear function from smooth tangent fields on M to smooth scalar fields. Thus it is a cotangent field on M with local coordinates given by applying the linear function to the canonical charts d dxl Ci d dx . The gradient is an example of this since we are taking grad 0 in the preceding example. Note that in general dotting with depends only on the tangent component of . This leads us to the next example. d If V is any tangent contravariant field then we can appeal to c above and obtain an associated covariant field. The coordinates of this field are not the same as those of V. To find them we write d V V See Note 4 . dx 31 Hence C d T i d d d V T V . dX dx dX dx Note that the tangent vectors d dx are not necessarily orthogonal so the dot products don t behave as simply as we might suspect. We let gịỊ 7 j 777 so that dX dx Cj g jV We shall see the quantities gjj again presently. Definition If V and W are contravariant or covariant vector fields on M and if a is a real number we can
