By R. A. Sharipov
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Extra info for Course of Differential Geometry
6) ϑ = arccos x3 / ((x1 )2 + (x2 )2 + (x3 )2 , ϕ = arctan(x2 /x1 ). Coordinate lines of spherical coordinates form three families. The first family is composed of the rays coming out from the point O; the second family is formed by circles that lie in various vertical planes passing through the axix Ox3 ; and the third family consists of horizontal circles whose centers are on the axis Ox3 . Exactly three coordinate lines pass through each regular point of the space E, one line from each family.
However, for ∇k the special name is reserved, it is called the operator of covariant differentiation or the covariant derivative. Below (in Chapter III) we shall see that the concept of covariant derivative can be extended so that it will not coincide with the partial derivative any more. Let A be a differentiable tensor field of the type (r, s) and let X be some arbitrary vector field. Let’s consider the tensor product ∇A ⊗ X. This is the tensor field of the type (r + 1, s + 1). The covariant differentiation adds one covariant index, while the tensor multiplication add one contravariant index.
Jm−1 n−1 k jn ... js−1 k=1 q=1 ∂xq proves the fifth property. This completes the proof of the theorem in whole. § 6. The metric tensor and the volume pseudotensor. Let O, e1 , e2 , e3 be some Cartesian coordinate system in the space E. The space E is equipped with the scalar product. Therefore, the basis e1 , e2 , e3 of any Cartesian coordinate system has its Gram matrix gij = (ei | ej ). 1) The gram matrix g is positive and non-degenerate: det g > 0. 2) follows from the Silvester criterion (see ).
Course of Differential Geometry by R. A. Sharipov