Advanced Calculus of real valued functions of real variable

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Let’s remember that we consider curvilinear coordinates in Euclidean space E. 5). 1. 4) and show that here it is given by formula gij = (Ei , Ej ). 1) In Cartesian coordinates all components of the metric tensor are constant since the basis vectors e1 , e2 , e3 are constant. 10) in Cartesian coordinates reduces to differentiation ∇p = ∂/∂xp . Therefore, ∇p gij = 0. 2) But ∇g is a tensor. If all of its components in some coordinate system are zero, then they are identically zero in any other coordinate system (explain why).

46 CHAPTER IV. TENSOR FIELDS IN CURVILINEAR COORDINATES. 2. 10) to a covector field and calculate the covariant derivative ∇p Xq . 3. 10) to an operator field and find q q ∇p Fm . Consider special case when ∇p is applied to the Kronecker symbol δm . 4. 10) to a bilinear form and find ∇p aqm . 5. 10) to a tensor product a ⊗ x for the case when x is a vector and a is a covector. Verify formula ∇(a ⊗ x) = ∇a ⊗ x + a ⊗ ∇x. 6. 10) to the contraction C(F) for the case when F is an operator field. Verify the formula ∇C(F) = C(∇F).

You certainly know the following formula: df (x1 (y), x2 (y), x3 (y)) = dy 3 fi (x1 (y), x2 (y), x3 (y)) i=1 dxi (y) ∂f , where fi = . dy ∂xi It’s for the differentiation of composite function. 8). § 25. Coordinate lines and the coordinate grid. 2) now assumed to contain differentiable functions. 1) i=1 is a differentiable function of three variables y 1 , y 2 , y 3 . 1) is called a basic vector-function of a curvilinear coordinate system. Let P0 be some fixed point of space given by its curvilinear coordinates y01 , y02 , y03 .