By R. A. Sharipov

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As a result we get the formula relating the frame vectors of two curvilinear coordinate systems: 3 3 i=1 q=1 ˜j = E ∂ui (x1 , x2 , x3 ) ∂xq (˜ u1 , u ˜2 , u ˜3 ) q j ∂x ∂u ˜ · Ei . 6), from this comparison for the components of S we get 3 Sji = q=1 ∂ui (x1 , x2 , x3 ) ∂xq (˜ u1 , u ˜2 , u ˜3 ) . 1). 4): Sji = ∂ui . 7). The theorem is proved. A remark on the orientation. 3) are continuously differentiable. 4) representing the inverse mappings are also continuously differentiable. 9) are continuous functions within the domain U .

This procedure can be repeated infinitely many times producing more and more dense networks in each step. Ultimately (in the continuum limit), one can think the coordinate network to be maximally dense. Such a network consist of two families of lines: the first family is given by the condition ϕ = const, the second one — by the similar condition ρ = const. § 1. SOME EXAMPLES OF CURVILINEAR COORDINATE SYSTEMS. 47 On Fig. 4 exactly two coordinate lines pass through each point of the map: one is from the first family and the other is from the second family.

Let’s consider a plane, choose some point O on it (this will be the pole) and some ray OX coming out from this point. For an arbitrary point A = O of that plane its position is determined by two parameters: the −→ length of its radius-vector ρ = |OA| and the value of the angle ϕ between the ray OX and the radius-vector of the point A. Certainly, one should also choose a positive (counterclockwise) direction to which the angle ϕ is laid (this is equivalent to choosing a preferable orientation on the plane).