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4 Curvilinear coordinate system

In many studies of tokamak plasmas, one need construct a curvilinear coordinate system based on a given magnetic cofiguration in order to make the problem amenable to analytical methods or numerical methods. Specifically, one of the coordinate surfaces of the constructed system will be required to coincide with magnetic surfaces. In addition, the magnetic field lines on a magnetic surface may be required to have some simple property (e.g., being a straight line) via carefully choosing the other two angular coordinates. Next, let us discuss some general properties about coordinates transformation.

  4.1 Coordinates transformation
  4.2 Jacobian
  4.3 Orthogonality relation between two sets of basis vectors
  4.4 An example: (ψ,𝜃,ζ) coordinates
  4.5 Gradient and directional derivative in general coordinates (ψ,𝜃,ζ)
  4.6 Divergence operator in general coordinates (ψ,𝜃,ζ)
  4.7 Laplacian operator in general coordinates (ψ,𝜃,ζ)
  4.8 Curl operator in general coordinates (ψ,𝜃,ζ)
  4.9 Metric tensor for general coordinate system
   Special case: metric tensor for (ψ,𝜃,ϕ) coordinate system

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