Curvilinear Coordinates - Vector and Tensor Algebra in Three-dimensional Curvilinear Coordinates - Vector Operations

Vector Operations

  1. Dot product: The scalar product of two vectors in curvilinear coordinates is
    \mathbf{u}\cdot\mathbf{v} = u^iv_i = u_iv^i = g_{ij}u^iv^j = g^{ij}u_iv_j
  2. Cross product: The cross product of two vectors is given by
    \mathbf{u}\times\mathbf{v} = \epsilon_{ijk}{u}_j{v}_k\mathbf{e}_i
    where is the permutation symbol and is a Cartesian basis vector. In curvilinear coordinates, the equivalent expression is
    \mathbf{u}\times\mathbf{v} = u^mv^n\mathbf{b}^s = \mathcal{E}_{smn}u^mv^n\mathbf{b}^s
    where is the third-order alternating tensor.

Read more about this topic:  Curvilinear Coordinates, Vector and Tensor Algebra in Three-dimensional Curvilinear Coordinates

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