Toroidal coordinates: Difference between revisions

Given the spatial dependence of a coordinate set <math>(\psi(\mathbf{x}),\theta(\mathbf{x}),\phi(\mathbf{x}))</math>

we can calculate the contravariant basis vectors  

\mathbf{e}^i = \{\nabla\psi, \nabla\theta, \nabla\phi\}  

and the dual covariant basis defined as

\mathbf{e}_i= \frac{\partial\mathbf{x}}{\partial{u^i}}

\to

= \frac{\mathbf{e}^j\times\mathbf{e}^k}{|\mathbf{e}^i\cdot\mathbf{e}^j\times\mathbf{e}^k|}

= \sqrt{g}\;\mathbf{e}^j\times\mathbf{e}^k ~,

where <math>(i,j,k)</math> are cyclic permutations of <math>(1,2,3)</math> and we have used the notation <math>(u^1, u^2, u^3) = (\psi,\theta,\phi)</math>. The Jacobian <math>\sqrt{g}</math> is defined below.

Any vector field <math>\mathbf{B}</math> can be represented as

\mathbf{B}  

= (\mathbf{B}\cdot\mathbf{e}^i)\mathbf{e}_i

= B^i\mathbf{e}_i

or

\mathbf{B}  

= (\mathbf{B}\cdot\mathbf{e}_i)\mathbf{e}^i

= B_i\mathbf{e}^i ~.

In particular any basis vector <math>\mathbf{e}_i = (\mathbf{e}_i\cdot\mathbf{e}_j)\mathbf{e}^j</math>. The metric tensor is defined as

g_{ij}  

= \mathbf{e}_i\cdot\mathbf{e}_j

g^j_i   

= \mathbf{e}_i\cdot\mathbf{e}^j = \delta_i^j ~.

=== Jacobian ===  

The Jacobian of the coordinate transformation <math>\mathbf{x}(\psi, \theta, \phi)</math> is defined as

J = \det\left(\frac{\partial(x,y,z)}{\partial(\psi,\theta,\phi)}\right) = \frac{\partial\mathbf{x}}{\partial{\psi}}\cdot\frac{\partial\mathbf{x}}{\partial{\theta}} \times \frac{\partial\mathbf{x}}{\partial{\phi}}

and that of the inverse transformation  

J^{-1} = \det\left(\frac{\partial(\psi,\theta,\phi)}{\partial(x,y,z)}\right) = \nabla{\psi}\cdot\nabla{\theta} \times \nabla{\phi}

It can be seen that<ref>W.D. D'haeseleer, ''Flux coordinates and magnetic field structure: a guide to a fundamental tool of plasma theory'', Springer series in computational physics, Springer-Verlag (1991) ISBN 3540524193</ref> <math>g \equiv \det(g_{ij}) = J^2 \Rightarrow J = \sqrt{g}</math>