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Toroidal coordinates: Difference between revisions
→General Curvilinear Coordinates
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Given the spatial dependence of a coordinate set <math>(\psi(\mathbf{x}),\theta(\mathbf{x}),\phi(\mathbf{x}))</math> | 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 | we can calculate the contravariant basis vectors | ||
:<math> | |||
\mathbf{e}^i = \{\nabla\psi, \nabla\theta, \nabla\phi\} | \mathbf{e}^i = \{\nabla\psi, \nabla\theta, \nabla\phi\} | ||
</math | </math> | ||
and the dual covariant basis defined as | and the dual covariant basis defined as | ||
:<math> | |||
\mathbf{e}_i= \frac{\partial\mathbf{x}}{\partial{u^i}} | \mathbf{e}_i= \frac{\partial\mathbf{x}}{\partial{u^i}} | ||
\to | \to | ||
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= \frac{\mathbf{e}^j\times\mathbf{e}^k}{|\mathbf{e}^i\cdot\mathbf{e}^j\times\mathbf{e}^k|} | = \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 ~, | = \sqrt{g}\;\mathbf{e}^j\times\mathbf{e}^k ~, | ||
</math | </math> | ||
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. | 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 | Any vector field <math>\mathbf{B}</math> can be represented as | ||
:<math> | |||
\mathbf{B} | \mathbf{B} | ||
= (\mathbf{B}\cdot\mathbf{e}^i)\mathbf{e}_i | = (\mathbf{B}\cdot\mathbf{e}^i)\mathbf{e}_i | ||
= B^i\mathbf{e}_i | = B^i\mathbf{e}_i | ||
</math | </math> | ||
or | or | ||
:<math> | |||
\mathbf{B} | \mathbf{B} | ||
= (\mathbf{B}\cdot\mathbf{e}_i)\mathbf{e}^i | = (\mathbf{B}\cdot\mathbf{e}_i)\mathbf{e}^i | ||
= B_i\mathbf{e}^i ~. | = B_i\mathbf{e}^i ~. | ||
</math | </math> | ||
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 | 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 | ||
:<math> | |||
g_{ij} | g_{ij} | ||
= \mathbf{e}_i\cdot\mathbf{e}_j | = \mathbf{e}_i\cdot\mathbf{e}_j | ||
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g^j_i | g^j_i | ||
= \mathbf{e}_i\cdot\mathbf{e}^j = \delta_i^j ~. | = \mathbf{e}_i\cdot\mathbf{e}^j = \delta_i^j ~. | ||
</math | </math> | ||
=== Jacobian === | === Jacobian === | ||
The Jacobian of the coordinate transformation <math>\mathbf{x}(\psi, \theta, \phi)</math> is defined as | The Jacobian of the coordinate transformation <math>\mathbf{x}(\psi, \theta, \phi)</math> is defined as | ||
:<math> | |||
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}} | 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}} | ||
</math | </math> | ||
and that of the inverse transformation | and that of the inverse transformation | ||
:<math> | |||
J^{-1} = \det\left(\frac{\partial(\psi,\theta,\phi)}{\partial(x,y,z)}\right) = \nabla{\psi}\cdot\nabla{\theta} \times \nabla{\phi} | J^{-1} = \det\left(\frac{\partial(\psi,\theta,\phi)}{\partial(x,y,z)}\right) = \nabla{\psi}\cdot\nabla{\theta} \times \nabla{\phi} | ||
</math | </math> | ||
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> | 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> | ||