Toroidal coordinates: Difference between revisions

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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  
<center><math>
:<math>
\mathbf{e}^i = \{\nabla\psi, \nabla\theta, \nabla\phi\}  
\mathbf{e}^i = \{\nabla\psi, \nabla\theta, \nabla\phi\}  
</math></center>
</math>
and the dual covariant basis defined as
and the dual covariant basis defined as
<center><math>
:<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></center>
</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
<center><math>
:<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></center>
</math>
or
or
<center><math>
:<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></center>
</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
<center><math>
:<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></center>
</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
<center><math>
:<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></center>
</math>
and that of the inverse transformation  
and that of the inverse transformation  
<center><math>
:<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></center>
</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>


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