Toroidal coordinates: Difference between revisions
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== General Curvilinear Coordinates == | == General Curvilinear Coordinates == | ||
Here we briefly review the basic definitions of a general curvilinear coordinate system for later convenience when discussing toroidal flux coordinates and magnetic coordinates. | |||
=== Function coordinates and basis vector === | |||
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> | |||
\mathbf{e}^i = \{\nabla\psi, \nabla\theta, \nabla\phi\} | |||
</math></center> | |||
and the dual covariant basis defined as | |||
<center><math> | |||
\mathbf{e}_i= \frac{\partial\mathbf{x}}{\partial{u^i}} | |||
\to | |||
\mathbf{e}_i\cdot\mathbf{e}^j | |||
= \delta_{i}^{j} \to \mathbf{e}_i | |||
= \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 ~, | |||
</math></center> | |||
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 | |||
<center><math> | |||
\mathbf{B} | |||
= (\mathbf{B}\cdot\mathbf{e}^i)\mathbf{e}_i | |||
= B^i\mathbf{e}_i | |||
</math></center> | |||
or | |||
<center><math> | |||
\mathbf{B} | |||
= (\mathbf{B}\cdot\mathbf{e}_i)\mathbf{e}^i | |||
= B_i\mathbf{e}^i ~. | |||
</math></center> | |||
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> | |||
g_{ij} | |||
= \mathbf{e}_i\cdot\mathbf{e}_j | |||
\; ; \; | |||
g^{ij} | |||
= \mathbf{e}^i\cdot\mathbf{e}^j | |||
\; ; \; | |||
g^j_i | |||
= \mathbf{e}_i\cdot\mathbf{e}^j = \delta_i^j ~. | |||
</math></center> | |||
=== Jacobian === | |||
The Jacobian of the coordinate transformation <math>\mathbf{x}(\psi, \theta, \phi)</math> is defined as | |||
<center><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}} | |||
</math></center> | |||
and that of the inverse transformation | |||
<center><math> | |||
J^{-1} = \det\left(\frac{\partial(\psi,\theta,\phi)}{\partial(x,y,z)}\right) = \nabla{\psi}\cdot\nabla{\theta} \times \nabla{\phi} | |||
</math></center> | |||
It can be seen that <math>g \equiv \det(g_{ij}) = J^2 \Rightarrow J = \sqrt{g}</math> | |||
== Magnetic == | == Magnetic == | ||