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Boozer coordinates: Difference between revisions
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→Form of the Jacobian for Boozer coordinates
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== Form of the Jacobian for Boozer coordinates == | == Form of the Jacobian for Boozer coordinates == | ||
Multiplying the covariant representation of the magnetic field by <math>\mathbf{B}\cdot</math> we get | Multiplying the [[Flux coordinates#Covariant Form|covariant representation]] of the magnetic field by <math>\mathbf{B}\cdot</math> we get | ||
:<math> | :<math> | ||
B^2 = \mathbf{B}\cdot\nabla\chi = \frac{I_{tor}}{2\pi}\mathbf{B}\cdot\nabla\theta + \frac{I_{pol}^d}{2\pi}\mathbf{B}\cdot\nabla\phi + \mathbf{B}\cdot\nabla\tilde\chi~. | B^2 = \mathbf{B}\cdot\nabla\chi = \frac{I_{tor}}{2\pi}\mathbf{B}\cdot\nabla\theta + \frac{I_{pol}^d}{2\pi}\mathbf{B}\cdot\nabla\phi + \mathbf{B}\cdot\nabla\tilde\chi~. | ||
</math> | </math> | ||
Now, using the known form of the contravariant components of the magnetic field for a magnetic coordinate system we get | Now, using the known form of the [[Flux coordinates#Magnetic coordinates|contravariant components]] of the magnetic field for a magnetic coordinate system we get | ||
:<math> | :<math> | ||
\mathbf{B}\cdot\nabla\tilde\chi = B^2 - \frac{1}{4\pi^2\sqrt{g}}\left(I_{tor}\Psi_{pol}' + I_{pol}^d\Psi_{tor}' \right)~, | \mathbf{B}\cdot\nabla\tilde\chi = B^2 - \frac{1}{4\pi^2\sqrt{g}}\left(I_{tor}\Psi_{pol}' + I_{pol}^d\Psi_{tor}' \right)~, | ||