Flux coordinates: Difference between revisions
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</math> | </math> | ||
=== Jacobian === | === Length element === | ||
The squared length of a differential change in the position vector is | |||
:<math> | |||
ds^2 = d\mathbf{x}\cdot d\mathbf{x} = \left(\frac{\partial\mathbf{x}}{\partial{u^i}}du^i\right)\cdot \left(\frac{\partial\mathbf{x}}{\partial{u^j}}du^j\right) = g_{ij}du^idu^j | |||
</math> | |||
=== Jacobian and volume element=== | |||
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> | :<math> | ||
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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 name='Dhaeseleer'></ref> <math>g \equiv \det(g_{ij}) = J^2 \Rightarrow J = \sqrt{g}</math> | It can be seen that <ref name='Dhaeseleer'></ref> <math>g \equiv \det(g_{ij}) = J^2 \Rightarrow J = \sqrt{g}</math>. | ||
The differential volume element in the curvilinear coordinates is <math>d\mathcal{V} = \sqrt{g} d\psi d\theta d\phi</math> | |||
=== Some surface elements === | === Some surface elements === | ||