Editing Jaycall/KillingVectorApproach (section)

==Metric Components, Orthogonality Condition, and Scale Factors==

For any coordinate system it is possible to express the components of the metric in terms of the partial derivatives defining the transformation from any other coordinate system.  Written in terms of the transformation from cylindrical coordinates, each metric component is

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g_{ij} = \left( \partial_i R \right) \left( \partial_j R \right) + \left( \partial_i z \right) \left( \partial_j z \right) + R^2 \left( \partial_i \phi \right) \left( \partial_j \phi \right) .
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The orthogonality condition is that the metric be diagonal.  That is, each off-diagonal component of the metric must equal zero.

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\left( \partial_i R \right) \left( \partial_j R \right) + \left( \partial_i z \right) \left( \partial_j z \right) + R^2 \left( \partial_i \phi \right) \left( \partial_j \phi \right) = 0_{ij}, \ \ \ \ \ i \neq j .
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If the coordinates should be orthogonal, then scale factors can also be defined in terms of the diagonal components of the metric such that <math>g_{ii} = {h_i}^2</math>.  This results in the following definition of the scale factors if the orthogonality condition is met.

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h_i = \left[ \left( \partial_i R \right)^2 + \left( \partial_i z \right)^2 + R^2 \left( \partial_i \phi \right)^2 \right]^{1/2} .
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