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In a few continuum classes I have seen indicial notation used to derive relations in Gibbs notation. However, Gibbs notation is valid for all coordinates while indicial notation is valid only for Cartesian coordinates. Is it safe to arrive at a relation in Gibbs notation based upon a proof in indicial notation? Is the result valid for all coordinate systems?

For example:

Showing

$\mathbf{u}\cdot \nabla \mathbf{u} = \frac{1}{2}\nabla(\mathbf{u}\cdot\mathbf{u})-\mathbf{u}\times (\nabla \times \mathbf{u})$

By using indicial/Einstein notation. I'm not at all concerned with showing the result above - just the generality of the approach of using Einstein notation in the middle.

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You can always prove an identity using whatever coordinates you want. That is, an equation that is stated independently of coordinates is true if it is verified using any specific set of coordinates. –  Eric O. Korman Nov 7 '11 at 3:02
    
Thank you, that would answer the question. I wasn't sure that this was always true. –  ccook Nov 7 '11 at 18:14

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As posted by Eric in comments:

You can always prove an identity using whatever coordinates you want. That is, an equation that is stated independently of coordinates is true if it is verified using any specific set of coordinates.

Its also worth noting that indicial notation can be used in curvilinear coordinates in general.

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