Or more generally, how does one derive the dot product in alternative coordinate systems? Since the unit vectors in polar coordinates are defined as $\hat{\theta}$ and $\hat{r}$. How can the dot product be defined when using these basis vectors without converting them to the cartesian unit vectors?
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Does this answer your question? – ViktorStein Jan 03 '20 at 10:54
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Unfortunately i am not familiar with metrics. – TWPebble Jan 03 '20 at 10:56
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I like to not limit myself to unit length basis vectors. It makes all kinds of formulas look much nicer in the end. It's more natural to let the basis vector $\hat\theta$ at point $(r, \theta)$ have length $r$. It's somewhat like using radians rather than degrees for angle measures because it makes combining calculus and trigonometry much nicer. – Arthur Jan 03 '20 at 11:51
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Observe that $\hat r$ and $\hat\theta$ are always orthogonal. That aside, I think you’d be better off applying to chain rule to the definition of $\operatorname{div}$ in the standard basis instead of trying to interpret the mnemonic notation $\nabla\cdot f$ as a “real” dot product. – amd Jan 03 '20 at 21:01
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1This might also be helpful, although it derives $\operatorname{grad}$ instead of $\operatorname{div}$. – amd Jan 03 '20 at 21:11