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Hello i'm an absolute beginner and i have some problem with del operator in polar coordinate and any help would be appreciated.

del operator in polar coordinate is defined as : $ ∇=(\frac{d}{dr}, \frac{1}{r} \frac{d }{d \theta}) $
assume a vector field F=(u,v) then the divergence of F should be : $∇.F=(\frac{d}{dr}, \frac{1}{r} \frac{d }{d \theta}).(u,v)=\frac{du}{dr}+\frac{1}{r} \frac{d v }{d \theta}=\frac{1}{r}(r\frac{du}{dr}+ \frac{d v }{d \theta})$ but i see $∇.F=\frac{1}{r}(\frac{d(ru)}{dr}+ \frac{d v }{d \theta})$ every where.

my question is some thing wrong with my product or it shouldn't be treated as a dot product (i read some people telling that divergence is not a real dot product)?

Tbt
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  • Hint . You should use the chain rule to express $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$ as linear combinations of $\frac{\partial}{\partial r}$ and $\frac{\partial}{\partial \theta}$. The "del" operator in polar coordinates is not what you were thinking. – Kurt G. Apr 17 '22 at 15:40
  • You have to note that the basis vectors change in polar coordinates. So $\mathrm{div}:F =\nabla\cdot F$ is only true in cartesian coordinates. You can see the general formula here. – Silas Apr 17 '22 at 16:24

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