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)?