# Divergence in spherical why take derivative first?

When we are going the divergence thereom we take the derivative before the dot product e.g. in spherical cordinates: $$\nabla\bullet \vec A =(\vec r\bullet \frac{\partial \vec A}{\partial \vec r}+\frac{\vec \theta}{r} \bullet \frac{\partial \vec A}{\partial \vec \theta}+\frac{\vec \phi}{r sin(\theta)} \bullet \frac{\partial \vec A}{\partial \vec \theta})$$ Why is this? i.e. why don't we do the dot product first?

• Do you have some definition of divergence outside of this formula? – Muphrid Feb 3 '15 at 15:31
• @Muphrid not that I can think of, apart from the cartisain form – Quantum spaghettification Feb 3 '15 at 15:37
• So, would a good answer be an explanation why this looks different from the Cartesian form? – Muphrid Feb 3 '15 at 15:56
• @Muphrid I found an another devition: it is the measure of the flux of $\vec A$ out of/ into a small volume sourding that point. However I still cannot see why we do the partial derivatives before? (I don't think an explainsion of why it looks different from the carteisain form would help.) – Quantum spaghettification Feb 5 '15 at 10:24