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i was hoping someone could either state an article or share some knowledge with a way (if any) to derive the vector calculus "del" relationships. i.e. $$ \nabla \cdot ( \rho \vec{V}) = \rho (\nabla \cdot \vec{V}) + \vec{V} \cdot (\nabla\rho)$$

now i do understand this to be like the product rule but is there any way to derive this in a similar fashion to other vector calculus identities, perhaps using the Kronecker delta or the permutation epsilon?

as an example, i can derive almost any trig identity using $$e^{i \theta}=\cos \theta+i \sin \theta$$ by making changes to [itex]\theta[/itex], perhaps letting [itex]\theta = \alpha + \beta[/itex] and then equating real and imaginary parts and doing a little bit of algebra.

is there anything like this for vector identities (perhaps not a base equation, but a method?)

thanks for your help!