Gradient of a dot product The wikipedia formula for the gradient of a dot product is given as
$$\nabla(a\cdot b) = (a\cdot\nabla)b +(b\cdot \nabla)a + a\times(\nabla\times b)+ b\times (\nabla \times a)$$
However, I also found the formula $$\nabla(a\cdot b) = (\nabla a)\cdot b + (\nabla b)\cdot a  $$
So... what is going on here?  The second formula seems much easier.  Are these equivalent?

Since there are not many signs that one may easily use in mathematical notations, many of these symbols are overloaded. In particular, the dot "$\cdot$" is used in the first formula to denote the scalar product of two vector fields in $\mathbb R^3$ called $a$ and $b$, while in the second formula it denotes the usual product of the functions $a$ and $b$. This means that both formulae are valid, but each one is so only in its proper context.
(It is scary to see that the answers and comments that were wrong collected the most votes!)
