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?
Best Answer
They are basically the same. For the first identity, you could refer to my proof using Levi-Civita notation here. And for the second, you should know that $\nabla a=\left(\frac{\partial a_j}{\partial x_i}\right)=\left(\frac{\partial a_i}{\partial x_j}\right)^T$ is a matrix and dot product is exactly matrix multiplication. So the proof is $$(\nabla a)\cdot b+(\nabla b)\cdot a=\left(\frac{\partial a_j}{\partial x_i}b_j+\frac{\partial b_j}{\partial x_i}a_j\right)e_i=\frac{\partial(a_jb_j)}{\partial x_i}e_i=\nabla(a\cdot b)$$