Formula of the gradient of vector dot product

Question

On Wikipedia in the article "Vector calculus identities" (https://en.m.wikipedia.org/wiki/Vector_calculus_identities) there are the following two formulas for computing the gradient of vector dot product: $\nabla (A \cdot B ) = (A \cdot \nabla) B + (B \cdot \nabla) A + A \times (\nabla \times B) + B \times (\nabla \times A)$ $\nabla (A \cdot B) = \nabla A \cdot B + \nabla B \cdot A$

Could you please explain what is the difference between terms $\nabla A \cdot B$ and $(B \cdot \nabla ) A$?

Answer 1

Gradient of a vector is a tensor of second complexity. Dot product of a second complexity tensor and a first complexity tensor (vector) is not commutative

$$\boldsymbol{\nabla} \boldsymbol{a} \cdot \boldsymbol{b} \neq \, \boldsymbol{b} \cdot \! \boldsymbol{\nabla} \boldsymbol{a}$$

The difference between them is (can be expressed as)

$$\boldsymbol{\nabla} \boldsymbol{a} \cdot \boldsymbol{b} \: - \: \boldsymbol{b} \cdot \! \boldsymbol{\nabla} \boldsymbol{a} = \, \boldsymbol{b} \times \bigl( \boldsymbol{\nabla} \! \times \boldsymbol{a} \bigr)$$

More details are in my answer to another question Gradient of a dot product

If you’re lost in the sea of brackets, here’s some help for you Scalar dot product with directional derivative

Answer 2

I think the confusion stems from the abuse (in my eyes) of $\nabla$ as a vector and not explicitly denoting what the operator is applied to.

Let us translate the expressions into array notation with \begin{align} \nabla"="\begin{bmatrix} \frac{\partial}{\partial x_1} \\ \frac{\partial}{\partial x_2} \\ \frac{\partial}{\partial x_3} \end{bmatrix}\,,\quad A"="\begin{bmatrix} A_1\\A_2\\A_3\end{bmatrix}\,,\quad B"="\begin{bmatrix} B_1\\B_2\\B_3\end{bmatrix}\\ \end{align}

Then what is meant by $B\cdot\nabla$ is the scalar expression $$ B\cdot\nabla=B_1\frac{\partial}{\partial x_1}+ B_2\frac{\partial}{\partial x_2}+ B_3\frac{\partial}{\partial x_3} $$ Now "multiplying" with, i.e. applying the resulting operator on the vector $A$ gives \begin{align} (B\cdot\nabla)A &=\left(B_1\frac{\partial}{\partial x_1}+ B_2\frac{\partial}{\partial x_2}+ B_3\frac{\partial}{\partial x_3}\right)\begin{bmatrix} A_1\\A_2\\A_3\end{bmatrix}\\ &=\begin{bmatrix} B_1\frac{\partial{A_1}}{\partial x_1}+B_2\frac{\partial{A_1}}{\partial x_2} +B_3\frac{\partial{A_1}}{\partial x_3}\\ B_1\frac{\partial{A_2}}{\partial x_1}+B_2\frac{\partial{A_2}}{\partial x_2} +B_3\frac{\partial{A_2}}{\partial x_3}\\ B_1\frac{\partial{A_3}}{\partial x_1}+B_2\frac{\partial{A_3}}{\partial x_2} +B_3\frac{\partial{A_3}}{\partial x_3}\\ \end{bmatrix} \end{align}