Formula of the gradient of vector dot product 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 $(B \cdot \nabla ) A$?
 A: 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}
A: 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
