$\newcommand{\bl}[1]{\boldsymbol{#1}} \newcommand{\e}{\bl=} \newcommand{\p}{\bl+} \newcommand{\m}{\bl-} \newcommand{\gr}{\bl>} \newcommand{\les}{\bl<} \newcommand{\greq}{\bl\ge} \newcommand{\leseq}{\bl\le} \newcommand{\plr}[1]{\left(#1\right)} \newcommand{\blr}[1]{\left[#1\right]} \newcommand{\lara}[1]{\langle#1\rangle} \newcommand{\lav}[1]{\langle#1|} \newcommand{\vra}[1]{|#1\rangle} \newcommand{\lavra}[2]{\langle#1|#2\rangle} \newcommand{\lavvra}[3]{\langle#1|\,#2\,|#3\rangle} \newcommand{\vp}{\vphantom{\dfrac{a}{b}}} \newcommand{\hp}[1]{\hphantom{#1}} \newcommand{\x}{\bl\times} \newcommand{\qqlraqq}{\qquad\bl{-\!\!\!-\!\!\!-\!\!\!\longrightarrow}\qquad} \newcommand{\qqLraqq}{\qquad\boldsymbol{\e\!\e\!\e\!\e\!\Longrightarrow}\qquad} \newcommand{\tl}[1]{\tag{#1}\label{#1}} $
Hint :
\begin{equation} \bl\upsilon \bl\cdot \bl\nabla\e \begin{bmatrix} v_1\vphantom{\dfrac{\tfrac{a}{b}}{\tfrac{a}{b}}}\\ v_2\vphantom{\dfrac{\tfrac{a}{b}}{\tfrac{a}{b}}}\\ v_3\vphantom{\dfrac{\tfrac{a}{b}}{\tfrac{a}{b}}}\\ \end{bmatrix} \begin{bmatrix} \dfrac{\partial}{\partial x_1} & \dfrac{\partial}{\partial x_2} & \dfrac{\partial}{\partial x_3} \vphantom{\dfrac{a}{\dfrac{a}{b}}} \end{bmatrix} \e \begin{bmatrix} v_1\dfrac{\partial}{\partial x_1} & v_1\dfrac{\partial}{\partial x_2} & v_1\dfrac{\partial}{\partial x_3} \vphantom{\dfrac{a}{\dfrac{a}{b}}}\\ v_2\dfrac{\partial}{\partial x_1} & v_2\dfrac{\partial}{\partial x_2} & v_2\dfrac{\partial}{\partial x_3} \vphantom{\dfrac{a}{\dfrac{a}{b}}}\\ v_3\dfrac{\partial}{\partial x_1} & v_3\dfrac{\partial}{\partial x_2} & v_3\dfrac{\partial}{\partial x_3} \vphantom{\dfrac{a}{\dfrac{a}{b}}} \end{bmatrix} \tl{01} \end{equation}\begin{align} \overbrace{ \begin{bmatrix} \mathrm a_1\dfrac{\partial \rm b_1}{\partial x_1}\boldsymbol{+}\mathrm a_2\dfrac{\partial \rm b_1}{\partial x_2}\boldsymbol{+}\mathrm a_3\dfrac{\partial \rm b_1}{\partial x_3}\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}}\\ \mathrm a_1\dfrac{\partial \rm b_2}{\partial x_1}\boldsymbol{+}\mathrm a_2\dfrac{\partial \rm b_2}{\partial x_2}\boldsymbol{+}\mathrm a_3\dfrac{\partial \rm b_2}{\partial x_3}\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}}\\ \mathrm a_1\dfrac{\partial \rm b_3}{\partial x_1}\boldsymbol{+}\mathrm a_2\dfrac{\partial \rm b_3}{\partial x_2}\boldsymbol{+}\mathrm a_3\dfrac{\partial \rm b_3}{\partial x_3}\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}} \end{bmatrix}}^{\left(\mathbf{a}\boldsymbol{\cdot}\boldsymbol{\nabla}\right)\mathbf{b}} &\boldsymbol{=} \begin{bmatrix} \mathbf{a}\boldsymbol{\cdot}\boldsymbol{\nabla}\mathrm b_1\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}}\\ \mathbf{a}\boldsymbol{\cdot}\boldsymbol{\nabla}\mathrm b_2\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}}\\ \mathbf{a}\boldsymbol{\cdot}\boldsymbol{\nabla}\mathrm b_3\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}} \end{bmatrix} \boldsymbol{=} \overbrace{ \begin{pmatrix} \mathrm a_1\dfrac{\partial \hphantom{\rm b_1}}{\partial x_1}\boldsymbol{+}\mathrm a_2\dfrac{\partial \hphantom{\rm b_1}}{\partial x_2}\boldsymbol{+}\mathrm a_3\dfrac{\partial \hphantom{\rm b_1}}{\partial x_3}\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}} \end{pmatrix}}^{\left(\mathbf{a}\boldsymbol{\cdot}\boldsymbol{\nabla}\right)} \overbrace{ \begin{bmatrix} \mathrm b_1\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}}\\ \mathrm b_2\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}}\\ \mathrm b_3\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}} \end{bmatrix}}^{\mathbf{b}} \label{vecform-12}\\ \overbrace{ \begin{bmatrix} \mathrm b_1\dfrac{\partial \rm a_1}{\partial x_1}\boldsymbol{+}\mathrm b_2\dfrac{\partial \rm a_1}{\partial x_2}\boldsymbol{+}\mathrm b_3\dfrac{\partial \rm a_1}{\partial x_3}\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}}\\ \mathrm b_1\dfrac{\partial \rm a_2}{\partial x_1}\boldsymbol{+}\mathrm b_2\dfrac{\partial \rm a_2}{\partial x_2}\boldsymbol{+}\mathrm b_3\dfrac{\partial \rm a_2}{\partial x_3}\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}}\\ \mathrm b_1\dfrac{\partial \rm a_3}{\partial x_1}\boldsymbol{+}\mathrm b_2\dfrac{\partial \rm a_3}{\partial x_2}\boldsymbol{+}\mathrm b_3\dfrac{\partial \rm a_3}{\partial x_3}\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}} \end{bmatrix}}^{\left(\mathbf{b}\boldsymbol{\cdot}\boldsymbol{\nabla}\right)\mathbf{a}} &\boldsymbol{=} \begin{bmatrix} \mathbf{b}\boldsymbol{\cdot}\boldsymbol{\nabla}\mathrm a_1\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}}\\ \mathbf{b}\boldsymbol{\cdot}\boldsymbol{\nabla}\mathrm a_2\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}}\\ \mathbf{b}\boldsymbol{\cdot}\boldsymbol{\nabla}\mathrm a_3\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}} \end{bmatrix} \boldsymbol{=} \overbrace{ \begin{pmatrix} \mathrm b_1\dfrac{\partial \hphantom{\rm b_1}}{\partial x_1}\boldsymbol{+}\mathrm b_2\dfrac{\partial \hphantom{\rm b_1}}{\partial x_2}\boldsymbol{+}\mathrm b_3\dfrac{\partial \hphantom{\rm b_1}}{\partial x_3}\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}} \end{pmatrix}}^{\left(\mathbf{b}\boldsymbol{\cdot}\boldsymbol{\nabla}\right)} \overbrace{ \begin{bmatrix} \mathrm a_1\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}}\\ \mathrm a_2\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}}\\ \mathrm a_3\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}} \end{bmatrix}}^{\mathbf{a}} \label{vecform-13} \end{align}