$\nabla \left({\mathbf f \cdot \mathbf g}\right) = \left({\mathbf g \cdot \nabla}\right) \mathbf f + \left({\mathbf f \cdot \nabla}\right) \mathbf g + \mathbf g \times \left({\nabla \times \mathbf f}\right) + \mathbf f \times \left({\nabla \times \mathbf g}\right)$

$\nabla \left({\mathbf f \cdot \mathbf g}\right) | = \nabla \left({f_x g_x + f_y g_y + f_z g_z}\right)$

$= \dfrac {\partial \left({f_x g_x + f_y g_y + f_z g_z}\right)} {\partial x} \mathbf i + \dfrac {\partial \left({f_x g_x + f_y g_y + f_z g_z}\right)} {\partial y} \mathbf j + \dfrac {\partial \left({f_x g_x + f_y g_y + f_z g_z}\right)} {\partial z} \mathbf k$

$= \left({f_x \dfrac {\partial g_x} {\partial x} + \dfrac {\partial f_x} {\partial x} g_x + f_y \dfrac {\partial g_y} {\partial x} + \dfrac {\partial f_y} {\partial x} g_y + f_z \dfrac {\partial g_z} {\partial x} + \dfrac {\partial f_z} {\partial x} g_z}\right) \mathbf i$

$+\left({f_x \dfrac {\partial g_x} {\partial y} + \dfrac {\partial f_x} {\partial y} g_x + f_y \dfrac {\partial g_y} {\partial y} + \dfrac {\partial f_y} {\partial y} g_y + f_z \dfrac {\partial g_z} {\partial y} + \dfrac {\partial f_z} {\partial y} g_z}\right) \mathbf j$

$+ \left({f_x \dfrac {\partial g_x} {\partial z} + \dfrac {\partial f_x} {\partial z} g_x + f_y \dfrac {\partial g_y} {\partial z} + \dfrac {\partial f_y} {\partial z} g_y + f_z \dfrac {\partial g_z} {\partial z} + \dfrac {\partial f_z} {\partial z} g_z}\right) \mathbf k$

$\left({\mathbf g \cdot \nabla}\right) \mathbf f = \left({g_x \dfrac \partial {\partial x} + g_y \dfrac \partial {\partial y} + g_z \dfrac \partial {\partial z} }\right) \mathbf f \left({\mathbf f \cdot \nabla}\right) \mathbf g = \left({f_x \dfrac \partial {\partial x} + f_y \dfrac \partial {\partial y} + f_z \dfrac \partial {\partial z} }\right) \mathbf g$

$=g_x \dfrac {\partial f_x} {\partial x} \mathbf i + g_y \dfrac {\partial f_y} {\partial y} \mathbf j + g_z \dfrac {\partial f_z} {\partial z} \mathbf k$ $= f_x \dfrac {\partial g_x} {\partial x} \mathbf i + f_y \dfrac {\partial g_y} {\partial y} \mathbf j + f_z \dfrac {\partial g_z} {\partial z} \mathbf k$