## Definition

Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions..

Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$.

Let $\mathbf f$ and $\mathbf g: \R^3 \to \R^3$ be vector-valued functions on $\R^3$:

$\mathbf f := \tuple {\map {f_x} {\mathbf x}, \map {f_y} {\mathbf x}, \map {f_z} {\mathbf x} }$
$\mathbf g := \tuple {\map {g_x} {\mathbf x}, \map {g_y} {\mathbf x}, \map {g_z} {\mathbf x} }$

Let $\nabla \mathbf f$ denote the gradient of $f$.

Then:

$\map \nabla {\mathbf f \cdot \mathbf g} = \paren {\mathbf g \cdot \nabla} \mathbf f + \paren {\mathbf f \cdot \nabla} \mathbf g + \mathbf g \times \paren {\nabla \times \mathbf f} + \mathbf f \times \paren {\nabla \times \mathbf g}$

where:

$\mathbf f \times \mathbf g$ denotes vector cross product
$\mathbf f \cdot \mathbf g$ denotes dot product

## Proof

 $\ds \map \nabla {\mathbf f \cdot \mathbf g}$ $=$ $\ds \map \nabla {f_x g_x + f_y g_y + f_z g_z}$ Definition of Dot Product $\ds$ $=$ $\ds \dfrac {\map \partial {f_x g_x + f_y g_y + f_z g_z} } {\partial x} \mathbf i + \dfrac {\map \partial {f_x g_x + f_y g_y + f_z g_z} } {\partial y} \mathbf j + \dfrac {\map \partial {f_x g_x + f_y g_y + f_z g_z} } {\partial z} \mathbf k$ Definition of Gradient Operator $\ds$ $=$ $\ds \paren {f_x \dfrac {\partial g_x} {\partial x} + \dfrac {\partial f_x} {\partial x} g_x + f_y \dfrac {\partial g_y} {\partial x} + \dfrac {\partial f_y} {\partial x} g_y + f_z \dfrac {\partial g_z} {\partial x} + \dfrac {\partial f_z} {\partial x} g_z} \mathbf i$ Product Rule for Derivatives $\ds$  $\, \ds + \,$ $\ds \paren {f_x \dfrac {\partial g_x} {\partial y} + \dfrac {\partial f_x} {\partial y} g_x + f_y \dfrac {\partial g_y} {\partial y} + \dfrac {\partial f_y} {\partial y} g_y + f_z \dfrac {\partial g_z} {\partial y} + \dfrac {\partial f_z} {\partial y} g_z} \mathbf j$ $\ds$  $\, \ds + \,$ $\ds \paren {f_x \dfrac {\partial g_x} {\partial z} + \dfrac {\partial f_x} {\partial z} g_x + f_y \dfrac {\partial g_y} {\partial z} + \dfrac {\partial f_y} {\partial z} g_y + f_z \dfrac {\partial g_z} {\partial z} + \dfrac {\partial f_z} {\partial z} g_z} \mathbf k$

Then:

 $\ds \mathbf g \times \paren {\nabla \times \mathbf f}$ $=$ $\ds \mathbf g \times \paren {\paren {\dfrac {\partial f_z} {\partial y} - \dfrac {\partial f_y} {\partial z} } \mathbf i + \paren {\dfrac {\partial f_x} {\partial z} - \dfrac {\partial f_z} {\partial x} } \mathbf j + \paren {\dfrac {\partial f_y} {\partial x} - \dfrac {\partial f_x} {\partial y} } \mathbf k}$ Definition of Curl Operator $\ds$ $=$ $\ds \paren {\map {g_y} {\dfrac {\partial f_y} {\partial x} - \dfrac {\partial f_x} {\partial y} } - \map {g_z} {\dfrac {\partial f_x} {\partial z} - \dfrac {\partial f_z} {\partial x} } } \mathbf i$ Definition 1 of Vector Cross Product $\ds$  $\, \ds + \,$ $\ds \paren {\map {g_z} {\dfrac {\partial f_z} {\partial y} - \dfrac {\partial f_y} {\partial z} } - \map {g_x} {\dfrac {\partial f_y} {\partial x} - \dfrac {\partial f_x} {\partial y} } } \mathbf j$ $\ds$  $\, \ds + \,$ $\ds \paren {\map {g_x} {\dfrac {\partial f_x} {\partial z} - \dfrac {\partial f_z} {\partial x} } - \map {g_y} {\dfrac {\partial f_z} {\partial y} - \dfrac {\partial f_y} {\partial z} } } \mathbf k$ $\ds$ $=$ $\ds \paren {g_y \dfrac {\partial f_y} {\partial x} - g_y \dfrac {\partial f_x} {\partial y} - g_z \dfrac {\partial f_x} {\partial z} + g_z \dfrac {\partial f_z} {\partial x} } \mathbf i$ expanding $\ds$  $\, \ds + \,$ $\ds \paren {g_z \dfrac {\partial f_z} {\partial y} - g_z \dfrac {\partial f_y} {\partial z} - g_x \dfrac {\partial f_y} {\partial x} + g_x \dfrac {\partial f_x} {\partial y} } \mathbf j$ $\ds$  $\, \ds + \,$ $\ds \paren {g_x \dfrac {\partial f_x} {\partial z} - g_x \dfrac {\partial f_z} {\partial x} - g_y \dfrac {\partial f_z} {\partial y} + g_y \dfrac {\partial f_y} {\partial z} } \mathbf k$

and similarly:

 $\ds \mathbf f \times \paren {\nabla \times \mathbf g}$ $=$ $\ds \mathbf f \times \paren {\paren {\dfrac {\partial g_z} {\partial y} - \dfrac {\partial g_y} {\partial z} } \mathbf i + \paren {\dfrac {\partial g_x} {\partial z} - \dfrac {\partial g_z} {\partial x} } \mathbf j + \paren {\dfrac {\partial g_y} {\partial x} - \dfrac {\partial g_x} {\partial y} } \mathbf k}$ Definition of Curl Operator $\ds$ $=$ $\ds \paren {\map {f_y} {\dfrac {\partial g_y} {\partial x} - \dfrac {\partial g_x} {\partial y} } - \map {f_z} {\dfrac {\partial g_x} {\partial z} - \dfrac {\partial g_z} {\partial x} } } \mathbf i$ Definition 1 of Vector Cross Product $\ds$  $\, \ds + \,$ $\ds \paren {\map {f_z} {\dfrac {\partial g_z} {\partial y} - \dfrac {\partial g_y} {\partial z} } - \map {f_x} {\dfrac {\partial g_y} {\partial x} - \dfrac {\partial g_x} {\partial y} } } \mathbf j$ $\ds$  $\, \ds + \,$ $\ds \paren {\map {f_x} {\dfrac {\partial g_x} {\partial z} - \dfrac {\partial g_z} {\partial x} } - \map {f_y} {\dfrac {\partial g_z} {\partial y} - \dfrac {\partial g_y} {\partial z} } } \mathbf k$ $\ds$ $=$ $\ds \paren {f_y \dfrac {\partial g_y} {\partial x} - f_y \dfrac {\partial g_x} {\partial y} - f_z \dfrac {\partial g_x} {\partial z} + f_z \dfrac {\partial g_z} {\partial x} } \mathbf i$ expanding $\ds$  $\, \ds + \,$ $\ds \paren {f_z \dfrac {\partial g_z} {\partial y} - f_z \dfrac {\partial g_y} {\partial z} - f_x \dfrac {\partial g_y} {\partial x} + f_x \dfrac {\partial g_x} {\partial y} }\mathbf j$ $\ds$  $\, \ds + \,$ $\ds \paren {f_x \dfrac {\partial g_x} {\partial z} - f_x \dfrac {\partial g_z} {\partial x} - f_y \dfrac {\partial g_z} {\partial y} + f_y \dfrac {\partial g_y} {\partial z} } \mathbf k$

Next:

 $\ds \paren {\mathbf g \cdot \nabla} \mathbf f$ $=$ $\ds \paren {g_x \dfrac \partial {\partial x} + g_y \dfrac \partial {\partial y} + g_z \dfrac \partial {\partial z} }\mathbf f$ Definition of Del Operator, Definition of Dot Product $\ds$ $=$ $\ds g_x \dfrac {\partial f_x} {\partial x} \mathbf i + g_y \dfrac {\partial f_y} {\partial y} \mathbf j + g_z \dfrac {\partial f_z} {\partial z} \mathbf k$ Definition of Gradient Operator

and:

 $\ds \paren {\mathbf f \cdot \nabla} \mathbf g$ $=$ $\ds \paren {f_x \dfrac \partial {\partial x} + f_y \dfrac \partial {\partial y} + f_z \dfrac \partial {\partial z} } \mathbf g$ Definition of Del Operator, Definition of Dot Product $\ds$ $=$ $\ds f_x \dfrac {\partial g_x} {\partial x} \mathbf i + f_y \dfrac {\partial g_y} {\partial y} \mathbf j + f_z \dfrac {\partial g_z} {\partial z} \mathbf k$ Definition of Gradient Operator