Gradient of Dot Product

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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
\(\ds \) \(=\) \(\ds \paren {g_x \dfrac {\partial f_x} {\partial x} + g_y \dfrac {\partial f_x} {\partial y} + g_z \dfrac {\partial f_x} {\partial z} } \mathbf i\) Definition of Gradient Operator
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {g_x \dfrac {\partial f_y} {\partial x} + g_y \dfrac {\partial f_y} {\partial y} + g_z \dfrac {\partial f_y} {\partial z} } \mathbf j\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {g_x \dfrac {\partial f_z} {\partial x} + g_y \dfrac {\partial f_z} {\partial y} + g_z \dfrac {\partial f_z} {\partial z} } \mathbf k\)


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
\(\ds \) \(=\) \(\ds \paren {f_x \dfrac {\partial g_x} {\partial x} + f_y \dfrac {\partial g_x} {\partial y} + f_z \dfrac {\partial g_x} {\partial z} } \mathbf i\) Definition of Gradient Operator
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {f_x \dfrac {\partial g_y} {\partial x} + f_y \dfrac {\partial g_y} {\partial y} + f_z \dfrac {\partial g_y} {\partial z} } \mathbf j\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {f_x \dfrac {\partial g_z} {\partial x} + f_y \dfrac {\partial g_z} {\partial y} + f_z \dfrac {\partial g_z} {\partial z} } \mathbf k\)

Combining in pairs:

\(\ds \paren {\mathbf g \cdot \nabla} \mathbf f + \mathbf g \times \paren {\nabla \times \mathbf f}\) \(=\) \(\ds \paren {g_y \dfrac {\partial f_y} {\partial x} - g_y \dfrac {\partial f_x} {\partial y} - g_z \dfrac {\partial f_x} {\partial z} + g_z \dfrac {\partial f_z} {\partial x} } \mathbf i + \paren {g_x \dfrac {\partial f_x} {\partial x} + g_y \dfrac {\partial f_x} {\partial y} + g_z \dfrac {\partial f_x} {\partial z} } \mathbf i\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {g_z \dfrac {\partial f_z} {\partial y} - g_z \dfrac {\partial f_y} {\partial z} - g_x \dfrac {\partial f_y} {\partial x} + g_x \dfrac {\partial f_x} {\partial y} } \mathbf j + \paren {g_x \dfrac {\partial f_y} {\partial x} + g_y \dfrac {\partial f_y} {\partial y} + g_z \dfrac {\partial f_y} {\partial z} } \mathbf j\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {g_x \dfrac {\partial f_x} {\partial z} - g_x \dfrac {\partial f_z} {\partial x} - g_y \dfrac {\partial f_z} {\partial y} + g_y \dfrac {\partial f_y} {\partial z} } \mathbf k + \paren{g_x \dfrac {\partial f_z} {\partial x} + g_y \dfrac {\partial f_z} {\partial y} + g_z \dfrac {\partial f_z} {\partial z} } \mathbf k\)
\(\ds \) \(=\) \(\ds \paren {g_x \dfrac {\partial f_x} {\partial x} + g_y \dfrac {\partial f_y} {\partial x} + g_z \dfrac {\partial f_z} {\partial x} } \mathbf i\) Canceling
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {g_x \dfrac {\partial f_x} {\partial y} + g_y \dfrac {\partial f_y} {\partial y} + g_z \dfrac {\partial f_z} {\partial y} } \mathbf j\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {g_x \dfrac {\partial f_x} {\partial z} + g_y \dfrac {\partial f_y} {\partial z} + g_z \dfrac {\partial f_z} {\partial z} } \mathbf k\)

and:

\(\ds \paren {\mathbf f \cdot \nabla} \mathbf g + \mathbf f \times \paren {\nabla \times \mathbf g}\) \(=\) \(\ds \paren {f_y \dfrac {\partial g_y} {\partial x} - f_y \dfrac {\partial g_x} {\partial y} - f_z \dfrac {\partial g_x} {\partial z} + f_z \dfrac {\partial g_z} {\partial x} } \mathbf i + \paren {f_x \dfrac {\partial g_x} {\partial x} + f_y \dfrac {\partial g_x} {\partial y} + f_z \dfrac {\partial g_x} {\partial z} } \mathbf i\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {f_z \dfrac {\partial g_z} {\partial y} - f_z \dfrac {\partial g_y} {\partial z} - f_x \dfrac {\partial g_y} {\partial x} + f_x \dfrac {\partial g_x} {\partial y} }\mathbf j + \paren {f_x \dfrac {\partial g_y} {\partial x} + f_y \dfrac {\partial g_y} {\partial y} + f_z \dfrac {\partial g_y} {\partial z} } \mathbf j\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {f_x \dfrac {\partial g_x} {\partial z} - f_x \dfrac {\partial g_z} {\partial x} - f_y \dfrac {\partial g_z} {\partial y} + f_y \dfrac {\partial g_y} {\partial z} } \mathbf k + \paren {f_x \dfrac {\partial g_z} {\partial x} + f_y \dfrac {\partial g_z} {\partial y} + f_z \dfrac {\partial g_z} {\partial z} } \mathbf k\)
\(\ds \) \(=\) \(\ds \paren {f_x \dfrac {\partial g_x} {\partial x} + f_y \dfrac {\partial g_y} {\partial x} + f_z \dfrac {\partial g_z} {\partial x} } \mathbf i\) Canceling
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {f_x \dfrac {\partial g_x} {\partial y} + f_y \dfrac {\partial g_y} {\partial y} + f_z \dfrac {\partial g_z} {\partial y} } \mathbf j\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {f_x \dfrac {\partial g_x} {\partial z} + f_y \dfrac {\partial g_y} {\partial z} + f_z \dfrac {\partial g_z} {\partial z} } \mathbf k\)

Finally combining all four terms:

\(\ds \paren {\mathbf g \cdot \nabla} \mathbf f + \paren {\mathbf f \cdot \nabla} \mathbf g + \mathbf g \times \paren {\nabla \times \mathbf f} + \mathbf f \times \paren {\nabla \times \mathbf g}\) \(=\) \(\ds \paren {f_x \dfrac {\partial g_x} {\partial x} + \dfrac {\partial f_x} {\partial x} g_x + f_y \dfrac {\partial g_y} {\partial x} + \dfrac {\partial f_y} {\partial x} g_y + f_z \dfrac {\partial g_z} {\partial x} + \dfrac {\partial f_z} {\partial x} g_z} \mathbf i\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {f_x \dfrac {\partial g_x} {\partial y} + \dfrac {\partial f_x} {\partial y} g_x + f_y \dfrac {\partial g_y} {\partial y} + \dfrac {\partial f_y} {\partial y} g_y + f_z \dfrac {\partial g_z} {\partial y} + \dfrac {\partial f_z} {\partial y} g_z} \mathbf j\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {f_x \dfrac {\partial g_x} {\partial z} + \dfrac {\partial f_x} {\partial z} g_x + f_y \dfrac {\partial g_y} {\partial z} + \dfrac {\partial f_y} {\partial z} g_y + f_z \dfrac {\partial g_z} {\partial z} + \dfrac {\partial f_z} {\partial z} g_z} \mathbf k\)

$\blacksquare$



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