Gradient of Dot Product

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Definition

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

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

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

$\mathbf f := \left({f_x \left({\mathbf x}\right), f_y \left({\mathbf x}\right), f_z \left({\mathbf x}\right)}\right)$
$\mathbf g := \left({g_x \left({\mathbf x}\right), g_y \left({\mathbf x}\right), g_z \left({\mathbf x}\right)}\right)$


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


Then:

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

where:

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


Proof

\(\displaystyle \nabla \left({\mathbf f \cdot \mathbf g}\right)\) \(=\) \(\displaystyle \nabla \left({f_x g_x + f_y g_y + f_z g_z}\right)\) Definition 1 of Dot Product
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\partial \left({f_x g_x + f_y g_y + f_z g_z}\right)} {\partial x} \mathbf i + \dfrac {\partial \left({f_x g_x + f_y g_y + f_z g_z}\right)} {\partial y} \mathbf j + \dfrac {\partial \left({f_x g_x + f_y g_y + f_z g_z}\right)} {\partial z} \mathbf k\) Definition of Gradient Operator
\(\displaystyle \) \(=\) \(\displaystyle \left({f_x \dfrac {\partial g_x} {\partial x} + \dfrac {\partial f_x} {\partial x} g_x + f_y \dfrac {\partial g_y} {\partial x} + \dfrac {\partial f_y} {\partial x} g_y + f_z \dfrac {\partial g_z} {\partial x} + \dfrac {\partial f_z} {\partial x} g_z}\right) \mathbf i\) Product Rule for Derivatives
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \left({f_x \dfrac {\partial g_x} {\partial y} + \dfrac {\partial f_x} {\partial y} g_x + f_y \dfrac {\partial g_y} {\partial y} + \dfrac {\partial f_y} {\partial y} g_y + f_z \dfrac {\partial g_z} {\partial y} + \dfrac {\partial f_z} {\partial y} g_z}\right) \mathbf j\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \left({f_x \dfrac {\partial g_x} {\partial z} + \dfrac {\partial f_x} {\partial z} g_x + f_y \dfrac {\partial g_y} {\partial z} + \dfrac {\partial f_y} {\partial z} g_y + f_z \dfrac {\partial g_z} {\partial z} + \dfrac {\partial f_z} {\partial z} g_z}\right) \mathbf k\)


Then:

\(\displaystyle \mathbf g \times \left({\nabla \times \mathbf f}\right)\) \(=\) \(\displaystyle \mathbf g \times \left({\left({\dfrac {\partial f_z} {\partial y} - \dfrac {\partial f_y} {\partial z} }\right) \mathbf i + \left({\dfrac {\partial f_x} {\partial z} - \dfrac {\partial f_z} {\partial x} }\right) \mathbf j + \left({\dfrac {\partial f_y} {\partial x} - \dfrac {\partial f_x} {\partial y} }\right) \mathbf k}\right)\) Definition of Curl Operator
\(\displaystyle \) \(=\) \(\displaystyle \left({g_y \left({\dfrac {\partial f_y} {\partial x} - \dfrac {\partial f_x} {\partial y} }\right) - g_z \left({\dfrac {\partial f_x} {\partial z} - \dfrac {\partial f_z} {\partial x} }\right)}\right) \mathbf i\) Definition 1 of Vector Cross Product
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \left({g_z \left({\dfrac {\partial f_z} {\partial y} - \dfrac {\partial f_y} {\partial z} }\right) - g_x \left({\dfrac {\partial f_y} {\partial x} - \dfrac {\partial f_x} {\partial y} }\right)}\right) \mathbf j\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \left({g_x \left({\dfrac {\partial f_x} {\partial z} - \dfrac {\partial f_z} {\partial x} }\right) - g_y \left({\dfrac {\partial f_z} {\partial y} - \dfrac {\partial f_y} {\partial z} }\right)}\right) \mathbf k\)
\(\displaystyle \) \(=\) \(\displaystyle \left({g_y \dfrac {\partial f_y} {\partial x} - g_y \dfrac {\partial f_x} {\partial y} - g_z \dfrac {\partial f_x} {\partial z} + g_z \dfrac {\partial f_z} {\partial x} }\right) \mathbf i\) expanding
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \left({g_z \dfrac {\partial f_z} {\partial y} - g_z \dfrac {\partial f_y} {\partial z} - g_x \dfrac {\partial f_y} {\partial x} + g_x \dfrac {\partial f_x} {\partial y} }\right) \mathbf j\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \left({g_x \dfrac {\partial f_x} {\partial z} - g_x \dfrac {\partial f_z} {\partial x} - g_y \dfrac {\partial f_z} {\partial y} + g_y \dfrac {\partial f_y} {\partial z} }\right) \mathbf k\)


and similarly:

\(\displaystyle \mathbf f \times \left({\nabla \times \mathbf g}\right)\) \(=\) \(\displaystyle \mathbf f \times \left({\left({\dfrac {\partial g_z} {\partial y} - \dfrac {\partial g_y} {\partial z} }\right) \mathbf i + \left({\dfrac {\partial g_x} {\partial z} - \dfrac {\partial g_z} {\partial x} }\right) \mathbf j + \left({\dfrac {\partial g_y} {\partial x} - \dfrac {\partial g_x} {\partial y} }\right) \mathbf k}\right)\) Definition of Curl Operator
\(\displaystyle \) \(=\) \(\displaystyle \left({f_y \left({\dfrac {\partial g_y} {\partial x} - \dfrac {\partial g_x} {\partial y} }\right) - f_z \left({\dfrac {\partial g_x} {\partial z} - \dfrac {\partial g_z} {\partial x} }\right)}\right) \mathbf i\) Definition 1 of Vector Cross Product
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \left({f_z \left({\dfrac {\partial g_z} {\partial y} - \dfrac {\partial g_y} {\partial z} }\right) - f_x \left({\dfrac {\partial g_y} {\partial x} - \dfrac {\partial g_x} {\partial y} }\right)}\right) \mathbf j\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \left({f_x \left({\dfrac {\partial g_x} {\partial z} - \dfrac {\partial g_z} {\partial x} }\right) - f_y \left({\dfrac {\partial g_z} {\partial y} - \dfrac {\partial g_y} {\partial z} }\right)}\right) \mathbf k\)
\(\displaystyle \) \(=\) \(\displaystyle \left({f_y \dfrac {\partial g_y} {\partial x} - f_y \dfrac {\partial g_x} {\partial y} - f_z \dfrac {\partial g_x} {\partial z} + f_z \dfrac {\partial g_z} {\partial x} }\right) \mathbf i\) expanding
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \left({f_z \dfrac {\partial g_z} {\partial y} - f_z \dfrac {\partial g_y} {\partial z} - f_x \dfrac {\partial g_y} {\partial x} + f_x \dfrac {\partial g_x} {\partial y} }\right) \mathbf j\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \left({f_x \dfrac {\partial g_x} {\partial z} - f_x \dfrac {\partial g_z} {\partial x} - f_y \dfrac {\partial g_z} {\partial y} + f_y \dfrac {\partial g_y} {\partial z} }\right) \mathbf k\)


Next:

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


and:

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



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