Vector analysis

1 Triple product

1.1 Scalar triple product

\boldsymbol{a}\dotproduct\left(\boldsymbol{b}\crossproduct\boldsymbol{c}\right% )=\boldsymbol{b}\dotproduct\left(\boldsymbol{c}\crossproduct\boldsymbol{a}% \right)=\boldsymbol{c}\dotproduct\left(\boldsymbol{a}\crossproduct\boldsymbol{% b}\right)

1.1.1 Proof

	$\displaystyle\boldsymbol{a}\dotproduct\left(\boldsymbol{b}\crossproduct% \boldsymbol{c}\right)$	$\displaystyle=\begin{pmatrix}a_{x}\\ a_{y}\\ a_{z}\end{pmatrix}\dotproduct\begin{pmatrix}b_{y}c_{z}-b_{z}c_{y}\\ b_{z}c_{x}-b_{x}c_{z}\\ b_{x}c_{y}-b_{y}c_{x}\end{pmatrix}$
		$\displaystyle=a_{x}\left(b_{y}c_{z}-b_{z}c_{y}\right)+a_{y}\left(b_{z}c_{x}-b_% {x}c_{z}\right)+a_{z}\left(b_{x}c_{y}-b_{y}c_{x}\right)$
		$\displaystyle=a_{x}b_{y}c_{z}+b_{x}c_{y}a_{z}+c_{x}a_{y}b_{z}-\left(c_{x}b_{y}% a_{z}+b_{x}a_{y}c_{z}+a_{x}c_{y}b_{z}\right)$
		$\displaystyle=\det\left(\begin{pmatrix}\boldsymbol{a}&\boldsymbol{b}&% \boldsymbol{c}\end{pmatrix}\right)=\det\left(\begin{pmatrix}\boldsymbol{b}&% \boldsymbol{c}&\boldsymbol{a}\end{pmatrix}\right)=\det\left(\begin{pmatrix}% \boldsymbol{c}&\boldsymbol{a}&\boldsymbol{b}\end{pmatrix}\right)$

1.2 Vector triple product

\boldsymbol{a}\crossproduct\left(\boldsymbol{b}\crossproduct\boldsymbol{c}% \right)=\left(\boldsymbol{a}\dotproduct\boldsymbol{c}\right)\boldsymbol{b}-% \left(\boldsymbol{a}\dotproduct\boldsymbol{b}\right)\boldsymbol{c}

1.2.1 Proof

	$\displaystyle\boldsymbol{a}\crossproduct\left(\boldsymbol{b}\crossproduct% \boldsymbol{c}\right)$	$\displaystyle=\begin{pmatrix}a_{x}\\ a_{y}\\ a_{z}\end{pmatrix}\crossproduct\left(\begin{pmatrix}b_{x}\\ b_{y}\\ b_{z}\end{pmatrix}\crossproduct\begin{pmatrix}c_{x}\\ c_{y}\\ c_{z}\end{pmatrix}\right)$
		$\displaystyle=\begin{pmatrix}a_{x}\\ a_{y}\\ a_{z}\end{pmatrix}\crossproduct\begin{pmatrix}b_{y}c_{z}-b_{z}c_{y}\\ b_{z}c_{x}-b_{x}c_{z}\\ b_{x}c_{y}-b_{y}c_{x}\end{pmatrix}$
		$\displaystyle=\begin{pmatrix}a_{y}\left(b_{x}c_{y}-b_{y}c_{x}\right)-a_{z}% \left(b_{z}c_{x}-b_{x}c_{z}\right)\\ a_{z}\left(b_{y}c_{z}-b_{z}c_{y}\right)-a_{x}\left(b_{x}c_{y}-b_{y}c_{x}\right% )\\ a_{x}\left(b_{z}c_{x}-b_{x}c_{z}\right)-a_{y}\left(b_{y}c_{z}-b_{z}c_{y}\right% )\end{pmatrix}$
		$\displaystyle=\begin{pmatrix}\left(a_{y}c_{y}+a_{z}c_{z}\right)b_{x}-\left(a_{% y}b_{y}+a_{z}b_{z}\right)c_{x}+a_{x}b_{x}c_{x}-a_{x}b_{x}c_{x}\\ \left(a_{z}c_{z}+a_{x}c_{x}\right)b_{y}-\left(a_{z}b_{z}+a_{x}b_{x}\right)c_{y% }+a_{y}b_{y}c_{y}-a_{y}b_{y}c_{y}\\ \left(a_{x}c_{x}+a_{y}c_{y}\right)b_{z}-\left(a_{x}b_{x}+a_{y}b_{y}\right)c_{z% }+a_{z}b_{z}c_{z}-a_{z}b_{z}c_{z}\end{pmatrix}$
		$\displaystyle=\begin{pmatrix}\left(a_{x}c_{x}+a_{y}c_{y}+a_{z}c_{z}\right)b_{x% }-\left(a_{x}b_{x}+a_{y}b_{y}+a_{z}b_{z}\right)c_{x}\\ \left(a_{x}c_{x}+a_{y}c_{y}+a_{z}c_{z}\right)b_{y}-\left(a_{x}b_{x}+a_{y}b_{y}% +a_{z}b_{z}\right)c_{y}\\ \left(a_{x}c_{x}+a_{y}c_{y}+a_{z}c_{z}\right)b_{z}-\left(a_{x}b_{x}+a_{y}b_{y}% +a_{z}b_{z}\right)c_{z}\end{pmatrix}$
		$\displaystyle=\left(\boldsymbol{a}\dotproduct\boldsymbol{c}\right)\boldsymbol{% b}-\left(\boldsymbol{a}\dotproduct\boldsymbol{b}\right)\boldsymbol{c}$

2 Differential operators

2.1 Nabla

\gradient=\left(\partialderivative{x},\partialderivative{y},\partialderivative% {z}\right)

2.2 Gradient

\boldsymbol{r}=\left(x,y,z\right)

\phi:\mathbb{R}^{3}\rightarrow\mathbb{R}

\text{grad}\left(\phi\left(\boldsymbol{r}\right)\right)=\gradient{\phi\left(% \boldsymbol{r}\right)}=\left(\partialderivative{\phi}{x},\partialderivative{% \phi}{y},\partialderivative{\phi}{z}\right)

2.3 Divergence

\boldsymbol{v}:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}

\boldsymbol{v}\left(\boldsymbol{r}\right)=\left(v_{x}\left(\boldsymbol{r}% \right),v_{y}\left(\boldsymbol{r}\right),v_{z}\left(\boldsymbol{r}\right)\right)

\text{div}\left(\boldsymbol{v}\left(\boldsymbol{r}\right)\right)=\divergence{% \boldsymbol{v}\left(\boldsymbol{r}\right)}=\partialderivative{v_{x}}{x}+% \partialderivative{v_{y}}{y}+\partialderivative{v_{z}}{z}

2.4 Curl

\text{curl}\left(\boldsymbol{v}\left(\boldsymbol{r}\right)\right)=\curl{% \boldsymbol{v}\left(\boldsymbol{r}\right)}=\begin{pmatrix}\frac{\partial{v_{z}% }}{\partial{y}}-\frac{\partial{v_{y}}}{\partial{z}}\\[3.00003pt] \frac{\partial{v_{x}}}{\partial{z}}-\frac{\partial{v_{z}}}{\partial{x}}\\[3.00% 003pt] \frac{\partial{v_{y}}}{\partial{x}}-\frac{\partial{v_{x}}}{\partial{y}}\end{pmatrix}

2.5 Laplacian

\Delta=\laplacian=\divergence\gradient=\frac{\partial^{2}{}}{\partial{x^{2}}}+% \frac{\partial^{2}{}}{\partial{y^{2}}}+\frac{\partial^{2}{}}{\partial{z^{2}}}

	$\displaystyle\laplacian{\boldsymbol{v}}$	$\displaystyle=\left(\frac{\partial^{2}{}}{\partial{x^{2}}}+\frac{\partial^{2}{% }}{\partial{y^{2}}}+\frac{\partial^{2}{}}{\partial{z^{2}}}\right)\left(v_{x},v% _{y},v_{z}\right)$
		$\displaystyle=\begin{pmatrix}\left(\frac{\partial^{2}{}}{\partial{x^{2}}}+% \frac{\partial^{2}{}}{\partial{y^{2}}}+\frac{\partial^{2}{}}{\partial{z^{2}}}% \right)v_{x}\\ \left(\frac{\partial^{2}{}}{\partial{x^{2}}}+\frac{\partial^{2}{}}{\partial{y^% {2}}}+\frac{\partial^{2}{}}{\partial{z^{2}}}\right)v_{y}\\ \left(\frac{\partial^{2}{}}{\partial{x^{2}}}+\frac{\partial^{2}{}}{\partial{y^% {2}}}+\frac{\partial^{2}{}}{\partial{z^{2}}}\right)v_{z}\end{pmatrix}$
		$\displaystyle=\left(\laplacian{v_{x}},\laplacian{v_{y}},\laplacian{v_{z}}\right)$

2.6 Properties of divergence and curl

$\displaystyle\text{div}\left(\phi\boldsymbol{v}\right)$	$\displaystyle=\phi\,\text{div}\,\boldsymbol{v}+\boldsymbol{v}\dotproduct% \gradient{\phi}$	(1)
$\displaystyle\text{div}\left(\boldsymbol{v}\crossproduct\boldsymbol{w}\right)$	$\displaystyle=\boldsymbol{w}\dotproduct\text{curl}\,\boldsymbol{v}-\boldsymbol% {v}\dotproduct\text{curl}\,\boldsymbol{w}$	(2)
$\displaystyle\text{curl}\left(\phi\boldsymbol{v}\right)$	$\displaystyle=\phi\,\text{curl}\,\boldsymbol{v}-\boldsymbol{v}\crossproduct% \gradient{\phi}$	(3)
$\displaystyle\text{div}\left(\gradient{\phi}\right)$	$\displaystyle=\laplacian{\phi}$	(4)
$\displaystyle\text{div}\left(\text{curl}\left(\boldsymbol{v}\right)\right)$	$\displaystyle=0$	(5)
$\displaystyle\text{curl}\left(\gradient{\phi}\right)$	$\displaystyle=\boldsymbol{0}$	(6)
$\displaystyle\text{curl}\left(\text{curl}\left(\boldsymbol{v}\right)\right)$	$\displaystyle=\gradient{\text{div}\,\boldsymbol{v}}-\laplacian{\boldsymbol{v}}$	(7)

2.6.1 Proof of (1)

	$\displaystyle\text{div}\left(\phi\boldsymbol{v}\right)$	$\displaystyle=\divergence\left(\phi v_{x},\phi v_{y},\phi v_{z}\right)$
		$\displaystyle=\frac{\partial{\phi v_{x}}}{\partial{x}}+\frac{\partial{\phi v_{% y}}}{\partial{y}}+\frac{\partial{\phi v_{z}}}{\partial{z}}$
		$\displaystyle=\left(\frac{\partial{\phi}}{\partial{x}}v_{x}+\phi\frac{\partial% {v_{x}}}{\partial{x}}\right)+\left(\frac{\partial{\phi}}{\partial{y}}v_{y}+% \phi\frac{\partial{v_{y}}}{\partial{y}}\right)+\left(\frac{\partial{\phi}}{% \partial{z}}v_{z}+\phi\frac{\partial{v_{z}}}{\partial{z}}\right)$
		$\displaystyle=\phi\left(\frac{\partial{v_{x}}}{\partial{x}}+\frac{\partial{v_{% y}}}{\partial{y}}+\frac{\partial{v_{z}}}{\partial{z}}\right)+\left(\frac{% \partial{\phi}}{\partial{x}}v_{x}+\frac{\partial{\phi}}{\partial{y}}v_{y}+% \frac{\partial{\phi}}{\partial{z}}v_{z}\right)$
		$\displaystyle=\phi\,\text{div}\,\boldsymbol{v}+\boldsymbol{v}\dotproduct% \gradient{\phi}$

2.6.2 Proof of (2)

	$\displaystyle\text{div}\left(\boldsymbol{v}\crossproduct\boldsymbol{w}\right)$	$\displaystyle=\divergence\left(\boldsymbol{v}\crossproduct\boldsymbol{w}\right)$
		$\displaystyle=\divergence\begin{pmatrix}v_{y}w_{z}-v_{z}w_{y}\\ v_{z}w_{x}-v_{x}w_{z}\\ v_{x}w_{y}-v_{y}w_{x}\end{pmatrix}$
		$\displaystyle=\frac{\partial{v_{y}w_{z}-v_{z}w_{y}}}{\partial{x}}+\frac{% \partial{v_{z}w_{x}-v_{x}w_{z}}}{\partial{y}}+\frac{\partial{v_{x}w_{y}-v_{y}w% _{x}}}{\partial{z}}$
		$\displaystyle=\left(\frac{\partial{v_{y}w_{z}}}{\partial{x}}-\frac{\partial{v_% {z}w_{y}}}{\partial{x}}\right)+\left(\frac{\partial{v_{z}w_{x}}}{\partial{y}}-% \frac{\partial{v_{x}w_{z}}}{\partial{y}}\right)+\left(\frac{\partial{v_{x}w_{y% }}}{\partial{z}}-\frac{\partial{v_{y}w_{x}}}{\partial{z}}\right)$
		$\displaystyle=\left(\frac{\partial{v_{y}w_{z}}}{\partial{x}}+\frac{\partial{v_% {z}w_{x}}}{\partial{y}}+\frac{\partial{v_{x}w_{y}}}{\partial{z}}\right)-\left(% \frac{\partial{v_{z}w_{y}}}{\partial{x}}+\frac{\partial{v_{x}w_{z}}}{\partial{% y}}+\frac{\partial{v_{y}w_{x}}}{\partial{z}}\right)$
		$\displaystyle=\left(\frac{\partial{v_{y}}}{\partial{x}}w_{z}+v_{y}\frac{% \partial{w_{z}}}{\partial{x}}+\frac{\partial{v_{z}}}{\partial{y}}w_{x}+v_{z}% \frac{\partial{w_{x}}}{\partial{y}}+\frac{\partial{v_{x}}}{\partial{z}}w_{y}+v% _{x}\frac{\partial{w_{y}}}{\partial{z}}\right)$
		$\displaystyle\quad-\left(\frac{\partial{v_{z}}}{\partial{x}}w_{y}+v_{z}\frac{% \partial{w_{y}}}{\partial{x}}+\frac{\partial{v_{x}}}{\partial{y}}w_{z}+v_{x}% \frac{\partial{w_{z}}}{\partial{y}}+\frac{\partial{v_{y}}}{\partial{z}}w_{x}+v% _{y}\frac{\partial{w_{x}}}{\partial{z}}\right)$
		$\displaystyle=\left\{\left(\frac{\partial{v_{z}}}{\partial{y}}-\frac{\partial{% v_{y}}}{\partial{z}}\right)w_{x}+\left(\frac{\partial{v_{x}}}{\partial{z}}-% \frac{\partial{v_{z}}}{\partial{x}}\right)w_{y}+\left(\frac{\partial{v_{y}}}{% \partial{x}}-\frac{\partial{v_{x}}}{\partial{y}}\right)w_{z}\right\}$
		$\displaystyle\quad-\left\{\left(\frac{\partial{w_{z}}}{\partial{y}}-\frac{% \partial{w_{y}}}{\partial{z}}\right)v_{x}+\left(\frac{\partial{w_{x}}}{% \partial{z}}-\frac{\partial{w_{z}}}{\partial{x}}\right)v_{y}+\left(\frac{% \partial{w_{y}}}{\partial{x}}-\frac{\partial{w_{x}}}{\partial{y}}\right)v_{z}\right\}$
		$\displaystyle=\boldsymbol{w}\dotproduct\text{curl}\,\boldsymbol{v}-\boldsymbol% {v}\dotproduct\text{curl}\,\boldsymbol{w}$

2.6.3 Proof of (3)

	$\displaystyle\text{curl}\left(\phi\boldsymbol{v}\right)$	$\displaystyle=\curl\left(\phi v_{x},\phi v_{y},\phi v_{z}\right)$
		$\displaystyle=\begin{pmatrix}\frac{\partial{\phi v_{z}}}{\partial{y}}-\frac{% \partial{\phi v_{y}}}{\partial{z}}\\[3.00003pt] \frac{\partial{\phi v_{x}}}{\partial{z}}-\frac{\partial{\phi v_{z}}}{\partial{% x}}\\[3.00003pt] \frac{\partial{\phi v_{y}}}{\partial{x}}-\frac{\partial{\phi v_{x}}}{\partial{% y}}\end{pmatrix}$
		$\displaystyle=\begin{pmatrix}\left(\frac{\partial{\phi}}{\partial{y}}v_{z}+% \phi\frac{\partial{v_{z}}}{\partial{y}}\right)-\left(\frac{\partial{\phi}}{% \partial{z}}v_{y}+\phi\frac{\partial{v_{y}}}{\partial{z}}\right)\\ \left(\frac{\partial{\phi}}{\partial{z}}v_{x}+\phi\frac{\partial{v_{x}}}{% \partial{z}}\right)-\left(\frac{\partial{\phi}}{\partial{x}}v_{z}+\phi\frac{% \partial{v_{z}}}{\partial{x}}\right)\\ \left(\frac{\partial{\phi}}{\partial{x}}v_{y}+\phi\frac{\partial{v_{y}}}{% \partial{x}}\right)-\left(\frac{\partial{\phi}}{\partial{y}}v_{x}+\phi\frac{% \partial{v_{x}}}{\partial{y}}\right)\end{pmatrix}$
		$\displaystyle=\phi\begin{pmatrix}\frac{\partial{v_{z}}}{\partial{y}}-\frac{% \partial{v_{y}}}{\partial{z}}\\[3.00003pt] \frac{\partial{v_{x}}}{\partial{z}}-\frac{\partial{v_{z}}}{\partial{x}}\\[3.00% 003pt] \frac{\partial{v_{y}}}{\partial{x}}-\frac{\partial{v_{x}}}{\partial{y}}\end{% pmatrix}-\begin{pmatrix}\frac{\partial{\phi}}{\partial{z}}v_{y}-\frac{\partial% {\phi}}{\partial{y}}v_{z}\\[3.00003pt] \frac{\partial{\phi}}{\partial{x}}v_{z}-\frac{\partial{\phi}}{\partial{z}}v_{x% }\\[3.00003pt] \frac{\partial{\phi}}{\partial{y}}v_{x}-\frac{\partial{\phi}}{\partial{x}}v_{y% }\end{pmatrix}$
		$\displaystyle=\phi\,\text{curl}\,\boldsymbol{v}-\boldsymbol{v}\crossproduct% \gradient{\phi}$

2.6.4 Proof of (4)

	$\displaystyle\text{div}\left(\gradient{\phi}\right)$	$\displaystyle=\divergence\left(\frac{\partial{\phi}}{\partial{x}},\frac{% \partial{\phi}}{\partial{y}},\frac{\partial{\phi}}{\partial{z}}\right)$
		$\displaystyle=\frac{\partial^{2}{\phi}}{\partial{x^{2}}}+\frac{\partial^{2}{% \phi}}{\partial{y^{2}}}+\frac{\partial^{2}{\phi}}{\partial{z^{2}}}=\laplacian{\phi}$

2.6.5 Proof of (5)

	$\displaystyle\text{div}\left(\text{curl}\left(\boldsymbol{v}\right)\right)$	$\displaystyle=\divergence\begin{pmatrix}\frac{\partial{v_{z}}}{\partial{y}}-% \frac{\partial{v_{y}}}{\partial{z}}\\[3.00003pt] \frac{\partial{v_{x}}}{\partial{z}}-\frac{\partial{v_{z}}}{\partial{x}}\\[3.00% 003pt] \frac{\partial{v_{y}}}{\partial{x}}-\frac{\partial{v_{x}}}{\partial{y}}\end{pmatrix}$
		$\displaystyle=\frac{\partial{}}{\partial{x}}\left(\frac{\partial{v_{z}}}{% \partial{y}}-\frac{\partial{v_{y}}}{\partial{z}}\right)+\frac{\partial{}}{% \partial{y}}\left(\frac{\partial{v_{x}}}{\partial{z}}-\frac{\partial{v_{z}}}{% \partial{x}}\right)+\frac{\partial{}}{\partial{z}}\left(\frac{\partial{v_{y}}}% {\partial{x}}-\frac{\partial{v_{x}}}{\partial{y}}\right)$
		$\displaystyle=\frac{\partial^{2}{v_{z}}}{\partial{x\partial{y}}}-\frac{% \partial^{2}{v_{y}}}{\partial{x\partial{z}}}+\frac{\partial^{2}{v_{x}}}{% \partial{y\partial{z}}}-\frac{\partial^{2}{v_{z}}}{\partial{y\partial{x}}}+% \frac{\partial^{2}{v_{y}}}{\partial{z\partial{x}}}-\frac{\partial^{2}{v_{x}}}{% \partial{z\partial{y}}}$
		$\displaystyle=\left(\frac{\partial^{2}{v_{x}}}{\partial{y\partial{z}}}-\frac{% \partial^{2}{v_{x}}}{\partial{z\partial{y}}}\right)+\left(\frac{\partial^{2}{v% _{y}}}{\partial{z\partial{x}}}-\frac{\partial^{2}{v_{y}}}{\partial{x\partial{z% }}}\right)+\left(\frac{\partial^{2}{v_{z}}}{\partial{x\partial{y}}}-\frac{% \partial^{2}{v_{z}}}{\partial{y\partial{x}}}\right)=0$

(assuming $\boldsymbol{v}$ is at least of class $C^{2}$ )

2.6.6 Proof of (6)

	$\displaystyle\text{curl}\left(\gradient{\phi}\right)$	$\displaystyle=\curl\left(\frac{\partial{\phi}}{\partial{x}},\frac{\partial{% \phi}}{\partial{y}},\frac{\partial{\phi}}{\partial{z}}\right)$
		$\displaystyle=\begin{pmatrix}\frac{\partial^{2}{\phi}}{\partial{y\partial{z}}}% -\frac{\partial^{2}{\phi}}{\partial{z\partial{y}}}\\[3.00003pt] \frac{\partial^{2}{\phi}}{\partial{z\partial{x}}}-\frac{\partial^{2}{\phi}}{% \partial{x\partial{z}}}\\[3.00003pt] \frac{\partial^{2}{\phi}}{\partial{x\partial{y}}}-\frac{\partial^{2}{\phi}}{% \partial{y\partial{x}}}\end{pmatrix}=\boldsymbol{0}$

(assuming $\phi$ is at least of class $C^{2}$ )

2.6.7 Proof of (7)

	$\displaystyle\text{curl}\left(\text{curl}\left(\boldsymbol{v}\right)\right)$	$\displaystyle=\curl\begin{pmatrix}\frac{\partial{v_{z}}}{\partial{y}}-\frac{% \partial{v_{y}}}{\partial{z}}\\ \frac{\partial{v_{x}}}{\partial{z}}-\frac{\partial{v_{z}}}{\partial{x}}\\ \frac{\partial{v_{y}}}{\partial{x}}-\frac{\partial{v_{x}}}{\partial{y}}\end{pmatrix}$
		$\displaystyle=\begin{pmatrix}\frac{\partial{}}{\partial{y}}\left(\frac{% \partial{v_{y}}}{\partial{x}}-\frac{\partial{v_{x}}}{\partial{y}}\right)-\frac% {\partial{}}{\partial{z}}\left(\frac{\partial{v_{x}}}{\partial{z}}-\frac{% \partial{v_{z}}}{\partial{x}}\right)\\ \frac{\partial{}}{\partial{z}}\left(\frac{\partial{v_{z}}}{\partial{y}}-\frac{% \partial{v_{y}}}{\partial{z}}\right)-\frac{\partial{}}{\partial{x}}\left(\frac% {\partial{v_{y}}}{\partial{x}}-\frac{\partial{v_{x}}}{\partial{y}}\right)\\ \frac{\partial{}}{\partial{x}}\left(\frac{\partial{v_{x}}}{\partial{z}}-\frac{% \partial{v_{z}}}{\partial{x}}\right)-\frac{\partial{}}{\partial{y}}\left(\frac% {\partial{v_{z}}}{\partial{y}}-\frac{\partial{v_{y}}}{\partial{z}}\right)\end{pmatrix}$
		$\displaystyle=\begin{pmatrix}\frac{\partial^{2}{v_{y}}}{\partial{y\partial{x}}% }-\frac{\partial^{2}{v_{x}}}{\partial{y^{2}}}-\frac{\partial^{2}{v_{x}}}{% \partial{z^{2}}}+\frac{\partial^{2}{v_{z}}}{\partial{z\partial{x}}}\\ \frac{\partial^{2}{v_{z}}}{\partial{z\partial{y}}}-\frac{\partial^{2}{v_{y}}}{% \partial{z^{2}}}-\frac{\partial^{2}{v_{y}}}{\partial{x^{2}}}+\frac{\partial^{2% }{v_{x}}}{\partial{x\partial{y}}}\\ \frac{\partial^{2}{v_{x}}}{\partial{x\partial{z}}}-\frac{\partial^{2}{v_{z}}}{% \partial{x^{2}}}-\frac{\partial^{2}{v_{z}}}{\partial{y^{2}}}+\frac{\partial^{2% }{v_{y}}}{\partial{y\partial{z}}}\end{pmatrix}$
		$\displaystyle=\begin{pmatrix}\frac{\partial^{2}{v_{x}}}{\partial{x^{2}}}+\frac% {\partial^{2}{v_{y}}}{\partial{y\partial{x}}}+\frac{\partial^{2}{v_{z}}}{% \partial{z\partial{x}}}-\frac{\partial^{2}{v_{x}}}{\partial{x^{2}}}-\frac{% \partial^{2}{v_{x}}}{\partial{y^{2}}}-\frac{\partial^{2}{v_{x}}}{\partial{z^{2% }}}\\ \frac{\partial^{2}{v_{x}}}{\partial{x\partial{y}}}+\frac{\partial^{2}{v_{y}}}{% \partial{y^{2}}}+\frac{\partial^{2}{v_{z}}}{\partial{z\partial{y}}}-\frac{% \partial^{2}{v_{y}}}{\partial{x^{2}}}-\frac{\partial^{2}{v_{y}}}{\partial{y^{2% }}}-\frac{\partial^{2}{v_{y}}}{\partial{z^{2}}}\\ \frac{\partial^{2}{v_{x}}}{\partial{x\partial{z}}}+\frac{\partial^{2}{v_{y}}}{% \partial{y\partial{z}}}+\frac{\partial^{2}{v_{z}}}{\partial{z^{2}}}-\frac{% \partial^{2}{v_{z}}}{\partial{x^{2}}}-\frac{\partial^{2}{v_{z}}}{\partial{y^{2% }}}-\frac{\partial^{2}{v_{z}}}{\partial{z^{2}}}\end{pmatrix}$
		$\displaystyle=\begin{pmatrix}\frac{\partial{}}{\partial{x}}\left(\frac{% \partial{v_{x}}}{\partial{x}}+\frac{\partial{v_{y}}}{\partial{y}}+\frac{% \partial{v_{z}}}{\partial{z}}\right)\\ \frac{\partial{}}{\partial{y}}\left(\frac{\partial{v_{x}}}{\partial{x}}+\frac{% \partial{v_{y}}}{\partial{y}}+\frac{\partial{v_{z}}}{\partial{z}}\right)\\ \frac{\partial{}}{\partial{z}}\left(\frac{\partial{v_{x}}}{\partial{x}}+\frac{% \partial{v_{y}}}{\partial{y}}+\frac{\partial{v_{z}}}{\partial{z}}\right)\end{% pmatrix}-\left(\laplacian{v_{x}},\laplacian{v_{y}},\laplacian{v_{z}}\right)$
		$\displaystyle=\gradient{\text{div}\,\boldsymbol{v}}-\laplacian{\boldsymbol{v}}$

(assuming $\boldsymbol{v}$ is at least of class $C^{2}$ )

3 Integral theorems

3.1 Gradient theorem

C:\boldsymbol{r_{0}}\rightarrow\boldsymbol{r_{1}}

\int_{C}\gradient{\phi(\boldsymbol{r})}\dotproduct d\boldsymbol{r}=\phi(% \boldsymbol{r_{1}})-\phi(\boldsymbol{r_{0}})

3.1.1 Proof

if $\boldsymbol{r}$ is a function of $t$ ; $\boldsymbol{r}(t)$ , where $\boldsymbol{r}(t_{0})=\boldsymbol{r_{0}},\;\boldsymbol{r}(t_{1})=\boldsymbol{r% _{1}}$ ,
by the multivariate chain rule,

\frac{d}{dt}\phi\left(\boldsymbol{r}(t)\right)=\gradient{\phi\left(\boldsymbol% {r}(t)\right)}\dotproduct\frac{d\boldsymbol{r}}{dt}(t)

	$\displaystyle\int_{C}\gradient{\phi(\boldsymbol{r})}\dotproduct d\boldsymbol{r}$	$\displaystyle=\int_{t_{0}}^{t_{1}}\gradient{\phi\left(\boldsymbol{r}(t)\right)% }\dotproduct\frac{d\boldsymbol{r}}{dt}(t)\,dt$
		$\displaystyle=\int_{t_{0}}^{t_{1}}\frac{d}{dt}\phi\left(\boldsymbol{r}(t)% \right)dt$
		$\displaystyle=\phi\left(\boldsymbol{r}(t_{1})\right)-\phi\left(\boldsymbol{r}(% t_{0})\right)$
		$\displaystyle=\phi\left(\boldsymbol{r_{1}}\right)-\phi\left(\boldsymbol{r_{0}}\right)$

3.2 Divergence theorem

d\boldsymbol{S}=\boldsymbol{\hat{n}}dS

\oint_{S}\boldsymbol{F}\dotproduct d\boldsymbol{S}=\int_{V}\left(\divergence{% \boldsymbol{F}}\right)dV

3.3 Curl theorem

\oint_{C}\boldsymbol{F}\dotproduct d\boldsymbol{r}=\int_{S}\left(\curl{% \boldsymbol{F}}\right)\dotproduct d\boldsymbol{S}