Appendix M Vector identities

In this chapter we show the derivation of some vector quantities in cartesian tensor notation.

M.1 $(\boldsymbol{u}\cdot\nabla)\boldsymbol{v}$

For some arbitraty vectors $u_{i},v_{i}$ we can write

	$\displaystyle u_{j}\frac{\partial}{\partial r_{j}}v_{i}$	$\displaystyle=\overbrace{u_{j}\frac{\partial}{\partial r_{i}}v_{j}-u_{j}\frac{% \partial}{\partial r_{i}}v_{j}}^{0}+u_{j}\frac{\partial}{\partial r_{j}}v_{i}$
		$\displaystyle=u_{j}\frac{\partial}{\partial r_{i}}v_{j}-\delta_{ik}\delta_{jl}% u_{j}\frac{\partial}{\partial r_{k}}v_{l}+\delta_{il}\delta_{jk}u_{j}\frac{% \partial}{\partial r_{k}}v_{l}$
		$\displaystyle=u_{j}\frac{\partial}{\partial r_{i}}v_{j}-(\delta_{ik}\delta_{jl% }-\delta_{il}\delta_{jk})u_{j}\frac{\partial}{\partial r_{k}}v_{l}$
		$\displaystyle=u_{j}\frac{\partial}{\partial r_{i}}v_{j}-\epsilon_{mij}\epsilon% _{mkl}u_{j}\frac{\partial}{\partial r_{k}}v_{l}$
		$\displaystyle=u_{j}\frac{\partial}{\partial r_{i}}v_{j}-\epsilon_{ijm}u_{j}% \epsilon_{mkl}\frac{\partial}{\partial r_{k}}v_{l}.$

In vector notation this can be expressed like

\displaystyle(\boldsymbol{u}\cdot\nabla)\boldsymbol{v}=\boldsymbol{u}\cdot(% \nabla\boldsymbol{v})-\boldsymbol{u}\times(\nabla\times\boldsymbol{v})

(M.1)

Inserting $u_{j}=v_{j}$ into equation (M.1) yields

\displaystyle v_{j}\frac{\partial}{\partial r_{j}}v_{i}=v_{j}\frac{\partial}{% \partial r_{i}}v_{j}-\epsilon_{ijm}v_{j}\epsilon_{mkl}\frac{\partial}{\partial r% _{k}}v_{l}

For $v_{j}\frac{\partial}{\partial r_{i}}v_{j}$ we can write

\displaystyle v_{j}\frac{\partial}{\partial r_{i}}v_{j}

\displaystyle=\frac{\partial}{\partial r_{i}}(v_{j}v_{j})-v_{j}\frac{\partial}% {\partial r_{i}}v_{j}

and therefore

\displaystyle v_{j}\frac{\partial}{\partial r_{i}}v_{j}=\frac{\partial}{% \partial r_{i}}{\left(\frac{1}{2}v_{j}v_{j}\right)}.

Using this we get for

\displaystyle v_{j}\frac{\partial}{\partial r_{j}}v_{i}=\frac{\partial}{% \partial r_{i}}{\left(\frac{1}{2}v_{j}v_{j}\right)}-\epsilon_{ijm}v_{j}% \epsilon_{mkl}\frac{\partial}{\partial r_{k}}v_{l}

or in vector notation

\displaystyle(\boldsymbol{v}\cdot\nabla)\boldsymbol{v}=\frac{1}{2}\nabla% \boldsymbol{v}^{2}-\boldsymbol{v}\times(\nabla\times\boldsymbol{v})

(M.2)

The $i$ -th component of the rotation of a cross product of two vectors is

	$\displaystyle{\left[\nabla\times{\left(\boldsymbol{u}\times\boldsymbol{v}% \right)}\right]}_{i}$	$\displaystyle=\epsilon_{ijk}\frac{\partial}{\partial r_{j}}\epsilon_{klm}u_{l}% v_{m}$
		$\displaystyle=\epsilon_{kij}\epsilon_{klm}\frac{\partial}{\partial r_{j}}(u_{l% }v_{m})$
		$\displaystyle=(\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})\frac{\partial}{% \partial r_{j}}(u_{l}v_{m})$
		$\displaystyle=\frac{\partial}{\partial r_{j}}(u_{i}v_{j})-\frac{\partial}{% \partial r_{j}}(u_{j}v_{i})$
		$\displaystyle=u_{i}\frac{\partial}{\partial r_{j}}v_{j}+v_{j}\frac{\partial}{% \partial r_{j}}u_{i}-u_{j}\frac{\partial}{\partial r_{j}}v_{i}-v_{i}\frac{% \partial}{\partial r_{j}}u_{j}$

It can be written in vector notation like

\displaystyle\nabla\times{\left(\boldsymbol{u}\times\boldsymbol{v}\right)}=% \boldsymbol{u}(\nabla\cdot\boldsymbol{v})-\boldsymbol{v}(\nabla\cdot% \boldsymbol{u})+(\boldsymbol{v}\cdot\nabla)\boldsymbol{u}-(\boldsymbol{u}\cdot% \nabla)\boldsymbol{v}

(M.3)