# Electric and Magnetic Fields due to Accelerated Charge

eb-8ef68b06-408f-4fca-9a0a-0f1c4a31a2ce
The Liénard-Wiechert potentials are given as:
$\begin{array}{rl}& \mathrm{\Phi }\left(\stackrel{\to }{r},t\right)=\frac{1}{4\pi {ϵ}_{0}}\frac{qc}{\left(Rc-\stackrel{\to }{R}\cdot \stackrel{\to }{v}\right)}\\ & \stackrel{\to }{A}\left(\stackrel{\to }{r},t\right)=\frac{\stackrel{\to }{v}}{{c}^{2}}\mathrm{\Phi }\left(\stackrel{\to }{r},t\right)\end{array}$$\begin{array}{r}\mathrm{\Phi }\left(\stackrel{\to }{r},t\right)=\frac{1}{4\pi {ϵ}_{0}}\frac{qc}{\left(Rc-\stackrel{\to }{R}\cdot \stackrel{\to }{v}\right)}\\ \stackrel{\to }{A}\left(\stackrel{\to }{r},t\right)=\frac{\stackrel{\to }{v}}{{c}^{2}}\mathrm{\Phi }\left(\stackrel{\to }{r},t\right)\end{array}${:[Phi( vec(r)”,”t)=(1)/(4piepsilon_(0))(qc)/((Rc-( vec(R))*( vec(v))))],[ vec(A)( vec(r)”,”t)=(( vec(v)))/(c^(2))Phi( vec(r)”,”t)]:}\begin{aligned} & \Phi(\vec{r}, t)=\frac{1}{4 \pi \epsilon_0} \frac{q c}{(R c-\vec{R} \cdot \vec{v})} \\ & \vec{A}(\vec{r}, t)=\frac{\vec{v}}{c^2} \Phi(\vec{r}, t) \end{aligned}$\begin{array}{rl}& \mathrm{\Phi }\left(\stackrel{\to }{r},t\right)=\frac{1}{4\pi {ϵ}_{0}}\frac{qc}{\left(Rc-\stackrel{\to }{R}\cdot \stackrel{\to }{v}\right)}\\ & \stackrel{\to }{A}\left(\stackrel{\to }{r},t\right)=\frac{\stackrel{\to }{v}}{{c}^{2}}\mathrm{\Phi }\left(\stackrel{\to }{r},t\right)\end{array}$
These equations can now be used to calculate the $\left(\stackrel{\mathbf{\to }}{\mathbf{E}}\mathbf{,}\stackrel{\mathbf{\to }}{\mathbf{B}}\right)$$\left(\stackrel{\mathbf{\to }}{\mathbf{E}}\mathbf{,}\stackrel{\mathbf{\to }}{\mathbf{B}}\right)$( vec(E), vec(B))(\mathbf{\vec{E}, \vec{B}})$\left(\stackrel{\mathbf{\to }}{\mathbf{E}}\mathbf{,}\stackrel{\mathbf{\to }}{\mathbf{B}}\right)$ due to a point charge in accelerated motion, using the equations:
$\begin{array}{rl}& \stackrel{\to }{E}=-\stackrel{\to }{\mathrm{\nabla }}\mathrm{\Phi }-\frac{\mathrm{\partial }\stackrel{\to }{A}}{\mathrm{\partial }t}\\ & \stackrel{\to }{B}=\stackrel{\to }{\mathrm{\nabla }}×\stackrel{\to }{A}\end{array}$$\begin{array}{r}\stackrel{\to }{E}=-\stackrel{\to }{\mathrm{\nabla }}\mathrm{\Phi }-\frac{\mathrm{\partial }\stackrel{\to }{A}}{\mathrm{\partial }t}\\ \stackrel{\to }{B}=\stackrel{\to }{\mathrm{\nabla }}×\stackrel{\to }{A}\end{array}${:[ vec(E)=- vec(grad)Phi-(del( vec(A)))/(del t)],[ vec(B)= vec(grad)xx vec(A)]:}\begin{aligned} & \vec{E}=-\vec{\nabla} \Phi-\frac{\partial \vec{A}}{\partial t} \\ & \vec{B}=\vec{\nabla} \times \vec{A} \end{aligned}$\begin{array}{rl}& \stackrel{\to }{E}=-\stackrel{\to }{\mathrm{\nabla }}\mathrm{\Phi }-\frac{\mathrm{\partial }\stackrel{\to }{A}}{\mathrm{\partial }t}\\ & \stackrel{\to }{B}=\stackrel{\to }{\mathrm{\nabla }}×\stackrel{\to }{A}\end{array}$
Calculation of $\stackrel{\mathbf{\to }}{\mathbf{E}}$$\stackrel{\mathbf{\to }}{\mathbf{E}}$vec(E)\mathbf{\vec{E}}$\stackrel{\mathbf{\to }}{\mathbf{E}}$ :
$\begin{array}{r}\stackrel{\to }{E}=-\stackrel{\to }{\mathrm{\nabla }}\mathrm{\Phi }-\frac{\mathrm{\partial }\stackrel{\to }{A}}{\mathrm{\partial }t}\end{array}$$\begin{array}{r}\stackrel{\to }{E}=-\stackrel{\to }{\mathrm{\nabla }}\mathrm{\Phi }-\frac{\mathrm{\partial }\stackrel{\to }{A}}{\mathrm{\partial }t}\end{array}${: vec(E)=- vec(grad)Phi-(del( vec(A)))/(del t):}\begin{aligned} \vec{E}=-\vec{\nabla} \Phi-\frac{\partial \vec{A}}{\partial t} \\ \end{aligned}$\begin{array}{r}\stackrel{\to }{E}=-\stackrel{\to }{\mathrm{\nabla }}\mathrm{\Phi }-\frac{\mathrm{\partial }\stackrel{\to }{A}}{\mathrm{\partial }t}\end{array}$
Here,
$\begin{array}{rl}\stackrel{\to }{\mathrm{\nabla }}\mathrm{\Phi }& =\frac{qc}{4\pi {ϵ}_{0}}\stackrel{\to }{\mathrm{\nabla }}\left[\left(Rc-\stackrel{\to }{R}\cdot \stackrel{\to }{v}{\right)}^{-1}\right]\\ & =\frac{qc}{4\pi {ϵ}_{0}}\frac{\left(-1\right)}{\left(Rc-\stackrel{\to }{R}\cdot \stackrel{\to }{v}{\right)}^{2}}\stackrel{\to }{\mathrm{\nabla }}\left(Rc-\stackrel{\to }{R}\cdot \stackrel{\to }{v}\right)\\ ⇒\stackrel{\to }{\mathrm{\nabla }}\mathrm{\Phi }& =\frac{qc}{4\pi {ϵ}_{0}}\frac{1}{\left(Rc-\stackrel{\to }{R}\cdot \stackrel{\to }{v}{\right)}^{2}}\left[\stackrel{\to }{\mathrm{\nabla }}\left(\stackrel{\to }{R}\cdot \stackrel{\to }{v}\right)-c\stackrel{\to }{\mathrm{\nabla }}R\right]\end{array}$$\begin{array}{r}\stackrel{\to }{\mathrm{\nabla }}\mathrm{\Phi }=\frac{qc}{4\pi {ϵ}_{0}}\stackrel{\to }{\mathrm{\nabla }}\left[\left(Rc-\stackrel{\to }{R}\cdot \stackrel{\to }{v}{\right)}^{-1}\right]\\ =\frac{qc}{4\pi {ϵ}_{0}}\frac{\left(-1\right)}{\left(Rc-\stackrel{\to }{R}\cdot \stackrel{\to }{v}{\right)}^{2}}\stackrel{\to }{\mathrm{\nabla }}\left(Rc-\stackrel{\to }{R}\cdot \stackrel{\to }{v}\right)\\ ⇒\stackrel{\to }{\mathrm{\nabla }}\mathrm{\Phi }=\frac{qc}{4\pi {ϵ}_{0}}\frac{1}{\left(Rc-\stackrel{\to }{R}\cdot \stackrel{\to }{v}{\right)}^{2}}\left[\stackrel{\to }{\mathrm{\nabla }}\left(\stackrel{\to }{R}\cdot \stackrel{\to }{v}\right)-c\stackrel{\to }{\mathrm{\nabla }}R\right]\end{array}${:[ vec(grad)Phi=(qc)/(4piepsilon_(0)) vec(grad)[(Rc-( vec(R))*( vec(v)))^(-1)]],[=(qc)/(4piepsilon_(0))((-1))/((Rc-( vec(R))*( vec(v)))^(2)) vec(grad)(Rc- vec(R)* vec(v))],[=> vec(grad)Phi=(qc)/(4piepsilon_(0))(1)/((Rc-( vec(R))*( vec(v)))^(2))[ vec(grad)( vec(R)* vec(v))-c vec(grad)R]]:}\begin{aligned} \vec{\nabla} \Phi & =\frac{q c}{4 \pi \epsilon_0} \vec{\nabla}\left[(R c-\vec{R} \cdot \vec{v})^{-1}\right] \\ & =\frac{q c}{4 \pi \epsilon_0} \frac{(-1)}{(R c-\vec{R} \cdot \vec{v})^2} \vec{\nabla}(R c-\vec{R} \cdot \vec{v}) \\ \Rightarrow \vec{\nabla} \Phi & =\frac{q c}{4 \pi \epsilon_0} \frac{1}{(R c-\vec{R} \cdot \vec{v})^2}[\vec{\nabla}(\vec{R} \cdot \vec{v})-c \vec{\nabla} R] \end{aligned}$\begin{array}{rl}\stackrel{\to }{\mathrm{\nabla }}\mathrm{\Phi }& =\frac{qc}{4\pi {ϵ}_{0}}\stackrel{\to }{\mathrm{\nabla }}\left[\left(Rc-\stackrel{\to }{R}\cdot \stackrel{\to }{v}{\right)}^{-1}\right]\\ & =\frac{qc}{4\pi {ϵ}_{0}}\frac{\left(-1\right)}{\left(Rc-\stackrel{\to }{R}\cdot \stackrel{\to }{v}{\right)}^{2}}\stackrel{\to }{\mathrm{\nabla }}\left(Rc-\stackrel{\to }{R}\cdot \stackrel{\to }{v}\right)\\ ⇒\stackrel{\to }{\mathrm{\nabla }}\mathrm{\Phi }& =\frac{qc}{4\pi {ϵ}_{0}}\frac{1}{\left(Rc-\stackrel{\to }{R}\cdot \stackrel{\to }{v}{\right)}^{2}}\left[\stackrel{\to }{\mathrm{\nabla }}\left(\stackrel{\to }{R}\cdot \stackrel{\to }{v}\right)-c\stackrel{\to }{\mathrm{\nabla }}R\right]\end{array}$
$\begin{array}{rl}\stackrel{\to }{\mathrm{\nabla }}\mathrm{\Phi }=\frac{qc}{4\pi {ϵ}_{0}}\frac{1}{\left(Rc-\stackrel{\to }{R}\cdot \stackrel{\to }{v}{\right)}^{2}}\left[\stackrel{\to }{\mathrm{\nabla }}\left(\stackrel{\to }{R}\cdot \stackrel{\to }{v}\right)-c\stackrel{\to }{\mathrm{\nabla }}R\right],& \dots \left(1\right)\end{array}$$\begin{array}{r}\stackrel{\to }{\mathrm{\nabla }}\mathrm{\Phi }=\frac{qc}{4\pi {ϵ}_{0}}\frac{1}{\left(Rc-\stackrel{\to }{R}\cdot \stackrel{\to }{v}{\right)}^{2}}\left[\stackrel{\to }{\mathrm{\nabla }}\left(\stackrel{\to }{R}\cdot \stackrel{\to }{v}\right)-c\stackrel{\to }{\mathrm{\nabla }}R\right],\dots \left(1\right)\end{array}${: vec(grad)Phi=(qc)/(4piepsilon_(0))(1)/((Rc-( vec(R))*( vec(v)))^(2))[ vec(grad)( vec(R)* vec(v))-c vec(grad)R]”,” dots(1):}\begin{aligned} \vec{\nabla} \Phi=\frac{q c}{4 \pi \epsilon_0} \frac{1}{(R c-\vec{R} \cdot \vec{v})^2}[\vec{\nabla}(\vec{R} \cdot \vec{v})-c \vec{\nabla} R], & \ldots\left(\mathrm{1}\right)\\ \end{aligned}$\begin{array}{rl}\stackrel{\to }{\mathrm{\nabla }}\mathrm{\Phi }=\frac{qc}{4\pi {ϵ}_{0}}\frac{1}{\left(Rc-\stackrel{\to }{R}\cdot \stackrel{\to }{v}{\right)}^{2}}\left[\stackrel{\to }{\mathrm{\nabla }}\left(\stackrel{\to }{R}\cdot \stackrel{\to }{v}\right)-c\stackrel{\to }{\mathrm{\nabla }}R\right],& \dots \left(1\right)\end{array}$
Since retarded time is given by:
$\begin{array}{rl}{t}_{r}& =t-\frac{R}{c}\\ ⇒R& =c\left(t-{t}_{r}\right)\end{array}$$\begin{array}{r}{t}_{r}=t-\frac{R}{c}\\ ⇒R=c\left(t-{t}_{r}\right)\end{array}${:[t_(r)=t-(R)/(c)],[=>R=c(t-t_(r))]:}\begin{aligned} t_r & =t-\frac{R}{c} \\ \Rightarrow R & =c\left(t-t_r\right) \end{aligned}$\begin{array}{rl}{t}_{r}& =t-\frac{R}{c}\\ ⇒R& =c\left(t-{t}_{r}\right)\end{array}$
This gives
$\stackrel{\to }{\mathrm{\nabla }}R=-c\stackrel{\to }{\mathrm{\nabla }}{t}_{r}$$\stackrel{\to }{\mathrm{\nabla }}R=-c\stackrel{\to }{\mathrm{\nabla }}{t}_{r}$vec(grad)R=-c vec(grad)t_(r)\vec{\nabla} R=-c \vec{\nabla} t_r$\stackrel{\to }{\mathrm{\nabla }}R=-c\stackrel{\to }{\mathrm{\nabla }}{t}_{r}$
Solving for 1st term in equation 1:\
Using the product rule:
$\begin{array}{r}\stackrel{\to }{\mathrm{\nabla }}\left(\stackrel{\to }{X}\cdot \stackrel{\to }{Y}\right)=\left(\stackrel{\to }{X}\cdot \stackrel{\to }{\mathrm{\nabla }}\right)\stackrel{\to }{Y}+\left(\stackrel{\to }{Y}\cdot \stackrel{\to }{\mathrm{\nabla }}\right)\stackrel{\to }{X}+\stackrel{\to }{X}×\left(\stackrel{\to }{\mathrm{\nabla }}×\stackrel{\to }{Y}\right)+\stackrel{\to }{Y}×\left(\stackrel{\to }{\mathrm{\nabla }}×\stackrel{\to }{X}\right)\\ ⇒\mathbf{\nabla }\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{v}}\right)=\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbf{\nabla }}\right)\stackrel{\to }{\mathbit{v}}+\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}\right)\stackrel{\to }{\mathbit{R}}+\stackrel{\to }{\mathbit{R}}×\left(\stackrel{\to }{\mathbf{\nabla }}×\stackrel{\to }{\mathbit{v}}\right)+\stackrel{\to }{\mathbit{v}}×\left(\stackrel{\to }{\mathbf{\nabla }}×\stackrel{\to }{\mathbit{R}}\right)\right),\dots \left(2\right)\end{array}$$\begin{array}{r}\stackrel{\to }{\mathrm{\nabla }}\left(\stackrel{\to }{X}\cdot \stackrel{\to }{Y}\right)=\left(\stackrel{\to }{X}\cdot \stackrel{\to }{\mathrm{\nabla }}\right)\stackrel{\to }{Y}+\left(\stackrel{\to }{Y}\cdot \stackrel{\to }{\mathrm{\nabla }}\right)\stackrel{\to }{X}+\stackrel{\to }{X}×\left(\stackrel{\to }{\mathrm{\nabla }}×\stackrel{\to }{Y}\right)+\stackrel{\to }{Y}×\left(\stackrel{\to }{\mathrm{\nabla }}×\stackrel{\to }{X}\right)\\ ⇒\mathbf{\nabla }\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{v}}\right)=\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbf{\nabla }}\right)\stackrel{\to }{\mathbit{v}}+\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}\right)\stackrel{\to }{\mathbit{R}}+\stackrel{\to }{\mathbit{R}}×\left(\stackrel{\to }{\mathbf{\nabla }}×\stackrel{\to }{\mathbit{v}}\right)+\stackrel{\to }{\mathbit{v}}×\left(\stackrel{\to }{\mathbf{\nabla }}×\stackrel{\to }{\mathbit{R}}\right)\right),\dots \left(2\right)\end{array}${:[ vec(grad)( vec(X)* vec(Y))=( vec(X)* vec(grad)) vec(Y)+( vec(Y)* vec(grad)) vec(X)+ vec(X)xx( vec(grad)xx vec(Y))+ vec(Y)xx( vec(grad)xx vec(X))],[=>grad( vec(R)* vec(v))=( vec(R)* vec(grad)) vec(v)+( vec(v)* vec(grad)) vec(R)+ vec(R)xx( vec(grad)xx vec(v))+ vec(v)xx( vec(grad)xx vec(R)))”,”dots(2)]:}\begin{aligned} \vec{\nabla}(\vec{X} \cdot \vec{Y})=(\vec{X} \cdot \vec{\nabla}) \vec{Y}+(\vec{Y} \cdot \vec{\nabla}) \vec{X}+\vec{X} \times(\vec{\nabla} \times \vec{Y})+\vec{Y} \times(\vec{\nabla} \times \vec{X}) \\ \Rightarrow{\boldsymbol{\nabla}}(\overrightarrow{\boldsymbol{R}} \cdot \overrightarrow{\boldsymbol{v}})=(\overrightarrow{\boldsymbol{R}} \cdot \overrightarrow{\boldsymbol{\nabla}}) \overrightarrow{\boldsymbol{v}}+(\overrightarrow{\boldsymbol{v}} \cdot \overrightarrow{\boldsymbol{\nabla}}) \overrightarrow{\boldsymbol{R}}+\overrightarrow{\boldsymbol{R}} \times(\overrightarrow{\boldsymbol{\nabla}} \times \overrightarrow{\boldsymbol{v}})+\overrightarrow{\boldsymbol{v}} \times(\overrightarrow{\boldsymbol{\nabla}} \times \overrightarrow{\boldsymbol{R}})),\ldots\left(\mathrm{2}\right)\\ \end{aligned}$\begin{array}{r}\stackrel{\to }{\mathrm{\nabla }}\left(\stackrel{\to }{X}\cdot \stackrel{\to }{Y}\right)=\left(\stackrel{\to }{X}\cdot \stackrel{\to }{\mathrm{\nabla }}\right)\stackrel{\to }{Y}+\left(\stackrel{\to }{Y}\cdot \stackrel{\to }{\mathrm{\nabla }}\right)\stackrel{\to }{X}+\stackrel{\to }{X}×\left(\stackrel{\to }{\mathrm{\nabla }}×\stackrel{\to }{Y}\right)+\stackrel{\to }{Y}×\left(\stackrel{\to }{\mathrm{\nabla }}×\stackrel{\to }{X}\right)\\ ⇒\mathbf{\nabla }\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{v}}\right)=\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbf{\nabla }}\right)\stackrel{\to }{\mathbit{v}}+\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}\right)\stackrel{\to }{\mathbit{R}}+\stackrel{\to }{\mathbit{R}}×\left(\stackrel{\to }{\mathbf{\nabla }}×\stackrel{\to }{\mathbit{v}}\right)+\stackrel{\to }{\mathbit{v}}×\left(\stackrel{\to }{\mathbf{\nabla }}×\stackrel{\to }{\mathbit{R}}\right)\right),\dots \left(2\right)\end{array}$
1st term in eq. (2):
$\begin{array}{rl}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathrm{\nabla }}\right)\stackrel{\to }{v}& =\left({R}_{x}\frac{\mathrm{\partial }}{\mathrm{\partial }x}+{R}_{y}\frac{\mathrm{\partial }}{\mathrm{\partial }y}+{R}_{z}\frac{\mathrm{\partial }}{\mathrm{\partial }z}\right)\stackrel{\to }{v}\left({t}_{r}\right)\\ & ={R}_{x}\frac{\mathrm{d}\stackrel{\to }{v}}{\text{}\mathrm{d}{t}_{r}}\frac{\mathrm{\partial }{t}_{r}}{\mathrm{\partial }x}+{R}_{y}\frac{\mathrm{d}\stackrel{\to }{v}}{\text{}\mathrm{d}{t}_{r}}\frac{\mathrm{\partial }{t}_{r}}{\mathrm{\partial }y}+{R}_{z}\frac{\mathrm{d}\stackrel{\to }{v}}{\text{}\mathrm{d}{t}_{r}}\frac{\mathrm{\partial }{t}_{r}}{\mathrm{\partial }z}\\ & =\stackrel{\to }{a}\left(\stackrel{\to }{R}\cdot \stackrel{\to }{\mathrm{\nabla }}{t}_{r}\right)\end{array}$$\begin{array}{r}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathrm{\nabla }}\right)\stackrel{\to }{v}=\left({R}_{x}\frac{\mathrm{\partial }}{\mathrm{\partial }x}+{R}_{y}\frac{\mathrm{\partial }}{\mathrm{\partial }y}+{R}_{z}\frac{\mathrm{\partial }}{\mathrm{\partial }z}\right)\stackrel{\to }{v}\left({t}_{r}\right)\\ ={R}_{x}\frac{\mathrm{d}\stackrel{\to }{v}}{\text{}\mathrm{d}{t}_{r}}\frac{\mathrm{\partial }{t}_{r}}{\mathrm{\partial }x}+{R}_{y}\frac{\mathrm{d}\stackrel{\to }{v}}{\text{}\mathrm{d}{t}_{r}}\frac{\mathrm{\partial }{t}_{r}}{\mathrm{\partial }y}+{R}_{z}\frac{\mathrm{d}\stackrel{\to }{v}}{\text{}\mathrm{d}{t}_{r}}\frac{\mathrm{\partial }{t}_{r}}{\mathrm{\partial }z}\\ =\stackrel{\to }{a}\left(\stackrel{\to }{R}\cdot \stackrel{\to }{\mathrm{\nabla }}{t}_{r}\right)\end{array}${:[( vec(R)* vec(grad)) vec(v)=(R_(x)(del)/(del x)+R_(y)(del)/(del y)+R_(z)(del)/(del z)) vec(v)(t_(r))],[=R_(x)(d( vec(v)))/((d)t_(r))(delt_(r))/(del x)+R_(y)(d( vec(v)))/((d)t_(r))(delt_(r))/(del y)+R_(z)(d( vec(v)))/((d)t_(r))(delt_(r))/(del z)],[= vec(a)(( vec(R))*( vec(grad))t_(r))]:}\begin{aligned} (\overrightarrow{\boldsymbol{R}} \cdot \vec{\nabla}) \vec{v} & =\left(R_x \frac{\partial}{\partial x}+R_y \frac{\partial}{\partial y}+R_z \frac{\partial}{\partial z}\right) \vec{v}\left(t_r\right) \\ & =R_x \frac{\mathrm{d} \vec{v}}{\mathrm{~d} t_r} \frac{\partial t_r}{\partial x}+R_y \frac{\mathrm{d} \vec{v}}{\mathrm{~d} t_r} \frac{\partial t_r}{\partial y}+R_z \frac{\mathrm{d} \vec{v}}{\mathrm{~d} t_r} \frac{\partial t_r}{\partial z} \\ & =\vec{a}\left(\vec{R} \cdot \vec{\nabla} t_r\right) \end{aligned}
2nd term in eq. (2):
$\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}\right)\stackrel{\to }{\mathbit{R}}=\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}\right)\stackrel{\to }{\mathbit{r}}-\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}\right)\stackrel{\to }{\mathbit{w}}\phantom{\rule{0ex}{0ex}}$$\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}\right)\stackrel{\to }{\mathbit{R}}=\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}\right)\stackrel{\to }{\mathbit{r}}-\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}\right)\stackrel{\to }{\mathbit{w}}\phantom{\rule{0ex}{0ex}}$( vec(v)* vec(grad)) vec(R)=( vec(v)* vec(grad)) vec(r)-( vec(v)* vec(grad)) vec(w)(\overrightarrow{\boldsymbol{v}} \cdot \overrightarrow{\boldsymbol{\nabla}}) \overrightarrow{\boldsymbol{R}}=(\overrightarrow{\boldsymbol{v}} \cdot \overrightarrow{\boldsymbol{\nabla}}) \overrightarrow{\boldsymbol{r}}-(\overrightarrow{\boldsymbol{v}} \cdot \overrightarrow{\boldsymbol{\nabla}}) \overrightarrow{\boldsymbol{w}}\\$\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}\right)\stackrel{\to }{\mathbit{R}}=\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}\right)\stackrel{\to }{\mathbit{r}}-\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}\right)\stackrel{\to }{\mathbit{w}}\phantom{\rule{0ex}{0ex}}$
As,
$\begin{array}{r}\stackrel{\to }{\mathbit{R}}=\stackrel{\to }{\mathbit{r}}-\stackrel{\to }{\mathbit{w}}\mathbf{\left(}{\mathbit{t}}_{\mathbit{r}}\mathbf{\right)}\end{array}$$\begin{array}{r}\stackrel{\to }{\mathbit{R}}=\stackrel{\to }{\mathbit{r}}-\stackrel{\to }{\mathbit{w}}\mathbf{\left(}{\mathbit{t}}_{\mathbit{r}}\mathbf{\right)}\end{array}${: vec(R)= vec(r)- vec(w)(t_(r)):}\begin{aligned} \overrightarrow{\boldsymbol{R}}= \overrightarrow{\boldsymbol{r}}- \overrightarrow{\boldsymbol{w}}\boldsymbol{(t_r)}\\ \end{aligned}$\begin{array}{r}\stackrel{\to }{\mathbit{R}}=\stackrel{\to }{\mathbit{r}}-\stackrel{\to }{\mathbit{w}}\mathbf{\left(}{\mathbit{t}}_{\mathbit{r}}\mathbf{\right)}\end{array}$
Here,
$\begin{array}{rl}\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}\right)\stackrel{\to }{\mathbit{r}}& =\left({v}_{x}\frac{\mathrm{\partial }}{\mathrm{\partial }x}+{v}_{y}\frac{\mathrm{\partial }}{\mathrm{\partial }y}+{v}_{z}\frac{\mathrm{\partial }}{\mathrm{\partial }z}\right)\left(x\stackrel{^}{\mathbit{i}}+y\stackrel{^}{\mathbit{j}}+z\stackrel{^}{\mathbit{k}}\right)\\ & ={v}_{x}\stackrel{^}{\mathbit{i}}+{v}_{y}\stackrel{^}{\mathbit{j}}+{v}_{z}\stackrel{^}{\mathbit{k}}=\stackrel{\to }{\mathbit{v}}\end{array}$$\begin{array}{r}\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}\right)\stackrel{\to }{\mathbit{r}}=\left({v}_{x}\frac{\mathrm{\partial }}{\mathrm{\partial }x}+{v}_{y}\frac{\mathrm{\partial }}{\mathrm{\partial }y}+{v}_{z}\frac{\mathrm{\partial }}{\mathrm{\partial }z}\right)\left(x\stackrel{^}{\mathbit{i}}+y\stackrel{^}{\mathbit{j}}+z\stackrel{^}{\mathbit{k}}\right)\\ ={v}_{x}\stackrel{^}{\mathbit{i}}+{v}_{y}\stackrel{^}{\mathbit{j}}+{v}_{z}\stackrel{^}{\mathbit{k}}=\stackrel{\to }{\mathbit{v}}\end{array}${:[( vec(v)* vec(grad)) vec(r)=(v_(x)(del)/(del x)+v_(y)(del)/(del y)+v_(z)(del)/(del z))(x hat(i)+y hat(j)+z hat(k))],[=v_(x) hat(i)+v_(y) hat(j)+v_(z) hat(k)= vec(v)]:}\begin{aligned} (\overrightarrow{\boldsymbol{v}} \cdot \overrightarrow{\boldsymbol{\nabla}}) \overrightarrow{\boldsymbol{r}} & =\left(v_x \frac{\partial}{\partial x}+v_y \frac{\partial}{\partial y}+v_z \frac{\partial}{\partial z}\right)(x \hat{\boldsymbol{i}}+y \hat{\boldsymbol{j}}+z \hat{\boldsymbol{k}}) \\ & =v_x \hat{\boldsymbol{i}}+v_y \hat{\boldsymbol{j}}+v_z \hat{\boldsymbol{k}}=\overrightarrow{\boldsymbol{v}} \end{aligned}$\begin{array}{rl}\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}\right)\stackrel{\to }{\mathbit{r}}& =\left({v}_{x}\frac{\mathrm{\partial }}{\mathrm{\partial }x}+{v}_{y}\frac{\mathrm{\partial }}{\mathrm{\partial }y}+{v}_{z}\frac{\mathrm{\partial }}{\mathrm{\partial }z}\right)\left(x\stackrel{^}{\mathbit{i}}+y\stackrel{^}{\mathbit{j}}+z\stackrel{^}{\mathbit{k}}\right)\\ & ={v}_{x}\stackrel{^}{\mathbit{i}}+{v}_{y}\stackrel{^}{\mathbit{j}}+{v}_{z}\stackrel{^}{\mathbit{k}}=\stackrel{\to }{\mathbit{v}}\end{array}$
Proceeding in a similar manner as for the 1st term in eq. (2):
$\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}\right)\stackrel{\to }{\mathbit{w}}=\stackrel{\to }{\mathbit{v}}\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)$$\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}\right)\stackrel{\to }{\mathbit{w}}=\stackrel{\to }{\mathbit{v}}\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)$( vec(v)* vec(grad)) vec(w)= vec(v)( vec(v)* vec(grad)t_(r))(\overrightarrow{\boldsymbol{v}} \cdot \overrightarrow{\boldsymbol{\nabla}}) \overrightarrow{\boldsymbol{w}}=\overrightarrow{\boldsymbol{v}}\left(\overrightarrow{\boldsymbol{v}} \cdot \overrightarrow{\boldsymbol{\nabla}} t_r\right)$\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}\right)\stackrel{\to }{\mathbit{w}}=\stackrel{\to }{\mathbit{v}}\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)$
Hence, we obtain
$\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}\right)\stackrel{\to }{\mathbit{R}}=\stackrel{\to }{\mathbit{v}}-\stackrel{\to }{\mathbit{v}}\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)$$\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}\right)\stackrel{\to }{\mathbit{R}}=\stackrel{\to }{\mathbit{v}}-\stackrel{\to }{\mathbit{v}}\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)$( vec(v)* vec(grad)) vec(R)= vec(v)- vec(v)( vec(v)* vec(grad)t_(r))(\overrightarrow{\boldsymbol{v}} \cdot \overrightarrow{\boldsymbol{\nabla}}) \overrightarrow{\boldsymbol{R}}=\overrightarrow{\boldsymbol{v}}-\overrightarrow{\boldsymbol{v}}\left(\overrightarrow{\boldsymbol{v}}\cdot \overrightarrow{\boldsymbol{\nabla}} t_r\right)$\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}\right)\stackrel{\to }{\mathbit{R}}=\stackrel{\to }{\mathbit{v}}-\stackrel{\to }{\mathbit{v}}\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)$
3rd term in eq. (2):
$\begin{array}{rl}\stackrel{\to }{\mathbf{\nabla }}×\stackrel{\to }{\mathbit{v}}& =\left(\frac{\mathrm{\partial }{v}_{z}}{\mathrm{\partial }y}-\frac{\mathrm{\partial }{v}_{y}}{\mathrm{\partial }z}\right)\stackrel{^}{\mathbit{i}}+\left(\frac{\mathrm{\partial }{v}_{x}}{\mathrm{\partial }z}-\frac{\mathrm{\partial }{v}_{z}}{\mathrm{\partial }x}\right)\stackrel{^}{\mathbit{j}}+\left(\frac{\mathrm{\partial }{v}_{y}}{\mathrm{\partial }x}-\frac{\mathrm{\partial }{v}_{x}}{\mathrm{\partial }y}\right)\stackrel{^}{\mathbit{k}}\\ & =\left(\frac{\mathrm{d}{v}_{z}}{\text{}\mathrm{d}{t}_{r}}\frac{\mathrm{\partial }{t}_{r}}{\mathrm{\partial }y}-\frac{\mathrm{d}{v}_{y}}{\text{}\mathrm{d}{t}_{r}}\frac{\mathrm{\partial }{t}_{r}}{\mathrm{\partial }z}\right)\stackrel{^}{\mathbit{i}}+\cdots \\ & =-\stackrel{\to }{\mathbit{a}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\end{array}$$\begin{array}{r}\stackrel{\to }{\mathbf{\nabla }}×\stackrel{\to }{\mathbit{v}}=\left(\frac{\mathrm{\partial }{v}_{z}}{\mathrm{\partial }y}-\frac{\mathrm{\partial }{v}_{y}}{\mathrm{\partial }z}\right)\stackrel{^}{\mathbit{i}}+\left(\frac{\mathrm{\partial }{v}_{x}}{\mathrm{\partial }z}-\frac{\mathrm{\partial }{v}_{z}}{\mathrm{\partial }x}\right)\stackrel{^}{\mathbit{j}}+\left(\frac{\mathrm{\partial }{v}_{y}}{\mathrm{\partial }x}-\frac{\mathrm{\partial }{v}_{x}}{\mathrm{\partial }y}\right)\stackrel{^}{\mathbit{k}}\\ =\left(\frac{\mathrm{d}{v}_{z}}{\text{}\mathrm{d}{t}_{r}}\frac{\mathrm{\partial }{t}_{r}}{\mathrm{\partial }y}-\frac{\mathrm{d}{v}_{y}}{\text{}\mathrm{d}{t}_{r}}\frac{\mathrm{\partial }{t}_{r}}{\mathrm{\partial }z}\right)\stackrel{^}{\mathbit{i}}+\cdots \\ =-\stackrel{\to }{\mathbit{a}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\end{array}${:[ vec(grad)xx vec(v)=((delv_(z))/(del y)-(delv_(y))/(del z)) hat(i)+((delv_(x))/(del z)-(delv_(z))/(del x)) hat(j)+((delv_(y))/(del x)-(delv_(x))/(del y)) hat(k)],[=((dv_(z))/((d)t_(r))(delt_(r))/(del y)-(dv_(y))/((d)t_(r))(delt_(r))/(del z)) hat(i)+cdots],[=- vec(a)xx vec(grad)t_(r)]:}\begin{aligned} \overrightarrow{\boldsymbol{\nabla}} \times \overrightarrow{\boldsymbol{v}} & =\left(\frac{\partial v_z}{\partial y}-\frac{\partial v_y}{\partial z}\right) \hat{\boldsymbol{i}}+\left(\frac{\partial v_x}{\partial z}-\frac{\partial v_z}{\partial x}\right) \hat{\boldsymbol{j}}+\left(\frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y}\right) \hat{\boldsymbol{k}} \\ & =\left(\frac{\mathrm{d} v_z}{\mathrm{~d} t_r} \frac{\partial t_r}{\partial y}-\frac{\mathrm{d} v_y}{\mathrm{~d} t_r} \frac{\partial t_r}{\partial z}\right) \hat{\boldsymbol{i}}+\cdots \\ & =-\overrightarrow{\boldsymbol{a}} \times \overrightarrow{\boldsymbol{\nabla}} t_r \end{aligned}
Therefore,
$\stackrel{\to }{\mathbit{R}}×\left(\stackrel{\to }{\mathrm{\nabla }}×\stackrel{\to }{\mathbit{v}}\right)=-\stackrel{\to }{\mathbit{R}}×\left(\stackrel{\to }{\mathbit{a}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)$$\stackrel{\to }{\mathbit{R}}×\left(\stackrel{\to }{\mathrm{\nabla }}×\stackrel{\to }{\mathbit{v}}\right)=-\stackrel{\to }{\mathbit{R}}×\left(\stackrel{\to }{\mathbit{a}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)$vec(R)xx( vec(grad)xx vec(v))=- vec(R)xx( vec(a)xx vec(grad)t_(r))\overrightarrow{\boldsymbol{R}} \times(\vec{\nabla} \times \overrightarrow{\boldsymbol{v}})=-\overrightarrow{\boldsymbol{R}} \times\left(\overrightarrow{\boldsymbol{a}} \times \overrightarrow{\boldsymbol{\nabla}} t_r\right)$\stackrel{\to }{\mathbit{R}}×\left(\stackrel{\to }{\mathrm{\nabla }}×\stackrel{\to }{\mathbit{v}}\right)=-\stackrel{\to }{\mathbit{R}}×\left(\stackrel{\to }{\mathbit{a}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)$
4th term in eq. (2): (Same as 3rd term in eq. 2)
$\begin{array}{rl}\stackrel{\to }{\mathbf{\nabla }}×\stackrel{\to }{\mathbit{R}}& =\stackrel{\to }{\mathbf{\nabla }}×\stackrel{\to }{\mathbit{r}}-\stackrel{\to }{\mathbf{\nabla }}×\stackrel{\to }{\mathbit{w}}\\ & =0-\left[-\stackrel{\to }{\mathbit{v}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right]\\ & =\stackrel{\to }{\mathbit{v}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\end{array}$$\begin{array}{r}\stackrel{\to }{\mathbf{\nabla }}×\stackrel{\to }{\mathbit{R}}=\stackrel{\to }{\mathbf{\nabla }}×\stackrel{\to }{\mathbit{r}}-\stackrel{\to }{\mathbf{\nabla }}×\stackrel{\to }{\mathbit{w}}\\ =0-\left[-\stackrel{\to }{\mathbit{v}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right]\\ =\stackrel{\to }{\mathbit{v}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\end{array}${:[ vec(grad)xx vec(R)= vec(grad)xx vec(r)- vec(grad)xx vec(w)],[=0-[- vec(v)xx vec(grad)t_(r)]],[= vec(v)xx vec(grad)t_(r)]:}\begin{aligned} \overrightarrow{\boldsymbol{\nabla}} \times \overrightarrow{\boldsymbol{R}} & =\overrightarrow{\boldsymbol{\nabla}} \times \overrightarrow{\boldsymbol{r}}-\overrightarrow{\boldsymbol{\nabla}} \times \overrightarrow{\boldsymbol{w}} \\ & =0-\left[-\overrightarrow{\boldsymbol{v}} \times \overrightarrow{\boldsymbol{\nabla}} t_r\right]\\ & =\overrightarrow{\boldsymbol{v}} \times \overrightarrow{\boldsymbol{\nabla}} t_r \end{aligned}$\begin{array}{rl}\stackrel{\to }{\mathbf{\nabla }}×\stackrel{\to }{\mathbit{R}}& =\stackrel{\to }{\mathbf{\nabla }}×\stackrel{\to }{\mathbit{r}}-\stackrel{\to }{\mathbf{\nabla }}×\stackrel{\to }{\mathbit{w}}\\ & =0-\left[-\stackrel{\to }{\mathbit{v}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right]\\ & =\stackrel{\to }{\mathbit{v}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\end{array}$
Therefore,
$\stackrel{\to }{\mathbit{v}}×\left(\stackrel{\to }{\mathbf{\nabla }}×\stackrel{\to }{\mathbit{R}}\right)=\stackrel{\to }{\mathbit{v}}×\left(\stackrel{\to }{\mathbit{v}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)$$\stackrel{\to }{\mathbit{v}}×\left(\stackrel{\to }{\mathbf{\nabla }}×\stackrel{\to }{\mathbit{R}}\right)=\stackrel{\to }{\mathbit{v}}×\left(\stackrel{\to }{\mathbit{v}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)$vec(v)xx( vec(grad)xx vec(R))= vec(v)xx( vec(v)xx vec(grad)t_(r))\overrightarrow{\boldsymbol{v}} \times(\overrightarrow{\boldsymbol{\nabla}} \times \overrightarrow{\boldsymbol{R}})=\overrightarrow{\boldsymbol{v}} \times\left(\overrightarrow{\boldsymbol{v}} \times \overrightarrow{\boldsymbol{\nabla}} t_r\right)$\stackrel{\to }{\mathbit{v}}×\left(\stackrel{\to }{\mathbf{\nabla }}×\stackrel{\to }{\mathbit{R}}\right)=\stackrel{\to }{\mathbit{v}}×\left(\stackrel{\to }{\mathbit{v}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)$
Then eq. (2) becomes,
$\begin{array}{rl}\stackrel{\to }{\mathrm{\nabla }}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{v}}\right)=& \stackrel{\to }{\mathbit{a}}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)+\stackrel{\to }{\mathbit{v}}-\stackrel{\to }{\mathbit{v}}\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)\\ & -\stackrel{\to }{\mathbit{R}}×\left(\stackrel{\to }{\mathbit{a}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)+\stackrel{\to }{\mathbit{v}}×\left(\stackrel{\to }{\mathbit{v}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)\end{array}$$\begin{array}{r}\stackrel{\to }{\mathrm{\nabla }}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{v}}\right)=\stackrel{\to }{\mathbit{a}}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)+\stackrel{\to }{\mathbit{v}}-\stackrel{\to }{\mathbit{v}}\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)\\ -\stackrel{\to }{\mathbit{R}}×\left(\stackrel{\to }{\mathbit{a}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)+\stackrel{\to }{\mathbit{v}}×\left(\stackrel{\to }{\mathbit{v}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)\end{array}${:[ vec(grad)( vec(R)* vec(v))= vec(a)( vec(R)* vec(grad)t_(r))+ vec(v)- vec(v)( vec(v)* vec(grad)t_(r))],[- vec(R)xx( vec(a)xx vec(grad)t_(r))+ vec(v)xx( vec(v)xx vec(grad)t_(r))]:}\begin{aligned} \vec{\nabla}(\overrightarrow{\boldsymbol{R}} \cdot \overrightarrow{\boldsymbol{v}})= & \overrightarrow{\boldsymbol{a}}\left(\overrightarrow{\boldsymbol{R}} \cdot \overrightarrow{\boldsymbol{\nabla}} t_r\right)+\overrightarrow{\boldsymbol{v}}-\overrightarrow{\boldsymbol{v}}\left(\overrightarrow{\boldsymbol{v}} \cdot \overrightarrow{\boldsymbol{\nabla}} t_r\right) \\ & -\overrightarrow{\boldsymbol{R}} \times\left(\overrightarrow{\boldsymbol{a}} \times \overrightarrow{\boldsymbol{\nabla}} t_r\right)+\overrightarrow{\boldsymbol{v}} \times\left(\overrightarrow{\boldsymbol{v}} \times \overrightarrow{\boldsymbol{\nabla}} t_r\right) \end{aligned}$\begin{array}{rl}\stackrel{\to }{\mathrm{\nabla }}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{v}}\right)=& \stackrel{\to }{\mathbit{a}}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)+\stackrel{\to }{\mathbit{v}}-\stackrel{\to }{\mathbit{v}}\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)\\ & -\stackrel{\to }{\mathbit{R}}×\left(\stackrel{\to }{\mathbit{a}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)+\stackrel{\to }{\mathbit{v}}×\left(\stackrel{\to }{\mathbit{v}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)\end{array}$
Using BAC-CAB rule:
$\begin{array}{rl}\stackrel{\to }{\mathbit{A}}×\left(\stackrel{\to }{\mathbit{B}}×\stackrel{\to }{\mathbit{C}}\right)& =\stackrel{\to }{\mathbit{B}}\left(\stackrel{\to }{\mathbit{A}}\cdot \stackrel{\to }{\mathbit{C}}\right)-\stackrel{\to }{\mathbit{C}}\left(\stackrel{\to }{\mathbit{A}}\cdot \stackrel{\to }{\mathbit{B}}\right)\\ \stackrel{\to }{\mathbit{R}}×\left(\stackrel{\to }{\mathbit{a}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)& =\stackrel{\to }{\mathbit{a}}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathrm{\nabla }}{t}_{r}\right)-\stackrel{\to }{\mathrm{\nabla }}{t}_{r}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{a}}\right)\\ \stackrel{\to }{\mathbit{v}}×\left(\stackrel{\to }{\mathbit{v}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)& =\stackrel{\to }{\mathbit{v}}\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)-{v}^{2}\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\end{array}$$\begin{array}{r}\stackrel{\to }{\mathbit{A}}×\left(\stackrel{\to }{\mathbit{B}}×\stackrel{\to }{\mathbit{C}}\right)=\stackrel{\to }{\mathbit{B}}\left(\stackrel{\to }{\mathbit{A}}\cdot \stackrel{\to }{\mathbit{C}}\right)-\stackrel{\to }{\mathbit{C}}\left(\stackrel{\to }{\mathbit{A}}\cdot \stackrel{\to }{\mathbit{B}}\right)\\ \stackrel{\to }{\mathbit{R}}×\left(\stackrel{\to }{\mathbit{a}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)=\stackrel{\to }{\mathbit{a}}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathrm{\nabla }}{t}_{r}\right)-\stackrel{\to }{\mathrm{\nabla }}{t}_{r}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{a}}\right)\\ \stackrel{\to }{\mathbit{v}}×\left(\stackrel{\to }{\mathbit{v}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)=\stackrel{\to }{\mathbit{v}}\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)-{v}^{2}\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\end{array}${:[ vec(A)xx( vec(B)xx vec(C))= vec(B)( vec(A)* vec(C))- vec(C)( vec(A)* vec(B))],[ vec(R)xx( vec(a)xx vec(grad)t_(r))= vec(a)( vec(R)*( vec(grad))t_(r))- vec(grad)t_(r)( vec(R)* vec(a))],[ vec(v)xx( vec(v)xx vec(grad)t_(r))= vec(v)( vec(v)* vec(grad)t_(r))-v^(2) vec(grad)t_(r)]:}\begin{aligned} \overrightarrow{\boldsymbol{A}} \times(\overrightarrow{\boldsymbol{B}} \times \overrightarrow{\boldsymbol{C}}) & =\overrightarrow{\boldsymbol{B}}(\overrightarrow{\boldsymbol{A}} \cdot \overrightarrow{\boldsymbol{C}})-\overrightarrow{\boldsymbol{C}}(\overrightarrow{\boldsymbol{A}} \cdot \overrightarrow{\boldsymbol{B}}) \\ \overrightarrow{\boldsymbol{R}} \times\left(\overrightarrow{\boldsymbol{a}} \times \overrightarrow{\boldsymbol{\nabla}} t_r\right) & =\overrightarrow{\boldsymbol{a}}\left(\overrightarrow{\boldsymbol{R}} \cdot \vec{\nabla} t_r\right)-\vec{\nabla} t_r(\overrightarrow{\boldsymbol{R}} \cdot \overrightarrow{\boldsymbol{a}}) \\ \overrightarrow{\boldsymbol{v}} \times\left(\overrightarrow{\boldsymbol{v}} \times \overrightarrow{\boldsymbol{\nabla}} t_r\right) & =\overrightarrow{\boldsymbol{v}}\left(\overrightarrow{\boldsymbol{v}} \cdot \overrightarrow{\boldsymbol{\nabla}} t_r\right)-v^2 \overrightarrow{\boldsymbol{\nabla}} t_r \end{aligned}$\begin{array}{rl}\stackrel{\to }{\mathbit{A}}×\left(\stackrel{\to }{\mathbit{B}}×\stackrel{\to }{\mathbit{C}}\right)& =\stackrel{\to }{\mathbit{B}}\left(\stackrel{\to }{\mathbit{A}}\cdot \stackrel{\to }{\mathbit{C}}\right)-\stackrel{\to }{\mathbit{C}}\left(\stackrel{\to }{\mathbit{A}}\cdot \stackrel{\to }{\mathbit{B}}\right)\\ \stackrel{\to }{\mathbit{R}}×\left(\stackrel{\to }{\mathbit{a}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)& =\stackrel{\to }{\mathbit{a}}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathrm{\nabla }}{t}_{r}\right)-\stackrel{\to }{\mathrm{\nabla }}{t}_{r}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{a}}\right)\\ \stackrel{\to }{\mathbit{v}}×\left(\stackrel{\to }{\mathbit{v}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)& =\stackrel{\to }{\mathbit{v}}\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)-{v}^{2}\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\end{array}$
Therefore,
$\begin{array}{rl}\stackrel{\to }{\mathrm{\nabla }}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{v}}\right)& =\stackrel{\to }{\mathbit{a}}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathrm{\nabla }}{t}_{r}\right)+\stackrel{\to }{\mathbit{v}}-\stackrel{\to }{\mathbit{v}}\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathrm{\nabla }}{t}_{r}\right)\\ & -\stackrel{\to }{\mathbit{a}}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathrm{\nabla }}{t}_{r}\right)+\stackrel{\to }{\mathrm{\nabla }}{t}_{r}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{a}}\right)+\stackrel{\to }{\mathbit{v}}\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)-{v}^{2}\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\\ ⇒\stackrel{\to }{\mathrm{\nabla }}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{v}}\right)& =\stackrel{\to }{\mathbit{v}}+\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{a}}-{v}^{2}\right)\stackrel{\to }{\mathrm{\nabla }}{t}_{r}\end{array}$$\begin{array}{r}\stackrel{\to }{\mathrm{\nabla }}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{v}}\right)=\stackrel{\to }{\mathbit{a}}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathrm{\nabla }}{t}_{r}\right)+\stackrel{\to }{\mathbit{v}}-\stackrel{\to }{\mathbit{v}}\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathrm{\nabla }}{t}_{r}\right)\\ -\stackrel{\to }{\mathbit{a}}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathrm{\nabla }}{t}_{r}\right)+\stackrel{\to }{\mathrm{\nabla }}{t}_{r}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{a}}\right)+\stackrel{\to }{\mathbit{v}}\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)-{v}^{2}\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\\ ⇒\stackrel{\to }{\mathrm{\nabla }}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{v}}\right)=\stackrel{\to }{\mathbit{v}}+\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{a}}-{v}^{2}\right)\stackrel{\to }{\mathrm{\nabla }}{t}_{r}\end{array}${:[ vec(grad)( vec(R)* vec(v))= vec(a)( vec(R)*( vec(grad))t_(r))+ vec(v)- vec(v)( vec(v)*( vec(grad))t_(r))],[- vec(a)( vec(R)*( vec(grad))t_(r))+ vec(grad)t_(r)( vec(R)* vec(a))+ vec(v)( vec(v)* vec(grad)t_(r))-v^(2) vec(grad)t_(r)],[=> vec(grad)( vec(R)* vec(v))= vec(v)+( vec(R)* vec(a)-v^(2)) vec(grad)t_(r)]:}\begin{aligned} \vec{\nabla}(\overrightarrow{\boldsymbol{R}} \cdot \overrightarrow{\boldsymbol{v}}) & =\overrightarrow{\boldsymbol{a}}\left(\overrightarrow{\boldsymbol{R}} \cdot \vec{\nabla} t_r\right)+\overrightarrow{\boldsymbol{v}}-\overrightarrow{\boldsymbol{v}}\left(\overrightarrow{\boldsymbol{v}} \cdot \vec{\nabla} t_r\right) \\ & -\overrightarrow{\boldsymbol{a}}\left(\overrightarrow{\boldsymbol{R}} \cdot \vec{\nabla} t_r\right)+\vec{\nabla} t_r(\overrightarrow{\boldsymbol{R}} \cdot \overrightarrow{\boldsymbol{a}})+\overrightarrow{\boldsymbol{v}}\left(\overrightarrow{\boldsymbol{v}} \cdot \overrightarrow{\boldsymbol{\nabla}} t_r\right)-v^2 \overrightarrow{\boldsymbol{\nabla}} t_r \\ \Rightarrow \vec{\nabla}(\overrightarrow{\boldsymbol{R}} \cdot \overrightarrow{\boldsymbol{v}}) & =\overrightarrow{\boldsymbol{v}}+\left(\overrightarrow{\boldsymbol{R}} \cdot \overrightarrow{\boldsymbol{a}}-v^2\right) \vec{\nabla} t_r \end{aligned}$\begin{array}{rl}\stackrel{\to }{\mathrm{\nabla }}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{v}}\right)& =\stackrel{\to }{\mathbit{a}}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathrm{\nabla }}{t}_{r}\right)+\stackrel{\to }{\mathbit{v}}-\stackrel{\to }{\mathbit{v}}\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathrm{\nabla }}{t}_{r}\right)\\ & -\stackrel{\to }{\mathbit{a}}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathrm{\nabla }}{t}_{r}\right)+\stackrel{\to }{\mathrm{\nabla }}{t}_{r}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{a}}\right)+\stackrel{\to }{\mathbit{v}}\left(\stackrel{\to }{\mathbit{v}}\cdot \stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)-{v}^{2}\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\\ ⇒\stackrel{\to }{\mathrm{\nabla }}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{v}}\right)& =\stackrel{\to }{\mathbit{v}}+\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{a}}-{v}^{2}\right)\stackrel{\to }{\mathrm{\nabla }}{t}_{r}\end{array}$
Hence, using above results in eq. (1):
$\begin{array}{rl}& \stackrel{\to }{\mathrm{\nabla }}\mathrm{\Phi }=\frac{qc}{4\pi {ϵ}_{0}}\frac{1}{\left(Rc-\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{v}}{\right)}^{2}}\left[\stackrel{\to }{\mathbit{v}}+\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{a}}-{v}^{2}\right)\stackrel{\to }{\mathrm{\nabla }}{t}_{r}+{c}^{2}\stackrel{\to }{\mathrm{\nabla }}{t}_{r}\right]\end{array}$$\begin{array}{r}\stackrel{\to }{\mathrm{\nabla }}\mathrm{\Phi }=\frac{qc}{4\pi {ϵ}_{0}}\frac{1}{\left(Rc-\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{v}}{\right)}^{2}}\left[\stackrel{\to }{\mathbit{v}}+\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{a}}-{v}^{2}\right)\stackrel{\to }{\mathrm{\nabla }}{t}_{r}+{c}^{2}\stackrel{\to }{\mathrm{\nabla }}{t}_{r}\right]\end{array}${: vec(grad)Phi=(qc)/(4piepsilon_(0))(1)/((Rc- vec(R)* vec(v))^(2))[ vec(v)+( vec(R)* vec(a)-v^(2))( vec(grad))t_(r)+c^(2)( vec(grad))t_(r)]:}\begin{aligned} & \vec{\nabla} \Phi=\frac{q c}{4 \pi \epsilon_0} \frac{1}{(R c-\overrightarrow{\boldsymbol{R}} \cdot \overrightarrow{\boldsymbol{v}})^2}\left[\overrightarrow{\boldsymbol{v}}+\left(\overrightarrow{\boldsymbol{R}} \cdot \overrightarrow{\boldsymbol{a}}-v^2\right) \vec{\nabla} t_r+c^2 \vec{\nabla} t_r\right] \\ \end{aligned}$\begin{array}{rl}& \stackrel{\to }{\mathrm{\nabla }}\mathrm{\Phi }=\frac{qc}{4\pi {ϵ}_{0}}\frac{1}{\left(Rc-\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{v}}{\right)}^{2}}\left[\stackrel{\to }{\mathbit{v}}+\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{a}}-{v}^{2}\right)\stackrel{\to }{\mathrm{\nabla }}{t}_{r}+{c}^{2}\stackrel{\to }{\mathrm{\nabla }}{t}_{r}\right]\end{array}$
$\stackrel{\to }{\mathbf{\nabla }}\mathrm{\Phi }=\frac{qc}{4\pi {ϵ}_{0}}\frac{1}{\left(Rc-\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{v}}{\right)}^{2}}\left[\stackrel{\to }{\mathbit{v}}+\left({c}^{2}-{v}^{2}+\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{a}}\right)\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right],\dots \left(3\right)\phantom{\rule{0ex}{0ex}}$$\stackrel{\to }{\mathbf{\nabla }}\mathrm{\Phi }=\frac{qc}{4\pi {ϵ}_{0}}\frac{1}{\left(Rc-\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{v}}{\right)}^{2}}\left[\stackrel{\to }{\mathbit{v}}+\left({c}^{2}-{v}^{2}+\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{a}}\right)\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right],\dots \left(3\right)\phantom{\rule{0ex}{0ex}}$vec(grad)Phi=(qc)/(4piepsilon_(0))(1)/((Rc- vec(R)* vec(v))^(2))[ vec(v)+(c^(2)-v^(2)+ vec(R)* vec(a)) vec(grad)t_(r)],dots(3)\overrightarrow{\boldsymbol{\nabla}} \Phi=\frac{q c}{4 \pi \epsilon_0} \frac{1}{(R c-\overrightarrow{\boldsymbol{R}} \cdot \overrightarrow{\boldsymbol{v}})^2}\left[\overrightarrow{\boldsymbol{v}}+\left(c^2-v^2+\overrightarrow{\boldsymbol{R}} \cdot \overrightarrow{\boldsymbol{a}}\right) \overrightarrow{\boldsymbol{\nabla}} t_r\right],\ldots\left(\mathrm{3}\right)\\$\stackrel{\to }{\mathbf{\nabla }}\mathrm{\Phi }=\frac{qc}{4\pi {ϵ}_{0}}\frac{1}{\left(Rc-\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{v}}{\right)}^{2}}\left[\stackrel{\to }{\mathbit{v}}+\left({c}^{2}-{v}^{2}+\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{a}}\right)\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right],\dots \left(3\right)\phantom{\rule{0ex}{0ex}}$
To find $\stackrel{\to }{\mathrm{\nabla }}{t}_{r}$$\stackrel{\to }{\mathrm{\nabla }}{t}_{r}$vec(grad)t_(r)\vec{\nabla} t_r$\stackrel{\to }{\mathrm{\nabla }}{t}_{r}$, we know:
$\begin{array}{rl}-c\stackrel{\to }{\mathbf{\nabla }}{t}_{r}& =\stackrel{\to }{\mathbf{\nabla }}R=\stackrel{\to }{\mathbf{\nabla }}\sqrt{\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{R}}}=\frac{1}{2\sqrt{\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{R}}}}\stackrel{\to }{\mathbf{\nabla }}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{R}}\right)\\ & =\frac{1}{R}\left[\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathrm{\nabla }}\right)\stackrel{\to }{\mathbit{R}}+\stackrel{\to }{\mathbit{R}}×\left(\stackrel{\to }{\mathbf{\nabla }}×\stackrel{\to }{\mathbit{R}}\right)\right]\end{array}$$\begin{array}{r}-c\stackrel{\to }{\mathbf{\nabla }}{t}_{r}=\stackrel{\to }{\mathbf{\nabla }}R=\stackrel{\to }{\mathbf{\nabla }}\sqrt{\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{R}}}=\frac{1}{2\sqrt{\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{R}}}}\stackrel{\to }{\mathbf{\nabla }}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{R}}\right)\\ =\frac{1}{R}\left[\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathrm{\nabla }}\right)\stackrel{\to }{\mathbit{R}}+\stackrel{\to }{\mathbit{R}}×\left(\stackrel{\to }{\mathbf{\nabla }}×\stackrel{\to }{\mathbit{R}}\right)\right]\end{array}${:[-c vec(grad)t_(r)= vec(grad)R= vec(grad)sqrt( vec(R)* vec(R))=(1)/(2sqrt( vec(R)* vec(R))) vec(grad)( vec(R)* vec(R))],[=(1)/(R)[( vec(R)* vec(grad)) vec(R)+ vec(R)xx( vec(grad)xx vec(R))]]:}\begin{aligned} -c \overrightarrow{\boldsymbol{\nabla}} t_r & =\overrightarrow{\boldsymbol{\nabla}} R=\overrightarrow{\boldsymbol{\nabla}} \sqrt{\overrightarrow{\boldsymbol{R}} \cdot \overrightarrow{\boldsymbol{R}}}=\frac{1}{2 \sqrt{\overrightarrow{\boldsymbol{R}} \cdot \overrightarrow{\boldsymbol{R}}}} \overrightarrow{\boldsymbol{\nabla}}(\overrightarrow{\boldsymbol{R}} \cdot \overrightarrow{\boldsymbol{R}}) \\ & =\frac{1}{R}[(\overrightarrow{\boldsymbol{R}} \cdot \vec{\nabla}) \overrightarrow{\boldsymbol{R}}+\overrightarrow{\boldsymbol{R}} \times(\overrightarrow{\boldsymbol{\nabla}} \times \overrightarrow{\boldsymbol{R}})] \end{aligned}$\begin{array}{rl}-c\stackrel{\to }{\mathbf{\nabla }}{t}_{r}& =\stackrel{\to }{\mathbf{\nabla }}R=\stackrel{\to }{\mathbf{\nabla }}\sqrt{\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{R}}}=\frac{1}{2\sqrt{\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{R}}}}\stackrel{\to }{\mathbf{\nabla }}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{R}}\right)\\ & =\frac{1}{R}\left[\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathrm{\nabla }}\right)\stackrel{\to }{\mathbit{R}}+\stackrel{\to }{\mathbit{R}}×\left(\stackrel{\to }{\mathbf{\nabla }}×\stackrel{\to }{\mathbit{R}}\right)\right]\end{array}$
It can be shown that
$\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathrm{\nabla }}\right)\stackrel{\to }{\mathbit{R}}=\stackrel{\to }{\mathbit{R}}-\stackrel{\to }{\mathbit{v}}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathrm{\nabla }}{t}_{r}\right)$$\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathrm{\nabla }}\right)\stackrel{\to }{\mathbit{R}}=\stackrel{\to }{\mathbit{R}}-\stackrel{\to }{\mathbit{v}}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathrm{\nabla }}{t}_{r}\right)$( vec(R)* vec(grad)) vec(R)= vec(R)- vec(v)( vec(R)*( vec(grad))t_(r))(\overrightarrow{\boldsymbol{R}} \cdot \vec{\nabla}) \overrightarrow{\boldsymbol{R}}=\overrightarrow{\boldsymbol{R}}-\overrightarrow{\boldsymbol{v}}\left(\overrightarrow{\boldsymbol{R}} \cdot \vec{\nabla} t_r\right)$\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathrm{\nabla }}\right)\stackrel{\to }{\mathbit{R}}=\stackrel{\to }{\mathbit{R}}-\stackrel{\to }{\mathbit{v}}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathrm{\nabla }}{t}_{r}\right)$
and $\stackrel{\to }{\mathrm{\nabla }}×\stackrel{\to }{\mathbit{R}}=\stackrel{\to }{\mathbit{v}}×\stackrel{\to }{\mathrm{\nabla }}{t}_{r}$$\stackrel{\to }{\mathrm{\nabla }}×\stackrel{\to }{\mathbit{R}}=\stackrel{\to }{\mathbit{v}}×\stackrel{\to }{\mathrm{\nabla }}{t}_{r}$vec(grad)xx vec(R)= vec(v)xx vec(grad)t_(r)\vec{\nabla} \times \overrightarrow{\boldsymbol{R}}=\overrightarrow{\boldsymbol{v}} \times \vec{\nabla} t_r$\stackrel{\to }{\mathrm{\nabla }}×\stackrel{\to }{\mathbit{R}}=\stackrel{\to }{\mathbit{v}}×\stackrel{\to }{\mathrm{\nabla }}{t}_{r}$\
Therefore,
$\begin{array}{rl}-c\stackrel{\to }{\mathbf{\nabla }}{t}_{r}& =\frac{1}{R}\left[\stackrel{\to }{\mathbit{R}}-\stackrel{\to }{\mathbit{v}}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)+\stackrel{\to }{\mathbit{R}}×\stackrel{\to }{\mathbit{v}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right]\end{array}$$\begin{array}{r}-c\stackrel{\to }{\mathbf{\nabla }}{t}_{r}=\frac{1}{R}\left[\stackrel{\to }{\mathbit{R}}-\stackrel{\to }{\mathbit{v}}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)+\stackrel{\to }{\mathbit{R}}×\stackrel{\to }{\mathbit{v}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right]\end{array}${:-c vec(grad)t_(r)=(1)/(R)[ vec(R)- vec(v)( vec(R)* vec(grad)t_(r))+ vec(R)xx vec(v)xx vec(grad)t_(r)]:}\begin{aligned} -c \overrightarrow{\boldsymbol{\nabla}} t_r & =\frac{1}{R}\left[\overrightarrow{\boldsymbol{R}}-\overrightarrow{\boldsymbol{v}}\left(\overrightarrow{\boldsymbol{R}} \cdot \overrightarrow{\boldsymbol{\nabla}} t_r\right)+\overrightarrow{\boldsymbol{R}} \times \overrightarrow{\boldsymbol{v}} \times \overrightarrow{\boldsymbol{\nabla}} t_r\right] \\ \end{aligned}$\begin{array}{rl}-c\stackrel{\to }{\mathbf{\nabla }}{t}_{r}& =\frac{1}{R}\left[\stackrel{\to }{\mathbit{R}}-\stackrel{\to }{\mathbit{v}}\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right)+\stackrel{\to }{\mathbit{R}}×\stackrel{\to }{\mathbit{v}}×\stackrel{\to }{\mathbf{\nabla }}{t}_{r}\right]\end{array}$
Using BAC-CAB rule:
$\begin{array}{rl}⇒-c\stackrel{\to }{\mathbf{\nabla }}{t}_{r}& =\frac{1}{R}\left[\stackrel{\to }{\mathbit{R}}-\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{v}}\right)\stackrel{\to }{\mathrm{\nabla }}{t}_{r}\right]\\ ⇒\stackrel{\to }{\mathbf{\nabla }}{t}_{r}& =\frac{-\stackrel{\to }{\mathbit{R}}}{Rc-\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{v}}},\dots \left(4\right)\end{array}$$\begin{array}{r}⇒-c\stackrel{\to }{\mathbf{\nabla }}{t}_{r}=\frac{1}{R}\left[\stackrel{\to }{\mathbit{R}}-\left(\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{v}}\right)\stackrel{\to }{\mathrm{\nabla }}{t}_{r}\right]\\ ⇒\stackrel{\to }{\mathbf{\nabla }}{t}_{r}=\frac{-\stackrel{\to }{\mathbit{R}}}{Rc-\stackrel{\to }{\mathbit{R}}\cdot \stackrel{\to }{\mathbit{v}}},\dots \left(4\right)\end{array}${:[=>-c vec(grad)t_(r)=(1)/(R)[ vec(R)-( vec(R)* vec(v))( vec(grad))t_(r)]],[=> vec(grad)t_(r)=(- vec(R))/(Rc- vec(R)* vec(v))”,”dots(4)]:}\begin{aligned} \Rightarrow-c \overrightarrow{\boldsymbol{\nabla}} t_r & =\frac{1}{R}\left[\overrightarrow{\boldsymbol{R}}-(\overrightarrow{\boldsymbol{R}} \cdot \overrightarrow{\boldsymbol{v}}) \vec{\nabla} t_r\right] \\ \Rightarrow \overrightarrow{\boldsymbol{\nabla}} t_r & =\frac{-\overrightarrow{\boldsymbol{R}}}{R c-\overrightarrow{\boldsymbol{R}} \cdot \overrightarrow{\boldsymbol{v}}},\ldots\left(\mathrm{4}\right)\\ \end{aligned}