# Invariance of Maxwell Field Equations under Lorentz Transformation

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The relativistic invariance of Maxwell’s field equations is to be established now. The field equations for free space are
$\begin{array}{rlr}\stackrel{\to }{\mathrm{\nabla }}\cdot \mathbf{E}& =0,& \dots \left(\mathrm{a}\right)\\ \stackrel{\to }{\mathrm{\nabla }}\cdot \mathbf{B}& =0,& \dots \left(\mathrm{b}\right)\\ \stackrel{\to }{\mathrm{\nabla }}×\mathbf{E}& =-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}& \dots \left(\mathrm{c}\right)\\ \stackrel{\to }{\mathrm{\nabla }}×\mathbf{B}& =\frac{1}{{c}^{2}}\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}& \dots \left(\mathrm{d}\right)\end{array}$$\begin{array}{r}\stackrel{\to }{\mathrm{\nabla }}\cdot \mathbf{E}=0,\dots \left(\mathrm{a}\right)\\ \stackrel{\to }{\mathrm{\nabla }}\cdot \mathbf{B}=0,\dots \left(\mathrm{b}\right)\\ \stackrel{\to }{\mathrm{\nabla }}×\mathbf{E}=-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}\dots \left(\mathrm{c}\right)\\ \stackrel{\to }{\mathrm{\nabla }}×\mathbf{B}=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\dots \left(\mathrm{d}\right)\end{array}${:[ vec(grad)*E=0″,” dots(a)],[ vec(grad)*B=0″,” dots(b)],[ vec(grad)xxE=-(delB)/(del t) dots(c)],[ vec(grad)xxB=(1)/(c^(2))(delE)/(del t) dots(d)]:}\begin{aligned} \vec{\nabla} \cdot \mathbf{E} & =0, & \ldots\left(\mathrm{a}\right) \\ \vec{\nabla} \cdot \mathbf{B} & =0, & \ldots\left(\mathrm{b}\right) \\ \vec{\nabla} \times \mathbf{E} & =-\frac{\partial \mathbf{B}}{\partial t} & \ldots\left(\mathrm{c}\right) \\ \vec{\nabla} \times \mathbf{B} & =\frac {1} {c^{2}}\frac{\partial \mathbf{E}}{\partial t} & \ldots\left(\mathrm{d}\right) \end{aligned}$\begin{array}{rlr}\stackrel{\to }{\mathrm{\nabla }}\cdot \mathbf{E}& =0,& \dots \left(\mathrm{a}\right)\\ \stackrel{\to }{\mathrm{\nabla }}\cdot \mathbf{B}& =0,& \dots \left(\mathrm{b}\right)\\ \stackrel{\to }{\mathrm{\nabla }}×\mathbf{E}& =-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}& \dots \left(\mathrm{c}\right)\\ \stackrel{\to }{\mathrm{\nabla }}×\mathbf{B}& =\frac{1}{{c}^{2}}\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}& \dots \left(\mathrm{d}\right)\end{array}$
because $\rho =0,J=0,$$\rho =0,J=0,$rho=0,J=0,\rho=0, J=0,$\rho =0,J=0,$ for free space.
Let a frame of reference ${F}^{\mathrm{\prime }}\left({x}^{\mathrm{\prime }},{y}^{\mathrm{\prime }},{z}^{\mathrm{\prime }},{t}^{\mathrm{\prime }}\right)$${F}^{\mathrm{\prime }}\left({x}^{\mathrm{\prime }},{y}^{\mathrm{\prime }},{z}^{\mathrm{\prime }},{t}^{\mathrm{\prime }}\right)$F^(‘)(x^(‘),y^(‘),z^(‘),t^(‘))F^{\prime}\left(x^{\prime}, y^{\prime}, z^{\prime}, t^{\prime}\right)${F}^{\mathrm{\prime }}\left({x}^{\mathrm{\prime }},{y}^{\mathrm{\prime }},{z}^{\mathrm{\prime }},{t}^{\mathrm{\prime }}\right)$ move with uniform velocity $v$$v$vv$v$ in positive $x$$x$xx$x$-direction with respect to frame $F\left(x,y,z,t\right)$$F\left(x,y,z,t\right)$F(x,y,z,t)F(x, y, z, t)$F\left(x,y,z,t\right)$. Then the eqs. (a-d) in $F$$F$FF$F$ should retain the same form in ${F}^{\mathrm{\prime }}$${F}^{\mathrm{\prime }}$F^(‘)F^{\prime}${F}^{\mathrm{\prime }}$ as given below by eqs. (a’-d’) to establish their invariance,
$\begin{array}{rlr}\left(\stackrel{\to }{\mathrm{\nabla }}\cdot \mathbf{E}{\right)}^{\mathrm{\prime }}& =0,& \dots \left({\mathrm{a}}^{\mathrm{\prime }}\right)\\ \left(\stackrel{\to }{\mathrm{\nabla }}\cdot \mathbf{H}{\right)}^{\mathrm{\prime }}& =0,& \dots \left({\mathrm{b}}^{\mathrm{\prime }}\right)\\ \left(\stackrel{\to }{\mathrm{\nabla }}×\mathbf{E}{\right)}^{\mathrm{\prime }}& =-\frac{\mathrm{\partial }{\mathbf{B}}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}},& \dots \left({\mathrm{c}}^{\mathrm{\prime }}\right)\\ \left(\stackrel{\to }{\mathrm{\nabla }}×\mathbf{B}{\right)}^{\mathrm{\prime }}& =\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{\mathbf{E}}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}},& \dots \left({\mathrm{d}}^{\mathrm{\prime }}\right)\end{array}$$\begin{array}{r}\left(\stackrel{\to }{\mathrm{\nabla }}\cdot \mathbf{E}{\right)}^{\mathrm{\prime }}=0,\dots \left({\mathrm{a}}^{\mathrm{\prime }}\right)\\ \left(\stackrel{\to }{\mathrm{\nabla }}\cdot \mathbf{H}{\right)}^{\mathrm{\prime }}=0,\dots \left({\mathrm{b}}^{\mathrm{\prime }}\right)\\ \left(\stackrel{\to }{\mathrm{\nabla }}×\mathbf{E}{\right)}^{\mathrm{\prime }}=-\frac{\mathrm{\partial }{\mathbf{B}}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}},\dots \left({\mathrm{c}}^{\mathrm{\prime }}\right)\\ \left(\stackrel{\to }{\mathrm{\nabla }}×\mathbf{B}{\right)}^{\mathrm{\prime }}=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{\mathbf{E}}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}},\dots \left({\mathrm{d}}^{\mathrm{\prime }}\right)\end{array}${:[( vec(grad)*E)^(‘)=0″,” dots(a^(‘))],[( vec(grad)*H)^(‘)=0″,” dots(b^(‘))],[( vec(grad)xxE)^(‘)=-(delB^(‘))/(delt^(‘))”,” dots(c^(‘))],[( vec(grad)xxB)^(‘)=(1)/(c^(2))(delE^(‘))/(delt^(‘))”,” dots(d^(‘))]:}\begin{aligned} (\vec{\nabla} \cdot \mathbf{E})^{\prime} & =0, & \ldots\left(\mathrm{a}^{\prime}\right) \\ (\vec{\nabla} \cdot \mathbf{H})^{\prime} & =0, & \ldots\left(\mathrm{b}^{\prime}\right) \\ (\vec{\nabla} \times \mathbf{E})^{\prime} & =-\frac{\partial \mathbf{B}^{\prime}}{\partial t^{\prime}}, & \ldots\left(\mathrm{c}^{\prime}\right) \\ (\vec{\nabla} \times \mathbf{B})^{\prime} & =\frac {1} {c^{2}}\frac{\partial \mathbf{E}^{\prime}}{\partial t^{\prime}}, & \ldots\left(\mathrm{d}^{\prime}\right) \end{aligned}$\begin{array}{rlr}\left(\stackrel{\to }{\mathrm{\nabla }}\cdot \mathbf{E}{\right)}^{\mathrm{\prime }}& =0,& \dots \left({\mathrm{a}}^{\mathrm{\prime }}\right)\\ \left(\stackrel{\to }{\mathrm{\nabla }}\cdot \mathbf{H}{\right)}^{\mathrm{\prime }}& =0,& \dots \left({\mathrm{b}}^{\mathrm{\prime }}\right)\\ \left(\stackrel{\to }{\mathrm{\nabla }}×\mathbf{E}{\right)}^{\mathrm{\prime }}& =-\frac{\mathrm{\partial }{\mathbf{B}}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}},& \dots \left({\mathrm{c}}^{\mathrm{\prime }}\right)\\ \left(\stackrel{\to }{\mathrm{\nabla }}×\mathbf{B}{\right)}^{\mathrm{\prime }}& =\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{\mathbf{E}}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}},& \dots \left({\mathrm{d}}^{\mathrm{\prime }}\right)\end{array}$
In order to show the relativistic invariance of eqs. (a-d), we write them in terms of components of $\mathbf{E}$$\mathbf{E}$E\mathbf{E}$\mathbf{E}$ and $\mathbf{B}$$\mathbf{B}$B\mathbf{B}$\mathbf{B}$ as
$\begin{array}{rlr}& \frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{E}_{y}}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{E}_{z}}{\mathrm{\partial }z}=0& \dots \left(\mathrm{a}1\right)\\ & \frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{B}_{y}}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{B}_{z}}{\mathrm{\partial }z}=0& \dots \left(\mathrm{b}1\right)\end{array}$$\begin{array}{r}\frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{E}_{y}}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{E}_{z}}{\mathrm{\partial }z}=0\dots \left(\mathrm{a}1\right)\\ \frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{B}_{y}}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{B}_{z}}{\mathrm{\partial }z}=0\dots \left(\mathrm{b}1\right)\end{array}${:[(delE_(x))/(del x)+(delE_(y))/(del y)+(delE_(z))/(del z)=0 dots(a1)],[(delB_(x))/(del x)+(delB_(y))/(del y)+(delB_(z))/(del z)=0 dots(b1)]:}\begin{aligned} & \frac{\partial E_x}{\partial x}+\frac{\partial E_y}{\partial y}+\frac{\partial E_z}{\partial z}=0 & \ldots\left(\mathrm{a1}\right) \\ & \frac{\partial B_x}{\partial x}+\frac{\partial B_y}{\partial y}+\frac{\partial B_z}{\partial z}=0 & \ldots\left(\mathrm{b1}\right) \\ \end{aligned}$\begin{array}{rlr}& \frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{E}_{y}}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{E}_{z}}{\mathrm{\partial }z}=0& \dots \left(\mathrm{a}1\right)\\ & \frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{B}_{y}}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{B}_{z}}{\mathrm{\partial }z}=0& \dots \left(\mathrm{b}1\right)\end{array}$
$\begin{array}{rlr}\left(\frac{\mathrm{\partial }{E}_{z}}{\mathrm{\partial }y}-\frac{\mathrm{\partial }{E}_{y}}{\mathrm{\partial }z}\right)& =-\frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }t},& \dots \left(\mathrm{c}1\right)\\ \left(\frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }z}-\frac{\mathrm{\partial }{E}_{z}}{\mathrm{\partial }x}\right)& =-\frac{\mathrm{\partial }{B}_{y}}{\mathrm{\partial }t},& \dots \left(\mathrm{c}2\right)\\ \left(\frac{\mathrm{\partial }{E}_{y}}{\mathrm{\partial }x}-\frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }y}\right)& =-\frac{\mathrm{\partial }{B}_{z}}{\mathrm{\partial }t},& \dots \left(\mathrm{c}3\right)\end{array}$$\begin{array}{r}\left(\frac{\mathrm{\partial }{E}_{z}}{\mathrm{\partial }y}-\frac{\mathrm{\partial }{E}_{y}}{\mathrm{\partial }z}\right)=-\frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }t},\dots \left(\mathrm{c}1\right)\\ \left(\frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }z}-\frac{\mathrm{\partial }{E}_{z}}{\mathrm{\partial }x}\right)=-\frac{\mathrm{\partial }{B}_{y}}{\mathrm{\partial }t},\dots \left(\mathrm{c}2\right)\\ \left(\frac{\mathrm{\partial }{E}_{y}}{\mathrm{\partial }x}-\frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }y}\right)=-\frac{\mathrm{\partial }{B}_{z}}{\mathrm{\partial }t},\dots \left(\mathrm{c}3\right)\end{array}${:[((delE_(z))/(del y)-(delE_(y))/(del z))=-(delB_(x))/(del t)”,” dots(c1)],[((delE_(x))/(del z)-(delE_(z))/(del x))=-(delB_(y))/(del t)”,” dots(c2)],[((delE_(y))/(del x)-(delE_(x))/(del y))=-(delB_(z))/(del t)”,” dots(c3)]:}\begin{aligned} \left(\frac{\partial E_z}{\partial y}-\frac{\partial E_y}{\partial z}\right) & =-\frac{\partial B_x}{\partial t}, & \ldots\left(\mathrm{c1}\right) \\ \left(\frac{\partial E_x}{\partial z}-\frac{\partial E_z}{\partial x}\right) & =-\frac{\partial B_y}{\partial t}, & \ldots\left(\mathrm{c2}\right) \\ \left(\frac{\partial E_y}{\partial x}-\frac{\partial E_x}{\partial y}\right) & =-\frac{\partial B_z}{\partial t}, & \ldots\left(\mathrm{c3}\right) \\ \end{aligned}$\begin{array}{rlr}\left(\frac{\mathrm{\partial }{E}_{z}}{\mathrm{\partial }y}-\frac{\mathrm{\partial }{E}_{y}}{\mathrm{\partial }z}\right)& =-\frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }t},& \dots \left(\mathrm{c}1\right)\\ \left(\frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }z}-\frac{\mathrm{\partial }{E}_{z}}{\mathrm{\partial }x}\right)& =-\frac{\mathrm{\partial }{B}_{y}}{\mathrm{\partial }t},& \dots \left(\mathrm{c}2\right)\\ \left(\frac{\mathrm{\partial }{E}_{y}}{\mathrm{\partial }x}-\frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }y}\right)& =-\frac{\mathrm{\partial }{B}_{z}}{\mathrm{\partial }t},& \dots \left(\mathrm{c}3\right)\end{array}$
and,
$\begin{array}{rl}\left(\frac{\mathrm{\partial }{B}_{z}}{\mathrm{\partial }y}-\frac{\mathrm{\partial }{B}_{y}}{\mathrm{\partial }z}\right)=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }t},& \dots \left(\mathrm{d}1\right)\\ \left(\frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }z}-\frac{\mathrm{\partial }{B}_{z}}{\mathrm{\partial }x}\right)=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{E}_{y}}{\mathrm{\partial }t},& \dots \left(\mathrm{d}2\right)\\ \left(\frac{\mathrm{\partial }{B}_{y}}{\mathrm{\partial }x}-\frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }y}\right)=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{E}_{z}}{\mathrm{\partial }t},& \dots \left(\mathrm{d}3\right)\end{array}$$\begin{array}{r}\left(\frac{\mathrm{\partial }{B}_{z}}{\mathrm{\partial }y}-\frac{\mathrm{\partial }{B}_{y}}{\mathrm{\partial }z}\right)=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }t},\dots \left(\mathrm{d}1\right)\\ \left(\frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }z}-\frac{\mathrm{\partial }{B}_{z}}{\mathrm{\partial }x}\right)=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{E}_{y}}{\mathrm{\partial }t},\dots \left(\mathrm{d}2\right)\\ \left(\frac{\mathrm{\partial }{B}_{y}}{\mathrm{\partial }x}-\frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }y}\right)=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{E}_{z}}{\mathrm{\partial }t},\dots \left(\mathrm{d}3\right)\end{array}${:[((delB_(z))/(del y)-(delB_(y))/(del z))=(1)/(c^(2))(delE_(x))/(del t)”,” dots(d1)],[((delB_(x))/(del z)-(delB_(z))/(del x))=(1)/(c^(2))(delE_(y))/(del t)”,” dots(d2)],[((delB_(y))/(del x)-(delB_(x))/(del y))=(1)/(c^(2))(delE_(z))/(del t)”,” dots(d3)]:}\begin{aligned} \left(\frac{\partial B_z}{\partial y}-\frac{\partial B_y}{\partial z}\right)=\frac {1} {c^{2}} \frac{\partial E_x}{\partial t}, & \ldots\left(\mathrm{d1}\right) \\ \left(\frac{\partial B_x}{\partial z}-\frac{\partial B_z}{\partial x}\right)=\frac {1} {c^{2}} \frac{\partial E_y}{\partial t}, & \ldots\left(\mathrm{d2}\right) \\ \left(\frac{\partial B_y}{\partial x}-\frac{\partial B_x}{\partial y}\right)=\frac {1} {c^{2}} \frac{\partial E_z}{\partial t}, & \ldots\left(\mathrm{d3}\right) \\ \end{aligned}$\begin{array}{rl}\left(\frac{\mathrm{\partial }{B}_{z}}{\mathrm{\partial }y}-\frac{\mathrm{\partial }{B}_{y}}{\mathrm{\partial }z}\right)=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }t},& \dots \left(\mathrm{d}1\right)\\ \left(\frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }z}-\frac{\mathrm{\partial }{B}_{z}}{\mathrm{\partial }x}\right)=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{E}_{y}}{\mathrm{\partial }t},& \dots \left(\mathrm{d}2\right)\\ \left(\frac{\mathrm{\partial }{B}_{y}}{\mathrm{\partial }x}-\frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }y}\right)=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{E}_{z}}{\mathrm{\partial }t},& \dots \left(\mathrm{d}3\right)\end{array}$
Physically these equations represent fields which are space-time dependent.
The Lorentz Transformation equations are given as:
$\begin{array}{rl}x\mathrm{\prime }& =\gamma \left(x-vt\right)\\ y\mathrm{\prime }& =y\\ z\mathrm{\prime }& =z\\ t\mathrm{\prime }& =\gamma \left(t-\frac{vx}{{c}^{2}}\right)\end{array}$$\begin{array}{r}x\mathrm{\prime }=\gamma \left(x-vt\right)\\ y\mathrm{\prime }=y\\ z\mathrm{\prime }=z\\ t\mathrm{\prime }=\gamma \left(t-\frac{vx}{{c}^{2}}\right)\end{array}${:[x’=gamma(x-vt)],[y’=y],[z’=z],[t’=gamma(t-(vx)/(c^(2)))]:}\begin{aligned} x{\prime} & = \gamma(x-vt) \\ y{\prime} & = y \\ z{\prime} & = z \\ t{\prime} & = \gamma(t-\frac {vx} {c^{2}}) \\ \end{aligned}$\begin{array}{rl}x\mathrm{\prime }& =\gamma \left(x-vt\right)\\ y\mathrm{\prime }& =y\\ z\mathrm{\prime }& =z\\ t\mathrm{\prime }& =\gamma \left(t-\frac{vx}{{c}^{2}}\right)\end{array}$
These equations give the following results:
$\begin{array}{rlr}\frac{\mathrm{\partial }}{\mathrm{\partial }x}& =\gamma \frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{\mathrm{\prime }}}-\gamma \frac{v}{{c}^{2}}\frac{\mathrm{\partial }}{\mathrm{\partial }{t}^{\mathrm{\prime }}},& \dots \left(\mathrm{e}\right)\\ \phantom{\rule{1em}{0ex}}\frac{\mathrm{\partial }}{\mathrm{\partial }y}& =\frac{\mathrm{\partial }}{\mathrm{\partial }{y}^{\mathrm{\prime }}},& \dots \left(\mathrm{f}\right)\\ \phantom{\rule{1em}{0ex}}\frac{\mathrm{\partial }}{\mathrm{\partial }z}& =\frac{\mathrm{\partial }}{\mathrm{\partial }{z}^{\mathrm{\prime }}},& \dots \left(\mathrm{g}\right)\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}& =\gamma \frac{\mathrm{\partial }}{\mathrm{\partial }{t}^{\mathrm{\prime }}}-\gamma v\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{\mathrm{\prime }}},& \dots \left(\mathrm{h}\right)\end{array}$$\begin{array}{r}\frac{\mathrm{\partial }}{\mathrm{\partial }x}=\gamma \frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{\mathrm{\prime }}}-\gamma \frac{v}{{c}^{2}}\frac{\mathrm{\partial }}{\mathrm{\partial }{t}^{\mathrm{\prime }}},\dots \left(\mathrm{e}\right)\\ \phantom{\rule{1em}{0ex}}\frac{\mathrm{\partial }}{\mathrm{\partial }y}=\frac{\mathrm{\partial }}{\mathrm{\partial }{y}^{\mathrm{\prime }}},\dots \left(\mathrm{f}\right)\\ \phantom{\rule{1em}{0ex}}\frac{\mathrm{\partial }}{\mathrm{\partial }z}=\frac{\mathrm{\partial }}{\mathrm{\partial }{z}^{\mathrm{\prime }}},\dots \left(\mathrm{g}\right)\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}=\gamma \frac{\mathrm{\partial }}{\mathrm{\partial }{t}^{\mathrm{\prime }}}-\gamma v\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{\mathrm{\prime }}},\dots \left(\mathrm{h}\right)\end{array}${:[(del)/(del x)=gamma(del)/(delx^(‘))-gamma(v)/(c^(2))(del)/(delt^(‘))”,” dots(e)],[quad(del)/(del y)=(del)/(dely^(‘))”,” dots(f)],[quad(del)/(del z)=(del)/(delz^(‘))”,” dots(g)],[(del)/(del t)=gamma(del)/(delt^(‘))-gamma v(del)/(delx^(‘))”,” dots(h)]:}\begin{aligned} \frac{\partial}{\partial x} & =\gamma \frac{\partial}{\partial x^{\prime}}-\gamma \frac{v}{c^2} \frac{\partial}{\partial t^{\prime}}, & \ldots\left(\mathrm{e}\right)\\ \quad \frac{\partial}{\partial y} & =\frac{\partial}{\partial y^{\prime}}, & \ldots\left(\mathrm{f}\right)\\ \quad \frac{\partial}{\partial z} & =\frac{\partial}{\partial z^{\prime}}, & \ldots\left(\mathrm{g}\right)\\ \frac{\partial}{\partial t} & =\gamma \frac{\partial}{\partial t^{\prime}}-\gamma v \frac{\partial}{\partial x^{\prime}}, & \ldots\left(\mathrm{h}\right)\\ \end{aligned}$\begin{array}{rlr}\frac{\mathrm{\partial }}{\mathrm{\partial }x}& =\gamma \frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{\mathrm{\prime }}}-\gamma \frac{v}{{c}^{2}}\frac{\mathrm{\partial }}{\mathrm{\partial }{t}^{\mathrm{\prime }}},& \dots \left(\mathrm{e}\right)\\ \phantom{\rule{1em}{0ex}}\frac{\mathrm{\partial }}{\mathrm{\partial }y}& =\frac{\mathrm{\partial }}{\mathrm{\partial }{y}^{\mathrm{\prime }}},& \dots \left(\mathrm{f}\right)\\ \phantom{\rule{1em}{0ex}}\frac{\mathrm{\partial }}{\mathrm{\partial }z}& =\frac{\mathrm{\partial }}{\mathrm{\partial }{z}^{\mathrm{\prime }}},& \dots \left(\mathrm{g}\right)\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}& =\gamma \frac{\mathrm{\partial }}{\mathrm{\partial }{t}^{\mathrm{\prime }}}-\gamma v\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{\mathrm{\prime }}},& \dots \left(\mathrm{h}\right)\end{array}$
Now substituting the results of transformation given by eq. (e), (f), (g) and (h) in eq. (b1) and (c1), we get
$\begin{array}{rl}\frac{\mathrm{\partial }{E}_{z}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{E}_{y}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}=-\gamma \frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}+\gamma v\frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }{x}^{\mathrm{\prime }}},& \dots \left(\mathrm{i}\right)\end{array}$$\begin{array}{r}\frac{\mathrm{\partial }{E}_{z}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{E}_{y}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}=-\gamma \frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}+\gamma v\frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }{x}^{\mathrm{\prime }}},\dots \left(\mathrm{i}\right)\end{array}${:(delE_(z))/(dely^(‘))-(delE_(y))/(delz^(‘))=-gamma(delB_(x))/(delt^(‘))+gammav(delB_(x))/(delx^(‘))”,” dots(i):}\begin{aligned} \frac{\partial E_z}{\partial y^{\prime}}-\frac{\partial E_y}{\partial z^{\prime}}=-\gamma \frac{\partial B_x}{\partial t^{\prime}}+{\gamma}{v} \frac{\partial B_x}{\partial x^{\prime}}, & \ldots\left(\mathrm{i}\right)\\ \end{aligned}$\begin{array}{rl}\frac{\mathrm{\partial }{E}_{z}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{E}_{y}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}=-\gamma \frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}+\gamma v\frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }{x}^{\mathrm{\prime }}},& \dots \left(\mathrm{i}\right)\end{array}$
and
$\begin{array}{rl}\gamma \frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }{x}^{\mathrm{\prime }}}-\gamma \frac{v}{{c}^{2}}\frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}+\frac{\mathrm{\partial }{B}_{y}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}+\frac{\mathrm{\partial }{B}_{z}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}=0,& \dots \left(\mathrm{j}\right)\end{array}$$\begin{array}{r}\gamma \frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }{x}^{\mathrm{\prime }}}-\gamma \frac{v}{{c}^{2}}\frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}+\frac{\mathrm{\partial }{B}_{y}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}+\frac{\mathrm{\partial }{B}_{z}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}=0,\dots \left(\mathrm{j}\right)\end{array}${:gamma(delB_(x))/(delx^(‘))-gamma(v)/(c^(2))(delB_(x))/(delt^(‘))+(delB_(y))/(dely^(‘))+(delB_(z))/(delz^(‘))=0″,” dots(j):}\begin{aligned} \gamma \frac{\partial B_x}{\partial x^{\prime}}-\gamma \frac{v}{c^2} \frac{\partial B_x}{\partial t^{\prime}}+\frac{\partial B_y}{\partial y^{\prime}}+\frac{\partial B_z}{\partial z^{\prime}}=0, & \ldots\left(\mathrm{j}\right) \end{aligned}$\begin{array}{rl}\gamma \frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }{x}^{\mathrm{\prime }}}-\gamma \frac{v}{{c}^{2}}\frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}+\frac{\mathrm{\partial }{B}_{y}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}+\frac{\mathrm{\partial }{B}_{z}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}=0,& \dots \left(\mathrm{j}\right)\end{array}$
Eliminating $\left(\mathrm{\partial }{B}_{x}/\mathrm{\partial }{x}^{\mathrm{\prime }}\right)$$\left(\mathrm{\partial }{B}_{x}/\mathrm{\partial }{x}^{\mathrm{\prime }}\right)$(delB_(x)//delx^(‘))\left(\partial B_x / \partial x^{\prime}\right)$\left(\mathrm{\partial }{B}_{x}/\mathrm{\partial }{x}^{\mathrm{\prime }}\right)$ from these equations, we get
$\begin{array}{rl}\frac{\mathrm{\partial }}{\mathrm{\partial }{y}^{\mathrm{\prime }}}\left[\gamma \left({E}_{z}+v{B}_{y}\right)\right]-\frac{\mathrm{\partial }}{\mathrm{\partial }{z}^{\mathrm{\prime }}}\left[\gamma \left({E}_{y}-v{B}_{z}\right)\right]=-\frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }{t}^{\mathrm{\prime }}},& \dots \left(\mathrm{k}\right)\end{array}$$\begin{array}{r}\frac{\mathrm{\partial }}{\mathrm{\partial }{y}^{\mathrm{\prime }}}\left[\gamma \left({E}_{z}+v{B}_{y}\right)\right]-\frac{\mathrm{\partial }}{\mathrm{\partial }{z}^{\mathrm{\prime }}}\left[\gamma \left({E}_{y}-v{B}_{z}\right)\right]=-\frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }{t}^{\mathrm{\prime }}},\dots \left(\mathrm{k}\right)\end{array}${:(del)/(dely^(‘))[gamma(E_(z)+vB_(y))]-(del)/(delz^(‘))[gamma(E_(y)-vB_(z))]=-(delB_(x))/(delt^(‘))”,” dots(k):}\begin{aligned} \frac{\partial}{\partial y^{\prime}}\left[\gamma\left(E_z+v B_y\right)\right]-\frac{\partial}{\partial z^{\prime}}\left[\gamma\left(E_y-v B_z\right)\right]=- \frac{\partial B_x}{\partial t^{\prime}}, & \ldots\left(\mathrm{k}\right)\\ \end{aligned}$\begin{array}{rl}\frac{\mathrm{\partial }}{\mathrm{\partial }{y}^{\mathrm{\prime }}}\left[\gamma \left({E}_{z}+v{B}_{y}\right)\right]-\frac{\mathrm{\partial }}{\mathrm{\partial }{z}^{\mathrm{\prime }}}\left[\gamma \left({E}_{y}-v{B}_{z}\right)\right]=-\frac{\mathrm{\partial }{B}_{x}}{\mathrm{\partial }{t}^{\mathrm{\prime }}},& \dots \left(\mathrm{k}\right)\end{array}$
As derived previously, we can write the components of the field in frame ${F}^{\mathrm{\prime }}$${F}^{\mathrm{\prime }}$F^(‘)F^{\prime}${F}^{\mathrm{\prime }}$ in the following way:
$\begin{array}{r}\gamma \left({E}_{z}+v{B}_{y}\right)={E}_{z}{}^{\mathrm{\prime }};\phantom{\rule{1em}{0ex}}\gamma \left({E}_{y}-v{B}_{z}\right)={E}_{y}{}^{\mathrm{\prime }};\phantom{\rule{1em}{0ex}}{E}_{x}={E}_{x}^{\mathrm{\prime }}\\ \gamma \left({B}_{z}-\frac{v}{{c}^{2}}{E}_{y}\right)={B}_{z}{}^{\mathrm{\prime }};\phantom{\rule{1em}{0ex}}\gamma \left({B}_{y}+\frac{v}{{c}^{2}}{E}_{z}\right)={B}_{y}{}^{\mathrm{\prime }};\phantom{\rule{1em}{0ex}}{B}_{x}={B}_{x}^{\mathrm{\prime }},& \dots \left(\mathrm{l}\right)\end{array}$$\begin{array}{r}\gamma \left({E}_{z}+v{B}_{y}\right)={E}_{z}{}^{\mathrm{\prime }};\phantom{\rule{1em}{0ex}}\gamma \left({E}_{y}-v{B}_{z}\right)={E}_{y}{}^{\mathrm{\prime }};\phantom{\rule{1em}{0ex}}{E}_{x}={E}_{x}^{\mathrm{\prime }}\\ \gamma \left({B}_{z}-\frac{v}{{c}^{2}}{E}_{y}\right)={B}_{z}{}^{\mathrm{\prime }};\phantom{\rule{1em}{0ex}}\gamma \left({B}_{y}+\frac{v}{{c}^{2}}{E}_{z}\right)={B}_{y}{}^{\mathrm{\prime }};\phantom{\rule{1em}{0ex}}{B}_{x}={B}_{x}^{\mathrm{\prime }},\dots \left(\mathrm{l}\right)\end{array}${:[gamma(E_(z)+vB_(y))=E_(z)^(‘);quad gamma(E_(y)-vB_(z))=E_(y)^(‘);quadE_(x)=E_(x)^(‘)],[gamma(B_(z)-(v)/(c^(2))E_(y))=B_(z)^(‘);quad gamma(B_(y)+(v)/(c^(2))E_(z))=B_(y)^(‘);quadB_(x)=B_(x)^(‘)”,” dots(l)]:}\begin{aligned} \gamma\left(E_z+v B_y\right)=E_z{ }^{\prime} ; \quad \gamma\left(E_y-v B_z\right)=E_y{ }^{\prime}; \quad E_x=E_x^{\prime}\\ \gamma\left(B_z-\frac {v}{c^2} E_y\right)=B_z{ }^{\prime} ; \quad \gamma\left(B_y+\frac {v}{c^2} E_z\right)=B_y{ }^{\prime}; \quad B_x=B_x^{\prime}, & \ldots\left(\mathrm{l}\right)\\ \end{aligned}$\begin{array}{r}\gamma \left({E}_{z}+v{B}_{y}\right)={E}_{z}{}^{\mathrm{\prime }};\phantom{\rule{1em}{0ex}}\gamma \left({E}_{y}-v{B}_{z}\right)={E}_{y}{}^{\mathrm{\prime }};\phantom{\rule{1em}{0ex}}{E}_{x}={E}_{x}^{\mathrm{\prime }}\\ \gamma \left({B}_{z}-\frac{v}{{c}^{2}}{E}_{y}\right)={B}_{z}{}^{\mathrm{\prime }};\phantom{\rule{1em}{0ex}}\gamma \left({B}_{y}+\frac{v}{{c}^{2}}{E}_{z}\right)={B}_{y}{}^{\mathrm{\prime }};\phantom{\rule{1em}{0ex}}{B}_{x}={B}_{x}^{\mathrm{\prime }},& \dots \left(\mathrm{l}\right)\end{array}$
Then eq. (k) assumes the form :
$\left[\frac{\mathrm{\partial }{E}_{z}{}^{\mathrm{\prime }}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{E}_{y}{}^{\mathrm{\prime }}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}\right]=-\frac{\mathrm{\partial }{B}_{x}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}},$$\left[\frac{\mathrm{\partial }{E}_{z}{}^{\mathrm{\prime }}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{E}_{y}{}^{\mathrm{\prime }}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}\right]=-\frac{\mathrm{\partial }{B}_{x}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}},$[(delE_(z)^(‘))/(dely^(‘))-(delE_(y)^(‘))/(delz^(‘))]=-(delB_(x)^(‘))/(delt^(‘)),\left[\frac{\partial E_z{ }^{\prime}}{\partial y^{\prime}}-\frac{\partial E_y{ }^{\prime}}{\partial z^{\prime}}\right]=-\frac{\partial B_x^{\prime}}{\partial t^{\prime}},$\left[\frac{\mathrm{\partial }{E}_{z}{}^{\mathrm{\prime }}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{E}_{y}{}^{\mathrm{\prime }}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}\right]=-\frac{\mathrm{\partial }{B}_{x}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}},$
which is similar to eq. (c1). Similarly, the other two sets assume the form;
$\begin{array}{rl}& \left[\frac{\mathrm{\partial }{E}_{x}^{\mathrm{\prime }}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{E}_{z}^{\mathrm{\prime }}}{\mathrm{\partial }{x}^{\mathrm{\prime }}}\right]=-\frac{\mathrm{\partial }{B}_{y}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}\\ & \left[\frac{\mathrm{\partial }{E}_{{y}^{\mathrm{\prime }}}^{\mathrm{\prime }}}{\mathrm{\partial }{x}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{E}_{x}^{\mathrm{\prime }}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}\right]=-\frac{\mathrm{\partial }{B}_{z}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}\end{array}$$\begin{array}{r}\left[\frac{\mathrm{\partial }{E}_{x}^{\mathrm{\prime }}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{E}_{z}^{\mathrm{\prime }}}{\mathrm{\partial }{x}^{\mathrm{\prime }}}\right]=-\frac{\mathrm{\partial }{B}_{y}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}\\ \left[\frac{\mathrm{\partial }{E}_{{y}^{\mathrm{\prime }}}^{\mathrm{\prime }}}{\mathrm{\partial }{x}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{E}_{x}^{\mathrm{\prime }}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}\right]=-\frac{\mathrm{\partial }{B}_{z}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}\end{array}${:[[(delE_(x)^(‘))/(delz^(‘))-(delE_(z)^(‘))/(delx^(‘))]=-(delB_(y)^(‘))/(delt^(‘))],[[(delE_(y^(‘))^(‘))/(delx^(‘))-(delE_(x)^(‘))/(dely^(‘))]=-(delB_(z)^(‘))/(delt^(‘))]:}\begin{aligned} & \left[\frac{\partial E_x^{\prime}}{\partial z^{\prime}}-\frac{\partial E_z^{\prime}}{\partial x^{\prime}}\right]=-\frac{\partial B_y^{\prime}}{\partial t^{\prime}} \\ & \left[\frac{\partial E_{y^{\prime}}^{\prime}}{\partial x^{\prime}}-\frac{\partial E_x^{\prime}}{\partial y^{\prime}}\right]=-\frac{\partial B_z^{\prime}}{\partial t^{\prime}} \end{aligned}$\begin{array}{rl}& \left[\frac{\mathrm{\partial }{E}_{x}^{\mathrm{\prime }}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{E}_{z}^{\mathrm{\prime }}}{\mathrm{\partial }{x}^{\mathrm{\prime }}}\right]=-\frac{\mathrm{\partial }{B}_{y}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}\\ & \left[\frac{\mathrm{\partial }{E}_{{y}^{\mathrm{\prime }}}^{\mathrm{\prime }}}{\mathrm{\partial }{x}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{E}_{x}^{\mathrm{\prime }}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}\right]=-\frac{\mathrm{\partial }{B}_{z}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}\end{array}$
or, in short;
$\left(\stackrel{\to }{\mathrm{\nabla }}×\mathbf{E}{\right)}^{\mathrm{\prime }}=\frac{\mathrm{\partial }{B}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}$$\left(\stackrel{\to }{\mathrm{\nabla }}×\mathbf{E}{\right)}^{\mathrm{\prime }}=\frac{\mathrm{\partial }{B}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}$( vec(grad)xxE)^(‘)=(delB^(‘))/(delt^(‘))(\vec{\nabla} \times \mathbf{E})^{\prime}=\frac{\partial B^{\prime}}{\partial t^{\prime}}$\left(\stackrel{\to }{\mathrm{\nabla }}×\mathbf{E}{\right)}^{\mathrm{\prime }}=\frac{\mathrm{\partial }{B}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}$
which is eq. (${c}^{\mathrm{\prime }}$${c}^{\mathrm{\prime }}$c^(‘)c^{\prime}${c}^{\mathrm{\prime }}$) in frame ${F}^{\mathrm{\prime }}$${F}^{\mathrm{\prime }}$F^(‘)F^{\prime}${F}^{\mathrm{\prime }}$ and bears the same form.
Similarly, substituting the results of transformation given by eq. (e), (f), (g) and (h) in eq. (d1) and (a1), we get
$\begin{array}{rl}\frac{\mathrm{\partial }{B}_{z}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{B}_{y}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}=\frac{\gamma }{{c}^{2}}\frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}-\frac{\gamma v}{{c}^{2}}\frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }{x}^{\mathrm{\prime }}},& \dots \left(\mathrm{m}\right)\end{array}$$\begin{array}{r}\frac{\mathrm{\partial }{B}_{z}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{B}_{y}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}=\frac{\gamma }{{c}^{2}}\frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}-\frac{\gamma v}{{c}^{2}}\frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }{x}^{\mathrm{\prime }}},\dots \left(\mathrm{m}\right)\end{array}${:(delB_(z))/(dely^(‘))-(delB_(y))/(delz^(‘))=(gamma)/(c^(2))(delE_(x))/(delt^(‘))-(gamma v)/(c^(2))(delE_(x))/(delx^(‘))”,” dots(m):}\begin{aligned} \frac{\partial B_z}{\partial y^{\prime}}-\frac{\partial B_y}{\partial z^{\prime}}=\frac{\gamma}{c^2} \frac{\partial E_x}{\partial t^{\prime}}-\frac{\gamma v}{c^2} \frac{\partial E_x}{\partial x^{\prime}}, & \ldots\left(\mathrm{m}\right) \\ \end{aligned}$\begin{array}{rl}\frac{\mathrm{\partial }{B}_{z}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{B}_{y}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}=\frac{\gamma }{{c}^{2}}\frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}-\frac{\gamma v}{{c}^{2}}\frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }{x}^{\mathrm{\prime }}},& \dots \left(\mathrm{m}\right)\end{array}$
and
$\begin{array}{rl}\gamma \frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }{x}^{\mathrm{\prime }}}-\gamma \frac{v}{{c}^{2}}\frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}+\frac{\mathrm{\partial }{E}_{y}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}+\frac{\mathrm{\partial }{E}_{z}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}=0,& \dots \left(\mathrm{n}\right)\end{array}$$\begin{array}{r}\gamma \frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }{x}^{\mathrm{\prime }}}-\gamma \frac{v}{{c}^{2}}\frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}+\frac{\mathrm{\partial }{E}_{y}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}+\frac{\mathrm{\partial }{E}_{z}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}=0,\dots \left(\mathrm{n}\right)\end{array}${:gamma(delE_(x))/(delx^(‘))-gamma(v)/(c^(2))(delE_(x))/(delt^(‘))+(delE_(y))/(dely^(‘))+(delE_(z))/(delz^(‘))=0″,” dots(n):}\begin{aligned} \gamma \frac{\partial E_x}{\partial x^{\prime}}-\gamma \frac{v}{c^2} \frac{\partial E_x}{\partial t^{\prime}}+\frac{\partial E_y}{\partial y^{\prime}}+\frac{\partial E_z}{\partial z^{\prime}}=0, & \ldots\left(\mathrm{n}\right) \end{aligned}$\begin{array}{rl}\gamma \frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }{x}^{\mathrm{\prime }}}-\gamma \frac{v}{{c}^{2}}\frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}+\frac{\mathrm{\partial }{E}_{y}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}+\frac{\mathrm{\partial }{E}_{z}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}=0,& \dots \left(\mathrm{n}\right)\end{array}$
Eliminating $\left(\mathrm{\partial }{E}_{x}/\mathrm{\partial }{x}^{\mathrm{\prime }}\right)$$\left(\mathrm{\partial }{E}_{x}/\mathrm{\partial }{x}^{\mathrm{\prime }}\right)$(delE_(x)//delx^(‘))\left(\partial E_x / \partial x^{\prime}\right)$\left(\mathrm{\partial }{E}_{x}/\mathrm{\partial }{x}^{\mathrm{\prime }}\right)$ from these equations, we get
$\begin{array}{rl}\frac{\mathrm{\partial }}{\mathrm{\partial }{y}^{\mathrm{\prime }}}\left[\gamma \left({B}_{z}-\frac{v}{{c}^{2}}{E}_{y}\right)\right]-\frac{\mathrm{\partial }}{\mathrm{\partial }{z}^{\mathrm{\prime }}}\left[\gamma \left({B}_{y}+\frac{v}{{c}^{2}}{E}_{z}\right)\right]=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }{t}^{\mathrm{\prime }}},& \dots \left(\mathrm{o}\right)\end{array}$$\begin{array}{r}\frac{\mathrm{\partial }}{\mathrm{\partial }{y}^{\mathrm{\prime }}}\left[\gamma \left({B}_{z}-\frac{v}{{c}^{2}}{E}_{y}\right)\right]-\frac{\mathrm{\partial }}{\mathrm{\partial }{z}^{\mathrm{\prime }}}\left[\gamma \left({B}_{y}+\frac{v}{{c}^{2}}{E}_{z}\right)\right]=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }{t}^{\mathrm{\prime }}},\dots \left(\mathrm{o}\right)\end{array}${:(del)/(dely^(‘))[gamma(B_(z)-(v)/(c^(2))E_(y))]-(del)/(delz^(‘))[gamma(B_(y)+(v)/(c^(2))E_(z))]=(1)/(c^(2))(delE_(x))/(delt^(‘))”,” dots(o):}\begin{aligned} \frac{\partial}{\partial y^{\prime}}\left[\gamma\left(B_z-\frac {v}{c^2} E_y\right)\right]-\frac{\partial}{\partial z^{\prime}}\left[\gamma\left(B_y+\frac {v}{c^2} E_z\right)\right]=\frac{1}{c^2}\frac{\partial E_x}{\partial t^{\prime}}, & \ldots\left(\mathrm{o}\right)\\ \end{aligned}$\begin{array}{rl}\frac{\mathrm{\partial }}{\mathrm{\partial }{y}^{\mathrm{\prime }}}\left[\gamma \left({B}_{z}-\frac{v}{{c}^{2}}{E}_{y}\right)\right]-\frac{\mathrm{\partial }}{\mathrm{\partial }{z}^{\mathrm{\prime }}}\left[\gamma \left({B}_{y}+\frac{v}{{c}^{2}}{E}_{z}\right)\right]=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{E}_{x}}{\mathrm{\partial }{t}^{\mathrm{\prime }}},& \dots \left(\mathrm{o}\right)\end{array}$
As derived previously, we can write the components of the field in frame ${F}^{\mathrm{\prime }}$${F}^{\mathrm{\prime }}$F^(‘)F^{\prime}${F}^{\mathrm{\prime }}$ in the following way:
$\begin{array}{r}\gamma \left({E}_{z}+v{B}_{y}\right)={E}_{z}{}^{\mathrm{\prime }};\phantom{\rule{1em}{0ex}}\gamma \left({E}_{y}-v{B}_{z}\right)={E}_{y}{}^{\mathrm{\prime }};\phantom{\rule{1em}{0ex}}{E}_{x}={E}_{x}^{\mathrm{\prime }}\\ \gamma \left({B}_{z}-\frac{v}{{c}^{2}}{E}_{y}\right)={B}_{z}{}^{\mathrm{\prime }};\phantom{\rule{1em}{0ex}}\gamma \left({B}_{y}+\frac{v}{{c}^{2}}{E}_{z}\right)={B}_{y}{}^{\mathrm{\prime }};\phantom{\rule{1em}{0ex}}{B}_{x}={B}_{x}^{\mathrm{\prime }},& \dots \left(\mathrm{l}\right)\end{array}$$\begin{array}{r}\gamma \left({E}_{z}+v{B}_{y}\right)={E}_{z}{}^{\mathrm{\prime }};\phantom{\rule{1em}{0ex}}\gamma \left({E}_{y}-v{B}_{z}\right)={E}_{y}{}^{\mathrm{\prime }};\phantom{\rule{1em}{0ex}}{E}_{x}={E}_{x}^{\mathrm{\prime }}\\ \gamma \left({B}_{z}-\frac{v}{{c}^{2}}{E}_{y}\right)={B}_{z}{}^{\mathrm{\prime }};\phantom{\rule{1em}{0ex}}\gamma \left({B}_{y}+\frac{v}{{c}^{2}}{E}_{z}\right)={B}_{y}{}^{\mathrm{\prime }};\phantom{\rule{1em}{0ex}}{B}_{x}={B}_{x}^{\mathrm{\prime }},\dots \left(\mathrm{l}\right)\end{array}${:[gamma(E_(z)+vB_(y))=E_(z)^(‘);quad gamma(E_(y)-vB_(z))=E_(y)^(‘);quadE_(x)=E_(x)^(‘)],[gamma(B_(z)-(v)/(c^(2))E_(y))=B_(z)^(‘);quad gamma(B_(y)+(v)/(c^(2))E_(z))=B_(y)^(‘);quadB_(x)=B_(x)^(‘)”,” dots(l)]:}\begin{aligned} \gamma\left(E_z+v B_y\right)=E_z{ }^{\prime} ; \quad \gamma\left(E_y-v B_z\right)=E_y{ }^{\prime}; \quad E_x=E_x^{\prime}\\ \gamma\left(B_z-\frac {v}{c^2} E_y\right)=B_z{ }^{\prime} ; \quad \gamma\left(B_y+\frac {v}{c^2} E_z\right)=B_y{ }^{\prime}; \quad B_x=B_x^{\prime}, & \ldots\left(\mathrm{l}\right)\\ \end{aligned}$\begin{array}{r}\gamma \left({E}_{z}+v{B}_{y}\right)={E}_{z}{}^{\mathrm{\prime }};\phantom{\rule{1em}{0ex}}\gamma \left({E}_{y}-v{B}_{z}\right)={E}_{y}{}^{\mathrm{\prime }};\phantom{\rule{1em}{0ex}}{E}_{x}={E}_{x}^{\mathrm{\prime }}\\ \gamma \left({B}_{z}-\frac{v}{{c}^{2}}{E}_{y}\right)={B}_{z}{}^{\mathrm{\prime }};\phantom{\rule{1em}{0ex}}\gamma \left({B}_{y}+\frac{v}{{c}^{2}}{E}_{z}\right)={B}_{y}{}^{\mathrm{\prime }};\phantom{\rule{1em}{0ex}}{B}_{x}={B}_{x}^{\mathrm{\prime }},& \dots \left(\mathrm{l}\right)\end{array}$
Then eq. (o) assumes the form :
$\left[\frac{\mathrm{\partial }{B}_{z}{}^{\mathrm{\prime }}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{B}_{y}{}^{\mathrm{\prime }}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}\right]=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{E}_{x}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}},$$\left[\frac{\mathrm{\partial }{B}_{z}{}^{\mathrm{\prime }}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{B}_{y}{}^{\mathrm{\prime }}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}\right]=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{E}_{x}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}},$[(delB_(z)^(‘))/(dely^(‘))-(delB_(y)^(‘))/(delz^(‘))]=(1)/(c^(2))(delE_(x)^(‘))/(delt^(‘)),\left[\frac{\partial B_z{ }^{\prime}}{\partial y^{\prime}}-\frac{\partial B_y{ }^{\prime}}{\partial z^{\prime}}\right]=\frac {1}{c^2}\frac{\partial E_x^{\prime}}{\partial t^{\prime}},$\left[\frac{\mathrm{\partial }{B}_{z}{}^{\mathrm{\prime }}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{B}_{y}{}^{\mathrm{\prime }}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}\right]=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{E}_{x}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}},$
which is similar to eq. (d1). Similarly, the other two sets assume the form;
$\begin{array}{rl}& \left[\frac{\mathrm{\partial }{B}_{x}^{\mathrm{\prime }}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{B}_{z}^{\mathrm{\prime }}}{\mathrm{\partial }{x}^{\mathrm{\prime }}}\right]=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{E}_{y}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}\\ & \left[\frac{\mathrm{\partial }{B}_{y}^{\mathrm{\prime }}}{\mathrm{\partial }{x}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{B}_{x}^{\mathrm{\prime }}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}\right]=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{E}_{z}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}\end{array}$$\begin{array}{r}\left[\frac{\mathrm{\partial }{B}_{x}^{\mathrm{\prime }}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{B}_{z}^{\mathrm{\prime }}}{\mathrm{\partial }{x}^{\mathrm{\prime }}}\right]=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{E}_{y}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}\\ \left[\frac{\mathrm{\partial }{B}_{y}^{\mathrm{\prime }}}{\mathrm{\partial }{x}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{B}_{x}^{\mathrm{\prime }}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}\right]=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{E}_{z}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}\end{array}${:[[(delB_(x)^(‘))/(delz^(‘))-(delB_(z)^(‘))/(delx^(‘))]=(1)/(c^(2))(delE_(y)^(‘))/(delt^(‘))],[[(delB_(y)^(‘))/(delx^(‘))-(delB_(x)^(‘))/(dely^(‘))]=(1)/(c^(2))(delE_(z)^(‘))/(delt^(‘))]:}\begin{aligned} & \left[\frac{\partial B_x^{\prime}}{\partial z^{\prime}}-\frac{\partial B_z^{\prime}}{\partial x^{\prime}}\right]=\frac{1}{c^2}\frac{\partial E_y^{\prime}}{\partial t^{\prime}} \\ & \left[\frac{\partial B_y^{\prime}}{\partial x^{\prime}}-\frac{\partial B_x^{\prime}}{\partial y^{\prime}}\right]=\frac{1}{c^2}\frac{\partial E_z^{\prime}}{\partial t^{\prime}} \end{aligned}$\begin{array}{rl}& \left[\frac{\mathrm{\partial }{B}_{x}^{\mathrm{\prime }}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{B}_{z}^{\mathrm{\prime }}}{\mathrm{\partial }{x}^{\mathrm{\prime }}}\right]=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{E}_{y}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}\\ & \left[\frac{\mathrm{\partial }{B}_{y}^{\mathrm{\prime }}}{\mathrm{\partial }{x}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{B}_{x}^{\mathrm{\prime }}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}\right]=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{E}_{z}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}\end{array}$
or, in short;
$\left(\stackrel{\to }{\mathrm{\nabla }}×\mathbf{B}{\right)}^{\mathrm{\prime }}=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{E}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}$$\left(\stackrel{\to }{\mathrm{\nabla }}×\mathbf{B}{\right)}^{\mathrm{\prime }}=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{E}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}$( vec(grad)xxB)^(‘)=(1)/(c^(2))(delE^(‘))/(delt^(‘))(\vec{\nabla} \times \mathbf{B})^{\prime}=\frac{1}{c^2}\frac{\partial E^{\prime}}{\partial t^{\prime}}$\left(\stackrel{\to }{\mathrm{\nabla }}×\mathbf{B}{\right)}^{\mathrm{\prime }}=\frac{1}{{c}^{2}}\frac{\mathrm{\partial }{E}^{\mathrm{\prime }}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}$
which is eq. (${d}^{\mathrm{\prime }}$${d}^{\mathrm{\prime }}$d^(‘)d^{\prime}${d}^{\mathrm{\prime }}$) in frame ${F}^{\mathrm{\prime }}$${F}^{\mathrm{\prime }}$F^(‘)F^{\prime}${F}^{\mathrm{\prime }}$ and bears the same form.
Using eq. (l) and the transformation equations (e-h) in eq. (a1), we get;
$\begin{array}{r}\gamma \frac{\mathrm{\partial }{E}_{x}^{\mathrm{\prime }}}{\mathrm{\partial }{x}^{\mathrm{\prime }}}+\frac{1}{\gamma }\frac{\mathrm{\partial }{E}_{y}^{\mathrm{\prime }}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}+\frac{1}{\gamma }\frac{\mathrm{\partial }{E}_{z}^{\mathrm{\prime }}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}+v\left(\frac{\mathrm{\partial }{B}_{z}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{B}_{y}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}\right)-\gamma \frac{v}{{c}^{2}}\frac{\mathrm{\partial }{E}_{{x}^{\mathrm{\prime }}}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}=0\end{array}$$\begin{array}{r}\gamma \frac{\mathrm{\partial }{E}_{x}^{\mathrm{\prime }}}{\mathrm{\partial }{x}^{\mathrm{\prime }}}+\frac{1}{\gamma }\frac{\mathrm{\partial }{E}_{y}^{\mathrm{\prime }}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}+\frac{1}{\gamma }\frac{\mathrm{\partial }{E}_{z}^{\mathrm{\prime }}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}+v\left(\frac{\mathrm{\partial }{B}_{z}}{\mathrm{\partial }{y}^{\mathrm{\prime }}}-\frac{\mathrm{\partial }{B}_{y}}{\mathrm{\partial }{z}^{\mathrm{\prime }}}\right)-\gamma \frac{v}{{c}^{2}}\frac{\mathrm{\partial }{E}_{{x}^{\mathrm{\prime }}}}{\mathrm{\partial }{t}^{\mathrm{\prime }}}=0\end{array}${:gamma(delE_(x)^(‘))/(delx^(‘))+(1)/(gamma)(delE_(y)^(‘))/(dely^(‘))+(1)/(gamma)(delE_(z)^(‘))/(delz^(‘))+v((delB_(z))/(dely^(‘))-(delB_(y))/(delz^(‘)))-gamma(v)/(c^(2))(delE_(x^(‘)))/(delt^(‘))=0:}\begin{aligned} {\gamma} \frac{\partial E_x^{\prime}}{\partial x^{\prime}}+\frac{1}{\gamma}\frac{\partial E_y^{\prime}}{\partial y^{\prime}}+\frac{1}{\gamma}\frac{\partial E_z^{\prime}}{\partial z^{\prime}}+v\left(\frac{\partial B_z}{\partial y^{\prime}}-\frac{\partial B_y}{\partial z^{\prime}}\right)-\gamma \frac{v}{c^2} \frac{\partial E_{x^{\prime}}}{\partial t^{\prime}}=0\\ \end{aligned}