- #1
genxium
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In Einstein's paper section 6 (I'm reading an English version online: https://www.fourmilab.ch/etexts/einstein/specrel/www/), it's said that one of the Maxwell Equations in frame [itex]K[/itex]
[tex]\frac{1}{c}\frac{\partial X}{\partial t} = \frac{\partial N}{\partial y} - \frac{\partial M}{\partial z}[/tex], where [itex]<X, Y, Z>[/itex] denotes the vector of the electric force and [itex]<L, M, N>[/itex] that of the magnetic force, can be "transformed" into frame [itex]K'[/itex] which is moving at a constant speed [itex]v[/itex] on the [itex]x-axis[/itex] with respect to [itex]K[/itex],
[tex]\frac{1}{c}\frac{\partial X}{\partial \tau} = \frac{\partial}{\partial \eta} \{ \beta \cdot (N - \frac{v}{c}Y) \} - \frac{\partial}{\partial \zeta} \{ \beta \cdot (M + \frac{v}{c}Z) \} [/tex], where [itex]\beta = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}[/itex]
I'm confused by the existence of terms [itex]\frac{\partial Y}{\partial \eta}[/itex] and [itex]\frac{\partial Z}{\partial \zeta}[/itex], because it's derived in section 3 that
[tex]\tau = \beta \cdot (t - \frac{vx}{c^2})[/tex]
[tex]\xi = \beta \cdot (x - vt)[/tex]
[tex]\eta = y[/tex]
[tex]\zeta = z[/tex]
I can't find a way to apply partial derivative operations to make [itex]\frac{\partial Y}{\partial \eta}[/itex] and [itex]\frac{\partial Z}{\partial \zeta}[/itex] come out. Could anyone give me some tips? Any help is appreciated.
[tex]\frac{1}{c}\frac{\partial X}{\partial t} = \frac{\partial N}{\partial y} - \frac{\partial M}{\partial z}[/tex], where [itex]<X, Y, Z>[/itex] denotes the vector of the electric force and [itex]<L, M, N>[/itex] that of the magnetic force, can be "transformed" into frame [itex]K'[/itex] which is moving at a constant speed [itex]v[/itex] on the [itex]x-axis[/itex] with respect to [itex]K[/itex],
[tex]\frac{1}{c}\frac{\partial X}{\partial \tau} = \frac{\partial}{\partial \eta} \{ \beta \cdot (N - \frac{v}{c}Y) \} - \frac{\partial}{\partial \zeta} \{ \beta \cdot (M + \frac{v}{c}Z) \} [/tex], where [itex]\beta = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}[/itex]
I'm confused by the existence of terms [itex]\frac{\partial Y}{\partial \eta}[/itex] and [itex]\frac{\partial Z}{\partial \zeta}[/itex], because it's derived in section 3 that
[tex]\tau = \beta \cdot (t - \frac{vx}{c^2})[/tex]
[tex]\xi = \beta \cdot (x - vt)[/tex]
[tex]\eta = y[/tex]
[tex]\zeta = z[/tex]
I can't find a way to apply partial derivative operations to make [itex]\frac{\partial Y}{\partial \eta}[/itex] and [itex]\frac{\partial Z}{\partial \zeta}[/itex] come out. Could anyone give me some tips? Any help is appreciated.