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:<math> \times </math> est [[productum vectorialis]] sive productum crucis.
==Aequatio Lorentziana
Aequatio Lorentziana forma vectorali (unitatibus Gaussiana CGSF) modo scriptae est sic
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:<math>\vec \mathbf v</math> est [[velocitas]] momentanea particulae (in centimetris per [[secundum]]), et
:<math> \times </math> est [[productum vectorialis]] sive productum crucis.
==Aequatio Lorentziana tensorali forma scripta=
Aequatio viris Lorentzianae scribere possumus in forma tensorali covariante (unitatibus MKSA) sic:
::<math> \frac{d p^\alpha}{d \tau} = q u_\beta F^{\alpha \beta} </math>
ubi
::<math>\tau </math> est [[tempus proprium]] particulae multiplicatum a c,
::''q'' est [[onus electricum]] particulae,
::''u'' est [[four-velocity|4-velocitas]] particulae, definita sicut:
::<math>u_\beta = \left(u_0, u_1, u_2, u_3 \right) = \gamma \left(c, v_x, v_y, v_z \right) \,</math> et
::''F'' est [[tensor campi electromagnetici]] definitus sicut:
::<math>F^{\alpha \beta} = \begin{bmatrix}
0 & -E_x/c & -E_y/c & -E_z/c \\
E_x/c & 0 & -B_z & B_y \\
E_y/c & B_z & 0 & -B_x \\
E_z/c & -B_y & B_x & 0
\end{bmatrix}
</math>.
==Demonstratio==
Pars <math>\mu =1</math> viris electromagnetica est:
::<math> \gamma \frac{d p^1}{d t} = \frac{d p^1}{d \tau} = q u_\beta F^{1 \beta} = q\left(-u^0 F^{10} + u^1 F^{11} + u^2 F^{12} + u^3 F^{13} \right) .\,</math>
Hic, <math> \tau </math> est [[tempus propium]] particulae. Elementos tensoris electromagnetici ''F'' substituimus et obtinemus:
::<math> \gamma \frac{d p^1}{d t} = q \left(-u^0 \left(\frac{-E_x}{c} \right) + u^2 (B_z) + u^3 (-B_y) \right) \,</math>
Si expressim introducimus partes [[quattuor-velocitas|quattuor-velocitatis]], obtinemus
::<math> \gamma \frac{d p^1}{d t} = q \gamma \left(c \left(\frac{E_x}{c} \right) + v_y B_z - v_z B_y \right) \,</math>
::<math> \gamma \frac{d p^1}{d t} = q \gamma \left( E_x + \left(\mathbf{v} \times \mathbf{B} \right)_x \right) .\,</math>
Calculatio partis <math>\mu = 2</math> aut <math>\mu = 3</math> est similis, postquam obtinemus:
::<math> \gamma \frac{d \mathbf{p} }{d t} = \frac{d \mathbf{p} }{d \tau} = q \gamma \left(\mathbf{E} + (\mathbf{v} \times \mathbf{B})\right) \,</math>,
quod est aequatio Lorentziana.
[[categoria:Physica]]
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