Page:Becquerel - Le Principe de relativité et la théorie de la gravitation, 1922.djvu/185

165

chapitre XIII. — notions de calcul tensoriel.

indices $\alpha$ et $\beta ,$ ce qui est légitime puisque ce sont des indices muets ; de plus, dans le dernier terme, nous avons écrit

${\frac {\partial }{\partial x'_{\lambda }}}={\frac {\partial x_{\gamma }}{\partial x'_{\lambda }}}{\frac {\partial }{\partial x_{\gamma }}}\cdot$

Nous avons de même les formules.

(31-13)

${\begin{aligned}{\frac {\partial g'_{\nu \lambda }}{\partial x'_{\mu }}}=g_{\alpha \beta }\left({\frac {\partial ^{2}x_{\alpha }}{\partial x'_{\nu }\,\partial x'_{\mu }}}{\frac {\partial x_{\beta }}{\partial x'_{\lambda }}}\right.&+\left.{\frac {\partial ^{2}x_{\alpha }}{\partial x'_{\lambda }\,\partial x'_{\mu }}}{\frac {\partial x_{\beta }}{\partial x'_{\nu }}}\right)\\&\quad \;\;\,+{\frac {\partial x_{\alpha }}{\partial x'_{\mu }}}{\frac {\partial x_{\beta }}{\partial x'_{\nu }}}{\frac {\partial x_{\gamma }}{\partial x'_{\lambda }}}{\frac {\partial g_{\beta \gamma }}{\partial x_{\alpha }}},\end{aligned}}$

(32-13)

${\begin{aligned}{\frac {\partial g'_{\mu \lambda }}{\partial x'_{\nu }}}=g_{\alpha \beta }\left({\frac {\partial ^{2}x_{\alpha }}{\partial x'_{\mu }\,\partial x'_{\nu }}}{\frac {\partial x_{\beta }}{\partial x'_{\lambda }}}\right.&+\left.{\frac {\partial ^{2}x_{\alpha }}{\partial x'_{\lambda }\,\partial x'_{\nu }}}{\frac {\partial x_{\beta }}{\partial x'_{\mu }}}\right)\\&\quad \;\;\,+{\frac {\partial x_{\alpha }}{\partial x'_{\mu }}}{\frac {\partial x_{\beta }}{\partial x'_{\nu }}}{\frac {\partial x_{\gamma }}{\partial x'_{\lambda }}}{\frac {\partial g_{\alpha \gamma }}{\partial x_{\beta }}}\cdot \end{aligned}}$

Nous avons interchangé dans les derniers termes les indices muets $\alpha ,$ $\beta ,$ $\gamma .$

Ajoutons (31) et (32) et retranchons (30), nous obtenons

(33-13)

${\begin{bmatrix}\mu \nu \\\lambda \end{bmatrix}}'=g_{\alpha \beta }{\frac {\partial ^{2}x_{\alpha }}{\partial x'_{\mu }\,\partial x'_{\nu }}}{\frac {\partial x_{\beta }}{\partial x'_{\lambda }}}+{\frac {\partial x_{\alpha }}{\partial x'_{\mu }}}{\frac {\partial x_{\beta }}{\partial x'_{\nu }}}{\frac {\partial x_{\gamma }}{\partial x'_{\lambda }}}{\begin{bmatrix}\alpha \beta \\\gamma \end{bmatrix}}$

et, en multipliant les deux membres par $g'^{\lambda \rho }{\frac {\partial x_{\varepsilon }}{\partial x'_{\rho }}},$

(34-13)	${\begin{Bmatrix}\!\mu \nu \!\\\rho \end{Bmatrix}}'{\frac {\partial x_{\varepsilon }}{\partial x'_{\rho }}}$	${}=\left(g'^{\lambda \rho }{\frac {\partial x_{\beta }}{\partial x'_{\lambda }}}{\frac {\partial x_{\varepsilon }}{\partial x'_{\rho }}}\right)g_{\alpha \beta }{\frac {\partial ^{2}x_{\alpha }}{\partial x'_{\mu }\,\partial x'_{\nu }}}$
		$\quad +\left(g'^{\lambda \rho }{\frac {\partial x_{\gamma }}{\partial x'_{\lambda }}}{\frac {\partial x_{\varepsilon }}{\partial x'_{\rho }}}\right){\frac {\partial x_{\alpha }}{\partial x'_{\mu }}}{\frac {\partial x_{\beta }}{\partial x'_{\nu }}}{\begin{bmatrix}\alpha \beta \\\gamma \end{bmatrix}}$
		${}=g^{\beta \varepsilon }g_{\alpha \beta }{\frac {\partial ^{2}x_{\alpha }}{\partial x'_{\mu }\,\partial x'_{\nu }}}+{\frac {\partial x_{\alpha }}{\partial x'_{\mu }}}{\frac {\partial x_{\beta }}{\partial x'_{\nu }}}g^{\gamma \varepsilon }{\begin{bmatrix}\alpha \beta \\\gamma \end{bmatrix}}$
		 [d’après (6-13)]
		${}={\frac {\partial ^{2}x_{\varepsilon }}{\partial x'_{\mu }\,\partial x'_{\nu }}}+{\frac {\partial x_{\alpha }}{\partial x'_{\mu }}}{\frac {\partial x_{\beta }}{\partial x'_{\nu }}}{\begin{Bmatrix}\!\alpha \beta \!\\\varepsilon \end{Bmatrix}}$
		 [d’après (14-13), (16-13) et (27-13)].

C’est la formule auxiliaire dont nous allons nous servir.

Soit maintenant $\mathrm {A} _{\mu }$ un quadrivecteur covariant : nous avons

$\mathrm {A} '_{\mu }={\frac {\partial x_{\sigma }}{\partial x'_{\mu }}}\mathrm {A} _{\sigma }$

et, par dérivation,

${\frac {\partial \mathrm {A} '_{\mu }}{\partial x'_{\nu }}}={\frac {\partial x_{\sigma }}{\partial x'_{\mu }}}{\frac {\partial x_{\tau }}{\partial x'_{\nu }}}{\frac {\partial \mathrm {A} _{\sigma }}{\partial x_{\tau }}}+{\frac {\partial ^{2}x_{\sigma }}{\partial x'_{\mu }\,\partial x'_{\nu }}}\mathrm {A} _{\sigma }.$