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Exemples d’équations avec le plugin Plugin LaTeX for WordPress

Introduction :

Exemples....

\alpha+\beta\geq\gamma

dU_i = d \left( \frac{\partial X_i}{ \partial x_j} \right).u_j + \frac{\partial X_i}{ \partial x_j}.du_j = \frac{\partial^2 X_i}{\partial x_k \partial x_j}.dx_k.u_j + \frac{\partial X_i}{ \partial x_j}.du_j = \frac{\partial X_i}{ \partial x_j}. \left( \frac{ \partial x_j}{\partial X_l}.\frac{\partial^2 X_l}{\partial x_k \partial x_m} .dx_k.u_m + du_j \right)

La métrique de Schwarzschild

ds^{2}=-(1-\frac{2GM}{c^{2}r})c^{2}dt^{2}+(1-\frac{2GM}{c^{2}r})^{(-1)}dr^{2}+r^{2}(d\theta^{2}+sin^{2}\theta d\phi^{2})

Transport de quadri-vecteurs Hypercomplexes

\begin{multline*}
\nabla_{\nu}.\nabla_{\mu}\overrightarrow{X}-\nabla_{\mu}.\nabla_{\nu}\overrightarrow{X}=\partial_{\nu}\partial_{\mu}x^{\alpha}.\varphi^{i}.h_{i}.\overrightarrow{e_{\alpha}}+\partial_{\mu}x^{\alpha}.\partial_{\nu}\varphi^{i}.h_{i}.\overrightarrow{e_{\alpha}}+\partial_{\mu}x^{\alpha}.\varphi^{i}.\partial_{\nu}h_{i}.\overrightarrow{e_{\alpha}}+\partial_{\mu}x^{\alpha}.\varphi^{i}.h_{i}.\partial_{\nu}\overrightarrow{e_{\alpha}}\\
+x^{\alpha}.[\partial_{\nu}\Phi_{\mu j}^{i}.\varphi^{j}.h_{i}+\partial_{\nu}\varphi^{i}.H_{\mu i}^{j}.h_{j}+\partial_{\nu}\varphi^{i}.h_{i}.\Gamma_{\mu\alpha}^{\beta}.\delta_{\beta}^{\alpha}+\Phi_{\mu j}^{i}.\partial_{\nu}\varphi^{j}.h_{i}+\varphi^{i}.\partial_{\nu}H_{\mu i}^{j}.h_{j}+\varphi^{i}.\partial_{\nu}h_{i}.\Gamma_{\mu\alpha}^{\beta}.\delta_{\beta}^{\alpha}\\
+\Phi_{\mu j}^{i}.\varphi^{j}.\partial_{\nu}h_{i}+\varphi^{i}.H_{\mu i}^{j}.\partial_{\nu}h_{j}+\varphi^{i}.h_{i}.\partial_{\nu}\Gamma_{\mu\alpha}^{\beta}.\delta_{\beta}^{\alpha}].\overrightarrow{e_{\alpha}}\\
+x^{\alpha}.(\Phi_{\mu j}^{i}.\varphi^{j}.h_{i}+\varphi^{i}.H_{\mu i}^{j}.h_{j}+\varphi^{i}.h_{i}.\Gamma_{\mu\alpha}^{\beta}.\delta_{\beta}^{\alpha}).\Gamma_{\nu\alpha}^{\beta}.\overrightarrow{e_{\beta}}\\
-\partial_{\mu}\partial_{\nu}x^{\alpha}.\varphi^{i}.h_{i}.\overrightarrow{e_{\alpha}}-\partial_{\nu}x^{\alpha}.H_{\mu i}^{j}.h_{j}.h_{i}.\overrightarrow{e_{\alpha}}-\partial_{\nu}x^{\alpha}.\varphi^{i}.H_{\mu i}^{j}.h_{j}.\overrightarrow{e_{\alpha}}-\partial_{\nu}x^{\alpha}.\varphi^{i}.h_{i}.\Gamma_{\mu\alpha}^{\beta}.\overrightarrow{e_{\beta}}\\
-\partial_{\mu}x^{\alpha}.(\Phi_{\nu j}^{i}.\varphi^{j}.h_{i}+\varphi^{i}.H_{\nu i}^{j}.h_{j}+\varphi^{i}.h_{i}.\Gamma_{\nu\alpha}^{\beta}.\delta_{\beta}^{\alpha}).\overrightarrow{e_{\alpha}}\\
-x^{\alpha}.[\partial_{\mu}\Phi_{\nu j}^{i}.\varphi^{j}.h_{i}+\partial_{\mu}\varphi^{i}.H_{\nu i}^{j}.h_{j}+\partial_{\mu}\varphi^{i}.h_{i}.\Gamma_{\nu\alpha}^{\beta}.\delta_{\beta}^{\alpha}+\Phi_{\nu j}^{i}.\partial_{\mu}\varphi^{j}.h_{i}+\varphi^{i}.\partial_{\mu}H_{\nu i}^{j}.h_{j}+\varphi^{i}.\partial_{\mu}h_{i}.\Gamma_{\nu\alpha}^{\beta}.\delta_{\beta}^{\alpha}\\
+\varphi^{i}.h_{i}.\Gamma_{\mu\alpha}^{\beta}.\partial_{\nu}\delta_{\beta}^{\alpha}+\Phi_{\nu j}^{i}.\varphi^{j}.\partial_{\mu}h_{i}+\varphi^{i}.H_{\nu i}^{j}.\partial_{\mu}h_{j}+\varphi^{i}.h_{i}.\partial_{\mu}\Gamma_{\nu\alpha}^{\beta}.\delta_{\beta}^{\alpha}+\varphi^{i}.h_{i}.\Gamma_{\nu\alpha}^{\beta}.\partial_{\mu}\delta_{\beta}^{\alpha}].\overrightarrow{e_{\alpha}}\\
-x^{\alpha}.(\Phi_{\nu j}^{i}.\varphi^{j}.h_{i}+\varphi^{i}.H_{\nu i}^{j}.h_{j}+\varphi^{i}.h_{i}.\Gamma_{\nu\alpha}^{\beta}.\delta_{\beta}^{\alpha}).\Gamma_{\mu\alpha}^{\beta}.\overrightarrow{e_{\beta}}
\end{multline*}

 

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