广义相对论的几何学|MATH11138 Geometry of General Relativity代写

这是一份ed.ac爱丁堡格大学MATH11138作业代写的成功案

广义相对论的几何学|MATH11138 Geometry of General Relativity代写

We can represent $V$ as a sum,
$$
\mathrm{V}=\pi V+V^{\perp},
$$
where $V^{\perp}$ is the component of $V$ normal to $T_{m}$. Now write $\partial / \partial x^{k}$ as $e_{k}$, and write
$$
\pi V=a^{1} e_{1}+\ldots+a^{n} e_{n},
$$
where the $a^{i}$ are the desired local coordinates. Then
$$
\begin{aligned}
V &=\pi V+V^{\perp} \
&=a^{1} e_{1}+\ldots+a^{n} e_{n}+V^{\perp}
\end{aligned}
$$


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MATH11138 COURSE NOTES :

which we can write in matrix form as
$$
\left[V e_{i}\right]=\left[a^{i}\right] g_{\text {事 }}
$$
whence
$$
\left[a^{i}\right]=\left[V \cdot e_{i}\right] g^{* } . $$ Finally, since $g^{ }$ is symmetric, we can transpose everything in sight to get $$ \left[a^{i}\right]=g^{ *}\left[V \cdot e_{i}\right],
$$



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