3-1. Tangent Vectors

Definition derivation

• $M$,
  • smooth manifold with or without nboundary
• $p \in M$,
• $v: C^{\infty}(M) \rightarrow \mathbb{R}$,

$v$ is said to be a derivation at $p$ if

\[\forall f, g \in C^{\infty}(M), \ v(fg) = f(p) vg + g(p) vf .\]

The set of all derivations of $C^{\infty}(M)$ at $p$, denoted by $T_{p}M$, is a vecvtor space called the tangent space to $M$ at $p$.

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Definition differential

• $M$,
  • smooth manifold with or without boundary
• $N$,
  • smooth manifold with or without boundary
• $F: M \rightarrow N$,
  • smooth map

A map $dF_{p}: T_{p}M \rightarrow T_{F(p)}N$ defined as

\[v \in T_{p}M, \ f \in C^{\infty}(N), \ dF_{p}(v)(f) := v(f \circ F)\]

is called differential of $F$ at $p$.

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Definition coordinate vectrors

• $M$,
• $(U, \phi)$,
  • a smooth coordiante chart on $M$
• $p \in U$,

\[\begin{eqnarray} \left. \frac{\partial}{\partial x^{i}} \right|_{p} & := & (d \phi_{p})^{-1} \left( \left. \frac{\partial}{\partial x^{i}} \right|_{\phi(p)} \right) \nonumber \\ & = & d (\phi^{-1}_{\phi(p)} \left( \left. \frac{\partial}{\partial x^{i}} \right|_{\phi(p)} \right) \nonumber \end{eqnarray}\]

is called the coordinate vectors at $p$.

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