So I was reading an article on Wikipedia on tangent spaces but I have a question.
So the definition is as follows,
Suppose that $M$ is a $C^{k}$ differentiable manifold (with smoothness $k\ge 1$ and that $x \in M$. Pick a coordinate chart $\varphi:U \rightarrow \mathbb {R} ^{n}$, where ${\displaystyle U}$ is an open subset of ${\displaystyle M}$ containing ${\displaystyle x}$. Suppose further that two curves ${\displaystyle \gamma _{1},\gamma _{2}:(-1,1)\to M}$ with ${\displaystyle {\gamma _{1}}(0)=x={\gamma _{2}}(0)}$ are given such that both ${\displaystyle \varphi \circ \gamma _{1},\varphi \circ \gamma _{2}:(-1,1)\to \mathbb {R} ^{n}}$ are differentiable
My question is how is ${\displaystyle \varphi \circ \gamma _{1},\varphi \circ \gamma _{2}:(-1,1)\to \mathbb {R} ^{n}}$?
How does this composition of $\varphi$ and $\gamma $ become from $(-1,1) \rightarrow \mathbb{R}^{n}$?
This is marginally imprecise as written. For the composition to make sense, we have to have $\gamma_1((-1,1)),\gamma_2((-1,1))\subseteq U$. However, note that $\gamma_1(0)=\gamma_2(0)=x\in U$, so, by continuity, we can find a small interval $0\in I\subseteq\mathbb{R}$, such that $\gamma_1(I),\gamma_2(I)\subseteq U$. Then, the compositions $\varphi\gamma_1,\varphi\gamma_2\colon I\rightarrow\mathbb{R}^n$ are defined. Since differentiability is a local property, the derivatives $(\varphi\gamma_1)^{\prime}(0),(\varphi\gamma_2)^{\prime}(0)$ do not depend on the chosen interval $I$ and they are all we need to define the tangent space.