Tangent space of circle at a point

Question

I am trying to compute tangent space of $S^1=\{(x,y)\in \mathbb{R}^2: x^2+y^2=1\}$ at a point say $(1,0)=p$.

As $S^1\subseteq \mathbb{R}^2$ we see that $\left\{\frac{\partial}{\partial x}\bigg|_p,\frac{\partial}{\partial y}\bigg|_p\right\}$ are elements of $T_pS^1$. As $S^1$ is one dimensional, we see that $\left\{\frac{\partial}{\partial x}\bigg|_p,\frac{\partial}{\partial y}\bigg|_p\right\}$ is a linearly dependent set when seen as elements of $T_pS^1$ i.e., for some $a,b\in \mathbb{R}$ we have $$a\frac{\partial}{\partial x}\bigg|_p+b\frac{\partial}{\partial y}\bigg|_p=0$$ as functions $C_p^{\infty}(S^1)\rightarrow \mathbb{R}$.

For all $f\in C_p^{\infty}(S^1)$ we have $$a\frac{\partial f}{\partial x}\bigg|_p+b\frac{\partial f}{\partial y}\bigg|_p=0.$$

I have no idea how to proceed from here.

Any reference which discuss this kind of examples are also welcome.

Answer 1

As I wrote in my comment you can do it using a parametrization, but you can do it in another way using the basis of $T_p\mathbb{R}^2$, namely $\left\{\frac{\partial}{\partial x}\bigg|_p,\frac{\partial}{\partial y}\bigg|_p\right\}$ and $f$ (defined below).
$S^1$ is the set of zeros of the map $f:\mathbb{R}^2\rightarrow\mathbb{R}$, $f(x,y)=x^2+y^2-1$. The differential of $f$ at $p\in\mathbb{R}^2$, $df_p$, is a linear map between the vector spaces $T_p\mathbb{R}^2$ and $T_p\mathbb{R}$. Now, the only critical point of $f$ is the origin, with critical value $-1$, this tells us that $S^1$ is an embedded submanifold of $\mathbb{R}^2$ (we already knew it) and that $\forall q \in S^1$, $T_qS^1=\ker df_q$ (a reference is Lee's book 'Introduction to smooth manifolds').
This is good since given $q\in S^1$, an element of $T_q\mathbb{R}^2$, $v=a\frac{\partial}{\partial x}\bigg|_q+b\frac{\partial}{\partial y}\bigg|_q$ is in $T_qS^1\iff df_q(v)=0$. As an example for $q=(1,0)$ you have $df_q=2 dx\bigg|_q+0dy\bigg|_q$ and the condition for being a tangent vector reads $2a=0$, thus a tanget vector is of the form $b\frac{\partial}{\partial y}\bigg|_q$, in accordance to what yo found using curves.

Answer 2

You should first for every point $x$ define a curve passing through that point. The equation $x^2 + y^2 = 1$ is not a curve yet. Fix a point where the equality is true, e.g. $x = (\cos(\theta), \sin(\theta))$ and try to find a curve $\gamma(t)$ passing through that point at $t_0$ that lives on $S^1$. (Can you find one?)

It's derivate $\gamma'(t_0)$ will span a one-dimensional space $T_xS^1$. As you have pointed out already, this one dimensional space is the whole tangent space at that point. $(x, T_xS^1)$ is the tangent bundle.