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Since $S^1$ is a compact 1-dimensional regular submanifold in $\mathbb{R}^2$ (it's $S^1 = f^{-1}(1)$ for $f : \mathbb{R}^2 \to \mathbb{R}$ given as $f(x,y) = x^2+y^2$), we can find the tangent space for $S^1$ in (1,0) as $$T_{(1,0)}(S^1) = Ker(d(f)_{(1,0)}).$$ Intuitively $T_{(1,0)}(S^1)$ is the plane $\{(1,y): y \in \mathbb{R}\}.$

But we get $$Ker(d(f)_{(1,0)}) = Ker((2x,2y)_{(1,0)}) = Ker((2,0)) = \{(0,y) : y \in \mathbb{R}\}.$$

What it is wrong? Thanks in advance!

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    $\begingroup$ You are calculating the tangent space as a subspace of $\mathbb{R}^2$. So, it passes through the origin. You can then translate it to the point $(1, 0)$. $\endgroup$ – Joe Johnson 126 Nov 18 '18 at 19:29
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    $\begingroup$ Your result is a subspace of tangent space $T_{(1,0)}R^2$. Nothing wrong with your calculation. The result represent the components of tangent vectors. That is the elements of the kernel should be in the form $0 \partial_x + s\partial_y$ for $s\in R$. Which is exactly what you wanted (up to the identification). $\endgroup$ – Sou Nov 18 '18 at 19:29
  • $\begingroup$ @KelvinLois: This looks like essentially an answer. Would you mind writing it up so that we can get this question off the Unanswered queue? $\endgroup$ – aleph_two Dec 29 '18 at 4:12

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