Differential of rotation matrix at the north pole of sphere

Question

Let T(p) rotate $p\in S^{2}$ by angle $\theta $ about the z-axis. The problem is to compute $dT_{(0,0,1)}$.

T can be represented by the usual 3x3 rotation matrix $A_{z}(\theta)$. So $T(p)=A_{z}p$.

Can you compute the $dT_{(0,0,1)}$?

Answer 1

On $\mathbb{R}^3$, because $A_z$ is linear and in $\mathbb{R}^3$ there is a natural identification of tangent spaces with the entire space, $dA_z = A_z$. Therefore, you know exactly what the action of $dA_z$ is on $T_{(0,0,1)}\mathbb{R}^3$.

Since $T_{(0,0,1)}S^2\subset T_{(0,0,1)}\mathbb{R}^3$, how can you deduce the action of $dA_z$ on it?