so I am given $f\in C^1(\mathbb{R}^3, \mathbb{R}^2)$, $f(x,y,z)=(x^2+y^2,y^2+z^2)$. I have to find the tangent space to $p=(1,0,2)\in M$ where $M:=f^{-1}(\{(1,4)\})$. I know that $M$ is a zylinder centered in the origin with radius $r=1$ and height $h=2$.
How can I compute the tangent space $T_p M$?
(1,4) is a regular value of $f$, therefore $M$ is a diffrentiable surface. Take $v \in T_pM$. $v$ is the derivative of a differentiable curve in $M$ that passes through $p$. So $Df_p(v)= (f \circ \lambda)'=0$ (since $\lambda$ lies in $M$ it is a constant function). So $ T_pM \subset ker(Df_p)$, but they have the same dimension (check) thus they are equal. So $T_pM = ker(Df_p)$.