Q1: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be given by $f(x,y) = x^2 + y^2 - 1$. We know that $M = f^{-1}[{ 0 } ]$ is a submanifold. What is the tangent space at the point $(0,1) \in M$?

A1: $T^{\mathrm{sub}}_p(M) = {0} \times \mathbb{R}$

A2: $T^{\mathrm{sub}}_p(M) = \mathbb{R}$

A3: $T^{\mathrm{sub}}_p(M) = \mathbb{R} \times {0}$

A4: $T^{\mathrm{sub}}_p(M) = \mathbb{R} \times \mathbb{R}$

A5: $T^{\mathrm{sub}}_p(M) = { 0 }$

Q2: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be given by $f(x,y) = x^2 + y^2 - 1$. We know that $M = f^{-1}[{ 0 } ]$ is a submanifold? Which vector is orthogonal to the tangent space $T^{\mathrm{sub}}_p(M)$.

A1: $\mathrm{grad}(f)(p)$

A2: $(1,1) - p$

A3: $(1,0) + p$

Q3: Let $M \subset \mathbb{R}^4$ be a submanifold of dimension 3. What is not correct for the tangent space $T^{\mathrm{sub}}_p(M)$?

A1: $T^{\mathrm{sub}}_p(M)$ is a vector space in $\mathbb{R}^3$

A2: dimension of $T^{\mathrm{sub}}_p(M)$ is 3

A3: $T^{\mathrm{sub}}_p(M)$ is a vector space in $\mathbb{R}^4$