Tangent space of a cubic

Question

my excercise is to find the tangent space of $$M=\{ (x,y,z) \in \mathbb{R}^3 \vert x^3 + y^3 + z^3 = 1 \}.$$ Can anyone help me?

Answer 1

This manifold is given as the level set of a graph; i.e, it is the points $F(x,y,z) = 0$ where $F(x,y,z) = x^3+y^3+z^3 - 1$. The tangent space at $p \in M$ is given by $\ker DF(p)$, where $DF$ is the derivative of $F$.