If $h: \mathbb{R}^3 \to \mathbb{R}^3$ defined by $h(x_1,x_2,x_3) = (x_1^3 - x_2 + x_3^2, x_2,x_1+x_2+x_3)$
I define $V(x)$ as a subspace formed by all directions tangent to some constrained set at the point $x$.
i.e
$V(x) : = \{ y \in \mathbb{R}^n : \nabla h_i(x)^ty = 0, i = 1, ... , m\}$
How would I compute $V(0,0,0)$ and $V(-1,0,1)$ ?
$\nabla h_i(x) = \{\langle (3x_1^2,0,1),(-1,1,1),(2x_3,0,1) \rangle \} $
So then $V(0,0,0) = \{ (\langle (0,0,1),(-1,1,1),(0,0,1) \rangle \} $
is this correct?