(All terminology is from Guillemin and Pollack.) A vector fied $v$ on a manifold $X\subset \mathbb R^N$ is a smooth map $v:X\to \mathbb R^N$ such that $v(x)\in T_x(X)$ for every $x$. The tangent bundle $T(X)$ is the subset of $X \times \mathbb{R}^N$ given by $$T(X)= \{(x,v) \in X \times \mathbb{R}^N : v \in T_x(X)\}.$$
If $x$ is a zero of a vector field $v$ as above, then it can be shown that $dv_x$ carries $T_x(X)$ into itself. A zero of $v$ is nondegenerate if $dv_x:T_x(X)\to T_x(X)$ is bijective.
A vector field $v$ on $X$ naturally defines a cross-sectional map $f_v:X\to T(X)$, by $f_v(x)=(x,v(x))$. Further, $f_v$ is an embedding, so its image $X_v$ is a submanifold of $T(X)$ diffeomorphic to $X$. Note that the zeros of $v$ correspond to the intersection points of $X_v$ with $X_0=\{(x,0)\}$.
- What is the tangent space of $X_v$ at a point $(x,v(x))$?
- Check that $x$ is a nondegenerate zero of $v$ if and only if $X_v\pitchfork X_0$ at $(x,0)$.
The first part is supposed to be easy, but it doesn't seem so to me. $T_{(x,v(x))}(X_v)$ is by definition the image of the differential of some local parametrization around $(x,v(x))$, but I don't know how to define such parametrizations explicitly and find the image.
The second part asks to show that $T_{(x,0)}(X_v)+T_{(x,0)}(X_0)=T_{(x,0)}(T(X))$. Here I see two problems. The first one is that unless I don't understand the above, I don't know what the first summand is. (The second summand should be a particular case of the first.) The second problem is that Guillemin and Pollack (as far as I can say) don't even mention that $T(X)$ is a manifold, so the RHS doesn't make sense...