I feel a bit confused with some notations and concepts.
I consider a vector field $X: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ and a function $G: \mathbb{R}^{n} \rightarrow \mathbb{R}$ such that
$$ dG(X)=0.$$
Now I have to show that every solution of the system
$$\dot{x}(t)= X(x(t)),$$
lies in the level set of $G$, i.e. the set $\{ x \in \mathbb{R}^{n} \mid G(x)=cst \}$.
My idea was to consider a solution curve $x(t)=(x_1(1),...,x_n(t))$ and to show that $$\frac{d}{dt}G(x(t))=0.$$
I am a bit confused with the notation $dG(X)$, what is meant ?
My calculations so fare are
$\frac{d}{dt}G(x(t))=\sum_{i=1}^{n}\frac{\partial G(x(t))}{\partial x_i}\dot{x}(t)=\sum_{i=1}^{n}\frac{\partial G(x(t))}{\partial x_i}X(x(t))=<\nabla G(x(t)),X(x(t))>$.
Is it then sufficient to conclude ? Thank you for your help and comments.
Yes, $$dG_x(X(x))=DG(x)X(x)=G'(x)X(x)$$ and $$\langle ∇G(x),X(x)\rangle$$ are the same. This is in fact the defining relation for the gradient $∇G$.