Let $M$ be a smooth manifold, $F:M\to M$ be a diffeomorphism, and $X$ be a vector field on $M$ i.e. $X\in TM$.
We say that $X$ is invariant if $$X(F(a))=F_*X(a)$$ for all $a\in M$ where $F^*$ is a pushforward math. I wanted to construct an example of a vector field in this case which is not invariant. Also, I wonder about other examples and about geometric meaning of an invariant vector field.
Consider $M=\mathbb{R}^2$, $F:\mathbb{R}^2\to \mathbb{R}^2$ where $F(x,y)=(y,x)$, and $X(x,y)=(x+1,y)$. Then we can see that this vector field is not invariant as $$X(F(a))=X(a_2,a_1)=(a_2+1,a_1)\text{ does not equal to}$$ $$F^*X(a)=(a_2,a_1+1).$$