Interpretation of the pushforward of a vector under a flow. Suppose $\theta_t(p)$ is the flow of some vector field. I don't really understand how to interpret the meaning of the pushforward $(\theta_t)_*X$ of some vector $X$. What is its intuitive meaning?
 A: Here's a simple example. Suppose we look at the vector field $\frac{\partial}{\partial y}$ on $M = \mathbb{R}^2$ (every vector is a unit vector in the positive $y$-direction). 
The corresponding flow is 
$\theta_t((x,y)) = (x,y+t)$
and the pushforward by the flow on the tangent space at $(x,y)$ is given by the matrix
$(\theta_t)_* = \begin{pmatrix} \partial(x)/\partial x & \partial(y+t)/\partial x \\ \partial(x)/\partial y & \partial(y + t)/\partial y \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$
(it does not depend on $(x,y)$ in this case). So any vector with basepoint at $(x,y)$ gets sent to the same vector with basepoint at $(x,y+t)$.
OK, maybe that was too boring. Suppose we look instead at $y\frac{\partial}{\partial y}$ on $M=\mathbb{R}^2$ (again every vector in the vector field is vertical, but now lengths of vectors are scaled by the $y$-coordinate).
The corresponding flow is
$\theta_t((x,y)) = (x, e^t y)$
and the pushforward at a point $(x,y)$ is
$(\theta_t)_* = \begin{pmatrix} 1 & 0 \\ 0 & e^t \end{pmatrix}$
So a vector starting at $(x,y)$ gets vertically scaled by $e^t$ when it gets to $(x,e^t y)$
