Finding diffeomorphism given vector fields Given a vector field how do you find the associated diffeomorphisms? Say I am given a vector field in Minkowski space
$$\xi = x \frac{\partial}{\partial t} + t \frac{\partial}{\partial x}.$$
How do I find the associated diffeomorphism, if one exist? I know in this example the Lorentz boost in the x-direction is the diffeomorphism, but I am having trouble understanding how to arrive at that answer. Additionally, how do you tell when they might not have an associated diffeomorphism? I am lead to believe that this vector field cannot have a diffeomorphism translating points forward
$$ \xi = e^{x}\frac{\partial}{\partial x}.$$
 A: Well, actually you are looking for a one-parameter group of diffeomorphisms (or isometries if referring to the boost vector field). This group is obtained by solving the differential equation $$\frac{dx}{ds}= X(x(s))\tag{1}$$
with a generic initial condition $z$ at $s=0$ in the manifold $M$ (Minkowski spacetime in your example).
 $X$ is your vector field on $M$. The solutions have the form $$M \times \mathbb R \ni (z,s) \mapsto \phi_s(z)$$
where $z$ is the initial condition, that is the point $x(0)= \phi_0(x) =z$ and the proper solution of (1) with that initial condition, $z$, is $x_s= \phi_s(z)$.
It turns out that 
$$\phi_0 = id\:, \quad \phi_s \circ \phi_r = \phi_{s+r}\:,\quad \phi_{-s}= (\phi_s)^{-1}\:.$$
Each $\phi_s : M \to M$ is a diffeomorphism. The one-parameter group of diffeomorphisms associated to $X$ is the family of diffeomorphisms $\{\phi_t\}_{t \in \mathbb R}$. 
Generally speaking, the group is only local, i.e., not defined for all values of $s$ (the $s$-domain depends on $z$), but I will not discuss this point in this elementary presentation. 
In the concrete case of the boost vector field, you have to solve the system
$$\frac{dt}{ds}= x(s)\:,\quad \frac{dx}{ds}= t(s)$$
This way you find that $\phi_s((t,x))= (t(s),x(s))$ with
$$t(s) = x\sinh(s)+ t\cosh(s)\:, \quad  x(s) = x\cosh(s)+ t\sinh(s)\:.$$
Ragarding your last question about the exponential vector field, you can solve it yourself now.
