I was brushing up on my complex arithmetic in preparation for a class in ODE's this semester and I found myself looking at Exercise 2.7.5 in Introduction to Complex Analysis for Engineers by Michael Alder, which reads
The exponential function is a procedure for turning vector fields into flows; if you take the vector field which is given by$V\begin{bmatrix} x \\ y \end{bmatrix}=\begin{bmatrix} 0 & -1\\ 1 & 0 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix}$ you call the matrix $A$ and then the flow is given as $e^{tA}$.
[...]
Draw a picture of the vector field. Identify the matrix as a complex number. Deduce that $e^{it}=\cos t+i\sin t$ is little more than the observation that exponentiation is about solving ODE's by Euler's method taken to the limit.
I would like very much to understand this very well. I've actually done most of it and perhaps the problem is that I haven't actually taken the ODE's class yet, but I've read ahead enough to understand most of what's being said.
I drew the vector field (by hand) and got some lovely circley looking things. When he says, "identify the matrix as a complex number," I understand that he is referring to the fact that in the book he defines a complex number $a+bi$ to be the matrix $\begin{bmatrix} a & -b\\ b & a \end{bmatrix}$and so $A$ is $i$. I also managed to do the exponentiation $e^{tA}$ and got
$\begin{bmatrix} \cos t & -\sin t\\ \sin t & \cos t \end{bmatrix}$
Which is, of course, the complex number $\cos t + i\sin t$.
So so far so good, I've shown that $e^{it}=\cos t + i\sin t$.
I'm just having problems understanding the last little bit, and maybe that's cause I haven't taken the ODE's class yet, but I see that there are tangent lines to radiuses of circles somewhere in there since the vector field makes tangent lines to circles around the origin and $e^{At}$ ends up being the rotation matrix with angle $t$, so we have some notion of a radius rotating around the origin somewhere? How does this relate to Euler's method for solving ODE's? Is the idea that there is an ODE which produces that vector field as a direction field, and $e^{it}$ gives solutions? I get a little bit lost at this point, maybe someone can help me finish putting the pieces together.