Integral Curves of a Vector Field

Question

How do I find the integral curves of a vector field and what are they intuitively?

eg. what are the integral curves of vector field

$X=\frac{1}{x}\frac{\partial}{\partial x}+\frac{1}{y}\frac{\partial}{\partial y}$?

Answer 1

You are trying to find a curve $\alpha(t) = \left(x(t), y(t)\right)$ such that the tangent vector field $\frac{d\alpha}{dt} = \left(\frac{dx}{dt}, \frac{dy}{dt}\right)$ along $\alpha$ agrees with the restriction of the vector field $X$ along $\alpha$ (i.e. $X(\alpha(t)) = \frac{d\alpha}{dt}$). As such, by equating coefficient functions of the indicated vector fields, you should find that you need to solve the following system of differential equations:

\begin{align*} \frac{dx}{dt} &= \frac{1}{x}\\ \frac{dy}{dt} &= \frac{1}{y}.\\ \end{align*}