Given a vector field, how does one compute the flow?

**ajskim** · October 13th 2010, 08:28 AM

Hi,
I need to be able to know how to compute the flow in R^3 and R^2 given some relatively simple vector fields. My issue is that whenever I read any literature on flows it is very abstract and generalized to the point that I'm not sure what to make of it.
Thanks

**TheEmptySet** · October 13th 2010, 09:48 AM

Here is the basic idea.

let $\vec{r}(t)=x(t)\vec{i}+y(t)\vec{j}+z(t)\vec{k}$ be the position of a particle at time t. Then its velocity is

$\vec{v}(t)=\frac{d}{dt}\vec{r}(t)=\frac{dx}{dt}\ve c{i}+\frac{dy}{dt}\vec{j}+\frac{dz}{dt}\vec{k}$

If your vector field $\vec{F}(x,y,z)$ is a velocity field then

$\vec{F}(\vec{r}(t))=\frac{dx}{dt}\vec{i}+\frac{dy} {dt}\vec{j}+\frac{dz}{dt}\vec{k}$

by equating the components of the vectors you will get a system of ODE's that are the flow lines of the vectorfield

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