What is the path of a particle in a vector field?

[Horv]

Hi all!

I want to ask about vector fields.
So if I had any kind of field for example $\bar{F}(x,y) = (0,x)$ which represents a river or somthing similar and I put into the river a particle, or point-like body how can I get the path, or curve (flwo line?) from the vector field? I mean that path which the particle or the body moves.

Thanks for the answers.

 

[HallsofIvy]

What vector field does that represent? The force vector (which would give acceleration) or the velocity vector? If the velocity vector, then it represents the two equations dx/dt= 0 and dy/dt= x. That is a system of differential equations and there are several ways of solving that system.

The simplest way would be to first differentiate both sides of dy/dt= x with respect to t: $d^2y/dt^2= dx/dt= 0$ so that y(t)= At+ B for any constants A and B. And then x(t)= dy/dt= A.