# Thread: vector field flow line calculation

1. ## vector field flow line calculation

Consider the vector field F(x,y,z)=(y/(x^2+y^2),-x/(x^2+y^2),0), (x,y) doesn't equal to 0.
Find the family of flowlines for F.

2. The vector given by the vector field at any point (x, y, z) is tangent to the flow line at that point. One thing I notice immediately is that that k component of the vector field is 0! That simplifies things a lot! It means that the flow is alway over a constant z and reduces this to a two dimensional problem- we only need to find x and y.

And in the xy-plane, if the vector fields is given by <A, B> that means that dx/dt is a multiple of A and that dy/dt is that same multiple of B. And that means that $\displaystyle \frac{dy}{dx}= \left(\frac{dy}{dt}\right)/\left(\frac{dx}{dt}\right)= \frac{B}{A}$.

The result of all that, for this problem, is that $\displaystyle \frac{dy}{dx}= \frac{-\frac{x}{x^2+ y^2}}{\frac{y}{x^2+ y^2}}= -\frac{x}{y}$.

That's a "separable" first order differential equation which should be easy to solve.