# Flow trajectories of a vector field with singular point

Definition: a flow line (of 2D vector field) is a curve that vector field is tangent everywhere to it. If the vector field is zero at some point (singular point) the definition is ambiguous. Does this mean a flow line that reaches a singular point should terminate there?

An example from physics: electric field lines of two positive same charges has a singular point at the middle of connecting line of two charges. Now does the field line statrting from one of the charges in the direction of the connecting line, terminate at the singular point? Because usually we assume that an electric field line cannot terminate at empty space?

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It does terminate at the singular point. From a more analytical point of view, the flow of a vector field $X$ is the solution $\gamma(t)$ of the differential equation $\gamma'(t)=X(\gamma(t))$. When $X(x_0)=0$ (i.e. $x_0$ is a singular point) then the solution to that equation with initial condition $x_0$ is constant. It can happen that solutions with nearby initial conditions (i.e. line fields passing through point nearby $x_0$ tend to $x_0$ in the future ($t \rightarrow \infty$) or in the past ($t \rightarrow - \infty$), in which case it "asymptotically terminates" at the singular point.
It may also have different behaviours, for example if $X(x,y)=(-y,x)$ then the line fields are circle centered around the singular point $(0,0)$.