Needs["VectorFieldPlots`"]

Needs["VectorAnalysis`"]

v1= y[t]
v2 = -x[t]

v={v1,v2}

VectorFieldPlot[v,{x[t],-1,1},{y[t],-1,1}]

tend=2 Pi -0.1

s1=NDSolve[
{x'[t]==v1,y'[t]==v2,
x[0]==0.5,y[0]==0.5},
{x,y},
{t,0,tend}
]

s2=NDSolve[
{x'[t]==v1,y'[t]==v2,
x[0]==0.4,y[0]==0.4},
{x,y},
{t,0,tend}
]

Show[ParametricPlot[Evaluate[{x[t],y[t]}/.s1],{t,0,tend},PlotRange->{{-1,1},{-1,1}}],ParametricPlot[Evaluate[{x[t],y[t]}/.s2],{t,0,tend},PlotRange->{{-1,1},{-1,1}}]]

v1=y
v2=-x

v={v1,v2,0}

Curl[v,Cartesian[x,y,z]]

Simplify[%]

Div[v,Cartesian[x,y,z]]