StreamPlot[{-Sin[x]Cos[y],Cos[x]Sin[y]},{x,-π,π},{y,-π,π}]

b[x_,y_]={((-y)/((x^2)+(y^2))),(x/((x^2)+(y^2)))}

p[x_,y_]=b[x,y][[1]]

q[x_,y_]=b[x,y][[2]]

Simplify[D[q[x,y],x]-D[p[x,y],y]]

Simplify[D[p[x,y],x]+D[q[x,y],y]]

jb[x_,y_]=Simplify[{{D[p[x,y],x],D[p[x,y],y]},{D[q[x,y],x],D[q[x,y],y]}}];

MatrixForm[jb[x,y]]

StreamPlot[-b[x-5,y-5]+b[x-5,y+5]+b[x+5,y-5]-b[x+5,y+5],{x,-10,10},{y,-10,10}]

β=StreamPlot[b[x,y],{x,-3,3},{y,-3,3}]

α[t_]={(1-2 Cos[t])Cos[t]+1/2,(1-2 Cos[t])Sin[t]};

γ=ParametricPlot[α[t],{t,0,2 π},PlotStyle->Thick];

Show[β,γ,Graphics[{Thick,Arrow[{α[π/8],α[π/8]+0.4*α'[π/8]}],Arrow[{α[3π/8],α[3π/8]+0.4*α'[3π/8]}],Arrow[{α[5π/8],α[5π/8]+0.4*α'[5π/8]}],Arrow[{α[9π/8],α[9π/8]+0.4*α'[9π/8]}]}]]

AbsoluteTiming[Integrate[b[α[t][[1]],α[t][[2]]].α'[t],{t,0,2 π}]]

FullSimplify[b[α[t][[1]],α[t][[2]]].α'[t]]

RegionPlot[(x^2)+(y^2)≤0.09,{x,-2,2},{y,-2,2}];

RegionPlot[x≥0&&(y^2)≤0.012,{x,-2,2},{y,-2,2}];

Show[%,%%,ParametricPlot[{x,0},{x,0,2},PlotStyle->Thick],Graphics[{PointSize[Large],Black,Point[{0,0}]}]]

RegionPlot[(x^2)+(y^2)≤0.09,{x,-2,2},{y,-2,2}];

RegionPlot[x≤0&&(y^2)≤0.012,{x,-2,2},{y,-2,2}];

Show[%,%%,ParametricPlot[{x,0},{x,-2,0},PlotStyle->Thick],Graphics[{PointSize[Large],Black,Point[{0,0}]}]]

Show[γ,Graphics[{PointSize[Large],Black,Point[α[2 π/3]],Point[α[-2 π/3]],Point[α[π/6]],Point[α[-π/6]]}]]