f[y_]=(Sinh[a*d]+Sinh[a*(y-d)]-Sinh[a*y])/(Cosh[a*(y-d)]-Cosh[a*y])

Integrate[f[y],y]

Simplify[%]

g[x_]=Sin[a*x]/Cos[a*x]

Integrate[g[x],x]

DSolve[y'[x]==Tan[a*x]*(Cosh[a*(y[x]-d)]-Cosh[a*y[x]])/(Sinh[a*d]+Sinh[a*(y[x]-d)]-Sinh[a*y[x]]),y,x]

ExT=-Cos[xT]*(1+(Sinh[yT-dT]-Sinh[yT])/Sinh[dT])

EyT=-Sin[xT]*(Cosh[yT-dT]-Cosh[yT])/Sinh[dT]

dT=2

LoadVectorFieldPlots`

plot1=PlotVectorField[{ExT,EyT},{xT,0,2*Pi},{yT,0,dT},Frame->True,FrameLabel->{ax,ay}]

plot2={GrayLevel[.8],Rectangle[{0,0},{2*Pi,dT}]}

Show[Graphics[plot2],plot1,Frame->True,FrameLabel->{ax,ay}]

g[x0_,y0_,y_,d_]=ArcCos[Exp[Log[Cos[x0]]+2*Cosh[d/2]*(ArcTanh[Cosh[(d-y0)/2]/Cosh[y0/2]]-ArcTanh[Cosh[(d-y)/2]/Cosh[y/2]])+Log[Sinh[d/2-y0]/(Sinh[d/2 -y])]]]

g1=Graphics[{Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{1.5875,.1},{0,0},{0,-1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{1.385,.1},{0,0},{0,-1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{1.78,.1},{0,0},{0,-1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{1.195,.1},{0,0},{0,-1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{1.98,.1},{0,0},{0,-1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{.61,.1},{0,0},{0,-1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{2.18,.1},{0,0},{0,-1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{.99,.1},{0,0},{0,-1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{2.57,.1},{0,0},{0,-1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{.8,.1},{0,0},{0,-1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{2.37,.1},{0,0},{0,-1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{2.763,.1},{0,0},{0,-1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{2.935,.1},{0,0},{0,-1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{.415,.1},{0,0},{0,-1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{.235,.1},{0,0},{0,-1}]}]

q1=ParametricPlot[{{g[0,0,y,2],y},{Graphics[{Text["X",{.5,.2}]}]},{g[Pi/16,0,y,2],y},{g[Pi/8,0,y,2],y},{g[3*Pi/16,0,y,2],y},{g[Pi/4,0,y,2],y},{g[5*Pi/16,0,y,2],y},{g[3*Pi/8,0,y,2],y},{g[7*Pi/16,0,y,2],y},{g[Pi/2,0,y,2],y},{g[9*Pi/16,0,y,2],y},{g[5*Pi/8,0,y,2],y},{g[11*Pi/16,0,y,2],y},{g[3*Pi/4,0,y,2],y},{g[13*Pi/16,0,y,2],y},{g[7*Pi/8,0,y,2],y},{g[15*Pi/16,0,y,2],y},{g[Pi,0,y,2],y}},{y,0,1}]

a1={Graphics[Arrow[{{.06,.06},{.05,.05}}]],Graphics[Arrow[{{3.09,.06},{3.10,.05}}]]}

s1=Show[g1,a1,q1]

g2=Graphics[{Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{1.57,1.9},{0,0},{0,1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{1.375,1.9},{0,0},{0,1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{1.77,1.9},{0,0},{0,1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{1.175,1.9},{0,0},{0,1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{1.96,1.9},{0,0},{0,1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{.59,1.9},{0,0},{0,1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{2.15,1.9},{0,0},{0,1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{.97,1.9},{0,0},{0,1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{2.55,1.9},{0,0},{0,1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{.78,1.9},{0,0},{0,1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{2.35,1.9},{0,0},{0,1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{2.72,1.9},{0,0},{0,1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{2.92,1.9},{0,0},{0,1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{.4,1.9},{0,0},{0,1}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{.20,1.9},{0,0},{0,1}]}]

q2=ParametricPlot[{{g[0,0,y,2],2-y},{g[Pi/16,0,y,2],2-y},{g[Pi/8,0,y,2],2-y},{g[3*Pi/16,0,y,2],2-y},{g[Pi/4,0,y,2],2-y},{g[5*Pi/16,0,y,2],2-y},{g[3*Pi/8,0,y,2],2-y},{g[7*Pi/16,0,y,2],2-y},{g[Pi/2,0,y,2],2-y},{g[9*Pi/16,0,y,2],2-y},{g[5*Pi/8,0,y,2],2-y},{g[11*Pi/16,0,y,2],2-y},{g[3*Pi/4,0,y,2],2-y},{g[13*Pi/16,0,y,2],2-y},{g[7*Pi/8,0,y,2],2-y},{g[15*Pi/16,0,y,2],2-y},{g[Pi,0,y,2],2-y}},{y,0,1}]

a2={Graphics[Arrow[{{.06,1.94},{.05,1.95}}]],Graphics[Arrow[{{3.09,1.94},{3.10,1.95}}]]}

s2=Show[g2,a2,q2]

Show[s1,s2]

Show[q1,q2, PlotRange->{0,2}]

g3=Graphics[{Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{1.5875+Pi,.1},{0,0},{0,-1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{1.385+Pi,.1},{0,0},{0,-1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{1.78+Pi,.1},{0,0},{0,-1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{1.195+Pi,.1},{0,0},{0,-1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{1.98+Pi,.1},{0,0},{0,-1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{.61+Pi,.1},{0,0},{0,-1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2.18+Pi,.1},{0,0},{0,-1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{.99+Pi,.1},{0,0},{0,-1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2.57+Pi,.1},{0,0},{0,-1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{.8+Pi,.1},{0,0},{0,-1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2.37+Pi,.1},{0,0},{0,-1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2.763+Pi,.1},{0,0},{0,-1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2.935+Pi,.1},{0,0},{0,-1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{.415+Pi,.1},{0,0},{0,-1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{.235+Pi,.1},{0,0},{0,-1}]}]

q3=ParametricPlot[{{Pi+g[0,0,y,2],y},{Graphics[{Text["X",{.5,.2}]}]},{Pi+g[Pi/16,0,y,2],y},{Pi+g[Pi/8,0,y,2],y},{Pi+g[3*Pi/16,0,y,2],y},{Pi+g[Pi/4,0,y,2],y},{Pi+g[5*Pi/16,0,y,2],y},{Pi+g[3*Pi/8,0,y,2],y},{Pi+g[7*Pi/16,0,y,2],y},{Pi+g[Pi/2,0,y,2],y},{Pi+g[9*Pi/16,0,y,2],y},{Pi+g[5*Pi/8,0,y,2],y},{Pi+g[11*Pi/16,0,y,2],y},{Pi+g[3*Pi/4,0,y,2],y},{Pi+g[13*Pi/16,0,y,2],y},{Pi+g[7*Pi/8,0,y,2],y},{Pi+g[15*Pi/16,0,y,2],y},{Pi+g[Pi,0,y,2],y}},{y,0,1}]

a3={Graphics[Arrow[{{.24+Pi,.23},{.25+Pi,.24}}]],Graphics[Arrow[{{2.91+Pi,.23},{2.90+Pi,.24}}]]}

Show[q3]

s3=Show[g3,q3,a3]

Show[s1,s2,s3]

Show[%,Frame->True,FrameTicks->True,FrameLabel->{ax,ay}]

g4=Graphics[{Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{1.56+Pi,2-.1},{0,0},{0,1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{1.37+Pi,2-.1},{0,0},{0,1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{1.76+Pi,2-.1},{0,0},{0,1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{1.17+Pi,2-.1},{0,0},{0,1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{1.95+Pi,2-.1},{0,0},{0,1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{.59+Pi,2-.1},{0,0},{0,1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2.15+Pi,2-.1},{0,0},{0,1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{.97+Pi,2-.1},{0,0},{0,1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2.54+Pi,2-.1},{0,0},{0,1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{.78+Pi,2-.1},{0,0},{0,1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2.34+Pi,2-.1},{0,0},{0,1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2.73+Pi,2-.1},{0,0},{0,1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2.915+Pi,2-.1},{0,0},{0,1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{.40+Pi,2-.1},{0,0},{0,1}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{.21+Pi,2-.1},{0,0},{0,1}]}]

q4=ParametricPlot[{{Pi+g[0,0,y,2],2-y},{Pi+g[Pi/16,0,y,2],2-y},{Pi+g[Pi/8,0,y,2],2-y},{Pi+g[3*Pi/16,0,y,2],2-y},{Pi+g[Pi/4,0,y,2],2-y},{Pi+g[5*Pi/16,0,y,2],2-y},{Pi+g[3*Pi/8,0,y,2],2-y},{Pi+g[7*Pi/16,0,y,2],2-y},{Pi+g[Pi/2,0,y,2],2-y},{Pi+g[9*Pi/16,0,y,2],2-y},{Pi+g[5*Pi/8,0,y,2],2-y},{Pi+g[11*Pi/16,0,y,2],2-y},{Pi+g[3*Pi/4,0,y,2],2-y},{Pi+g[13*Pi/16,0,y,2],2-y},{Pi+g[7*Pi/8,0,y,2],2-y},{Pi+g[15*Pi/16,0,y,2],2-y},{Pi+g[Pi,0,y,2],2-y}},{y,0,1}]

a4={Graphics[Arrow[{{.29+Pi,2-.27},{.30+Pi,2-.28}}]],Graphics[Arrow[{{2*Pi-.29,2-.27},{2*Pi-.30,2-.28}}]]}

s4=Show[g4,q4,a4]

Show[s1,s2,s3,s4]

Show[%,Frame->True,FrameTicks->True,FrameLabel->{ax,ay}]

g5=Graphics[{Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{0,.1},{-1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{0,.2},{-1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{0,.3},{-1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{0,.4},{-1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{0,.5},{-1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{0,.6},{-1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{0,.7},{-1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{0,.8},{-1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{0,.9},{-1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{0,1.},{-1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{0,1.1},{-1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{0,1.2},{-1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{0,1.3},{-1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{0,1.4},{-1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{0,1.5},{-1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{0,1.6},{-1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{0,1.7},{-1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{0,1.8},{-1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{0,1.9},{-1,0}]}]

q5=ParametricPlot[{{g[0,.1,y,2],y},{g[0,.2,y,2],y},{g[0,.3,y,2],y},{g[0,.4,y,2],y},{g[0,.5,y,2],y},{g[0,.6,y,2],y},{g[0,.7,y,2],y},{g[0,.8,y,2],y},{g[0,.9,y,2],y},{g[0,1,y,2],y},{g[0,1.1,y,2],y},{g[0,1.2,y,2],y},{g[0,1.3,y,2],y},{g[0,1.4,y,2],y},{g[0,1.5,y,2],y},{g[0,1.6,y,2],y},{g[0,1.7,y,2],y},{g[0,1.8,y,2],y},{g[0,1.9,y,2],y}},{y,.1,1.9}]

s5=Show[g5,q5]

Show[s1,s2,s3,s4,s5]

g6=Graphics[{Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{Pi,.1},{-1,0}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{Pi,.2},{-1,0}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{Pi,.3},{-1,0}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{Pi,.4},{-1,0}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{Pi,.5},{-1,0}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{Pi,.6},{-1,0}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{Pi,.7},{-1,0}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{Pi,.8},{-1,0}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{Pi,.9},{-1,0}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{Pi,1.},{-1,0}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{Pi,1.1},{-1,0}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{Pi,1.2},{-1,0}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{Pi,1.3},{-1,0}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{Pi,1.4},{-1,0}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{Pi,1.5},{-1,0}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{Pi,1.6},{-1,0}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{Pi,1.7},{-1,0}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{Pi,1.8},{-1,0}],Text[StyleForm[">",FontSize->14, FontWeight->"Bold"],{Pi,1.9},{-1,0}]}]

q6=ParametricPlot[{{Pi-g[0,.1,y,2],y},{Pi-g[0,.2,y,2],y},{Pi-g[0,.3,y,2],y},{Pi-g[0,.4,y,2],y},{Pi-g[0,.5,y,2],y},{Pi-g[0,.6,y,2],y},{Pi-g[0,.7,y,2],y},{Pi-g[0,.8,y,2],y},{Pi-g[0,.9,y,2],y},{Pi-g[0,1,y,2],y},{Pi-g[0,1.1,y,2],y},{Pi-g[0,1.2,y,2],y},{Pi-g[0,1.3,y,2],y},{Pi-g[0,1.4,y,2],y},{Pi-g[0,1.5,y,2],y},{Pi-g[0,1.6,y,2],y},{Pi-g[0,1.7,y,2],y},{Pi-g[0,1.8,y,2],y},{Pi-g[0,1.9,y,2],y}},{y,.1,1.9}]

q7=ParametricPlot[{{Pi+g[0,.1,y,2],y},{Pi+g[0,.2,y,2],y},{Pi+g[0,.3,y,2],y},{Pi+g[0,.4,y,2],y},{Pi+g[0,.5,y,2],y},{Pi+g[0,.6,y,2],y},{Pi+g[0,.7,y,2],y},{Pi+g[0,.8,y,2],y},{Pi+g[0,.9,y,2],y},{Pi+g[0,1,y,2],y},{Pi+g[0,1.1,y,2],y},{Pi+g[0,1.2,y,2],y},{Pi+g[0,1.3,y,2],y},{Pi+g[0,1.4,y,2],y},{Pi+g[0,1.5,y,2],y},{Pi+g[0,1.6,y,2],y},{Pi+g[0,1.7,y,2],y},{Pi+g[0,1.8,y,2],y},{Pi+g[0,1.9,y,2],y}},{y,.1,1.9}]

s6=Show[q6,q7,g6]

Show[s1,s2,s3,s4,s5,s6]

g8=Graphics[{Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2*Pi,.1},{1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2*Pi,.2},{1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2*Pi,.3},{1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2*Pi,.4},{1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2*Pi,.5},{1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2*Pi,.6},{1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2*Pi,.7},{1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2*Pi,.8},{1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2*Pi,.9},{1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2*Pi,1.},{1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2*Pi,1.1},{1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2*Pi,1.2},{1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2*Pi,1.3},{1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2*Pi,1.4},{1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2*Pi,1.5},{1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2*Pi,1.6},{1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2*Pi,1.7},{1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2*Pi,1.8},{1,0}],Text[StyleForm["<",FontSize->14, FontWeight->"Bold"],{2*Pi,1.9},{1,0}]}]

q8=ParametricPlot[{{2*Pi-g[0,.1,y,2],y},{2*Pi-g[0,.2,y,2],y},{2*Pi-g[0,.3,y,2],y},{2*Pi-g[0,.4,y,2],y},{2*Pi-g[0,.5,y,2],y},{2*Pi-g[0,.6,y,2],y},{2*Pi-g[0,.7,y,2],y},{2*Pi-g[0,.8,y,2],y},{2*Pi-g[0,.9,y,2],y},{2*Pi-g[0,1,y,2],y},{2*Pi-g[0,1.1,y,2],y},{2*Pi-g[0,1.2,y,2],y},{2*Pi-g[0,1.3,y,2],y},{2*Pi-g[0,1.4,y,2],y},{2*Pi-g[0,1.5,y,2],y},{2*Pi-g[0,1.6,y,2],y},{2*Pi-g[0,1.7,y,2],y},{2*Pi-g[0,1.8,y,2],y},{2*Pi-g[0,1.9,y,2],y}},{y,.1,1.9}]

s8=Show[g8,q8]

s7=Plot[{0,1,2},{x,0,2*Pi}]

Line1=Graphics[Line[{{0,0},{0,2}}]]

Line2=Graphics[Line[{{2*Pi,0},{2*Pi,2}}]]

Show[s1,s2,s3,s4,s5,s6,s7,s8,Line1,Line2,Frame->True,FrameLabel->{ax,ay}]