Problem 9.4.34 (Laplace’s PDE)
and 9.4.46 (chain rule)

u[x_,y_]=E^(x^2-y^2) Cos[2 x y]

D[u[x,y],x]

D[u[x,y],{x,2}]

D[u[x,y],y]

D[u[x,y],{y,2}]

D[u[x,y],{x,2}] + D[u[x,y],{y,2}]

Simplify[D[u[x,y],{x,2}] + D[u[x,y],{y,2}]]

Plot3D[u[x,y],{x,-Pi,Pi},{y,-Pi,Pi}]

Plot3D[u[x,y],{x,-Pi,Pi},{y,-Pi,Pi},PlotRange->All]

Plot3D[u[x,y],{x,-Pi/2,Pi/2},{y,-Pi/2,Pi/2},AspectRatio->Automatic]

Plot3D[u[x,y],{x,-Pi/2,Pi/2},{y,-Pi/2,Pi/2},AspectRatio->Automatic,PlotRange->All]

p[x_,t_]=t^2 ArcSin[x]
q[x_,t_]=x/t^2
r[x_,t_]=ArcTan[x/t]

tophalf=ParametricPlot3D[{p[x,t],q[x,t],r[x,t]},{x,-1,1},{t,0.5,1.5}]

bottomhalf=ParametricPlot3D[{p[x,t],q[x,t],r[x,t]},{x,-1,1},{t,-1.5,-0.5}]

Show[bottomhalf,tophalf]

Show[bottomhalf,tophalf,ParametricPlot3D[{{t,0,0},{0,t,0},{0,0,t}},{t,0,4},PlotStyle->{{Thick,Black},{Thick,Red},{Thick,Green}}]]

curve=ParametricPlot3D[{p[1,t],q[1,t],r[1,t]},{t,0.5,1.5},PlotStyle->{AbsoluteThickness[4.7],Blue}]

Show[bottomhalf,tophalf,ParametricPlot3D[{{t,0,0},{0,t,0},{0,0,t}},{t,0,4},PlotStyle->{{Thick,Black},{Thick,Red},{Thick,Green}}],curve]