Line Integrals

The following notebook must be evaluated in the order given.  This is because it reinitializes the variables x, y, and z.

Evaluate First

Needs["VectorFieldPlots`"]

The Line Integral

The Parametric Equations for the Line Integral Problem

x[t_]=Cos[t];
y[t_]=4Sin[t];
F[X_,Y_]:=Sin[X]+Cos[Y]+3;
Q1:=ParametricPlot3D[{x[t],y[t],0},{t,0,((3π)/2)}]
Q2:=ParametricPlot3D[{x[t],y[t],F[x[t],y[t]]+0.05},{t,0,((3π)/2)}]
Q3:=Plot3D[F[X,Y],{X,-2π,2π},{Y,-2π,2π},Mesh->False]
Q4:=ParametricPlot3D[{x[t],y[t],u F[x[t],y[t]]},{t,0,((3π)/2)},{u,0,1},Boxed->False,Axes->False,AspectRatio->1]

The Curve and Projection of the Curve onto the surface

Show[{Q1,Q2,Q3},Boxed->False,Axes->False,AspectRatio->1,PlotRange->All]

The Surface calculated by the Line Integral

Show[{Q1,Q2,Q4},Boxed->False,Axes->False,AspectRatio->1,PlotRange->All]

The Work Problem

The Parametric Equations for the Work Problem (and a Force Table)

x:=Cos[t];
y:=Sin[t];
z:=t;
Fx:=x y;
Fy:=Sin[x];
Fz:=y;
ForceOnCurve=Table[{{x,y,z},{Fx,Fy,Fz}},{t,0,6.2,0.2}];
P1=ParametricPlot3D[{x,y,z},{t,0,2π},Boxed->True,Ticks->False,AspectRatio->1,DisplayFunction->Identity];
x=.;y=.;z=.;
P2=PlotVectorField3D[{Fx,Fy,Fz},{x,-2,2},{y,-2,2},{z,0,2π},PlotPoints->5,VectorHeads->True,DisplayFunction->Identity];

The Space Curve

Show[{P1},DisplayFunction->$DisplayFunction,Boxed->True,Axes->False,AspectRatio->1];

The Space Curve in the Force Field

Show[{P1,P2},DisplayFunction->$DisplayFunction,Boxed->True,Axes->False,AspectRatio->1];

The Space Curve and Forces on the Curve

P3=ListPlotVectorField3D[ForceOnCurve,VectorHeads->True,DisplayFunction->Identity];
Table[Show[{P1,P3},DisplayFunction->$DisplayFunction,Boxed->False,Axes->False,AspectRatio->1,ViewPoint->{3 Cos[i], 3 Sin[i], 5Sin[i]}],{i,0,6.3,0.1}];