NotebookDirectory[]

Show[
ParametricPlot3D[{(2+Sin[z])Cos[t],(2+Sin[z])Sin[t],z},{z,0,2 Pi},{t,0,2 Pi}],
ParametricPlot3D[{(2+Sin[z])Cos[2z],(2+Sin[z])Sin[2 z],z},{z,0,2 Pi},PlotStyle->{Thickness[0.015]}]
]

The curve

rr[z_]={(2+Sin[z])Cos[2z],(2+Sin[z])Sin[2 z],z}

The velocity vector is

D[rr[z],z]

Illustrate the velocity vector on the graph

s=Pi; Show[
ParametricPlot3D[{(2+Sin[z])Cos[2z],(2+Sin[z])Sin[2 z],z},{z,0,2 Pi},PlotStyle->{Thickness[0.01]}],

Graphics3D[{
{PointSize[0.02],Cyan,Point[rr[s]]},
{Thickness[0.007],Cyan,Arrow[{rr[s],rr[s]+rr'[s]}]}
}],
PlotRange->{{-5,5},{-5,5},{0,8}}
]

Clear[s];Manipulate[
 Show[
ParametricPlot3D[{(2+Sin[z])Cos[2z],(2+Sin[z])Sin[2 z],z},{z,0,2 Pi},PlotStyle->{Thickness[0.01]}],

Graphics3D[{
{PointSize[0.02],Cyan,Point[rr[s]]},
{Thickness[0.007],Cyan,Arrow[{rr[s],rr[s]+rr'[s]}]}
}],
PlotRange->{{-7,6},{-7,6},{0,8}}
],
{s,0,2Pi}]

Now we need a vector field to calculate line integral along this curve. I will make this vector field a complicated one!

vF[{x_,y_,z_}]:=(1/1){x y,y z,z x}

Illustrate the vector field. The field is complicated so it is not easy to understand its action.

Show[
VectorPlot3D[vF[{x,y,z}],{x,-3,3},{y,-3,3},{z,0,2 Pi},VectorScale->Large],
ParametricPlot3D[{(2+Sin[z])Cos[2z],(2+Sin[z])Sin[2 z],z},{z,0,2 Pi},PlotStyle->{Thickness[0.015]}]
]

Now “feed the vector field to the curve”. What I mean by this is that we look at the action of the vector field along the curve.

Clear[s];Manipulate[
 Show[
ParametricPlot3D[{(2+Sin[z])Cos[2z],(2+Sin[z])Sin[2 z],z},{z,0,2 Pi},PlotStyle->{Thickness[0.01]}],

Graphics3D[{
{PointSize[0.02],Cyan,Point[rr[s]]},
{Thickness[0.007],Blue,Arrow[{rr[s],rr[s]+vF[rr[s]]}]}
}],
PlotRange->{{-7,6},{-7,6},{-3,14}}
],
{s,0,2Pi}]

Now show together the vector field along the curve (blue vectors) and the velocity vectors (cyan vectors)

Clear[s];Manipulate[
 Show[
ParametricPlot3D[{(2+Sin[z])Cos[2z],(2+Sin[z])Sin[2 z],z},{z,0,2 Pi},PlotStyle->{Thickness[0.01]}],

Graphics3D[{
{PointSize[0.02],Cyan,Point[rr[s]]},
{Thickness[0.007],Cyan,Arrow[{rr[s],rr[s]+rr'[s]}]},
{Thickness[0.007],Blue,Arrow[{rr[s],rr[s]+vF[rr[s]]}]}
}],
PlotRange->{{-7,6},{-7,6},{-3,14}}
],
{s,0,2Pi}]

Now calculate the dot product of the vector field calculated at a point on the curve and the velocity vector at the same point

FullSimplify[vF[rr[z]].D[rr[z],z]]

And integrate this quantity

Integrate[(2+Sin[z]) (z Cos[z] ((Sin[2 z])^2)+Cos[2 z] (z+2 (2+Sin[z]) Sin[2 z] (z-(2+Sin[z]) Sin[2 z]))+(1/8) (1+2 Csc[z]) ((Sin[4 z])^2)),{z,0,2 Pi}]

Below I show the plot of the dot product, so that we can see where it is negative and where it is positive.

Plot[(2+Sin[z]) (z Cos[z] ((Sin[2 z])^2)+Cos[2 z] (z+2 (2+Sin[z]) Sin[2 z] (z-(2+Sin[z]) Sin[2 z]))+(1/8) (1+2 Csc[z]) ((Sin[4 z])^2)),{z,0,2 Pi}]