NotebookDirectory[]

SetDirectory["C:\\Dropbox\\Work\\myweb\\Courses\\Math_pages\\Math_225"]

Problem 33

ContourPlot[x^3+y^3==3 x y,{x,-3,3},{y,-3,3}]

Show[ParametricPlot[{((3 t)/(1+t^3)),((3 t^2)/(1+t^3))},{t,-.9999,100},PlotRange-> {{-3,3},{-3,3}}],ParametricPlot[{((3 t)/(1+t^3)),((3 t^2)/(1+t^3))},{t,-100,-1.0009},PlotStyle->Red,PlotRange-> {{-3,3},{-3,3}}]]

Integrate[{0,((3 t)/(1+t^3))}.D[{((3 t)/(1+t^3)),((3 t^2)/(1+t^3))},t],{t,0,Infinity}]

Problem 29

Needs["VectorAnalysis`"]

vp={1.290787372567624`,1.9329830867147417`,0.19350541412247907`}

Show[
ParametricPlot3D[{t Cos[2 Pi t],0 , t Sin[2 Pi t]},{t,0,2},PlotStyle-> {{Thickness[0.007],RGBColor[0.5,0,0.5]}},PlotPoints->256],ParametricPlot3D[{t Cos[2 Pi t],t , t Sin[2 Pi t]},{t,0,2},PlotStyle-> {{Thickness[0.007],RGBColor[0,0,0.7]}},PlotPoints->256],ParametricPlot3D[{2, t,0 },{t,0,2},PlotStyle-> {{Thickness[0.007],RGBColor[0.5,0.5,0]}}],

Graphics3D[{{RGBColor[0.5,0,0.5],Arrow@Table[{{(t-.05) Cos[2 Pi (t-.05)],0 , (t-.05) Sin[2 Pi (t-.05)]},{t Cos[2 Pi t],0 , t Sin[2 Pi t]}},{t,0.7,2,.3}]},{RGBColor[0,0,0.5],Arrow@Table[{{(t-.05) Cos[2 Pi (t-.05)],t-0.05 , (t-.05) Sin[2 Pi (t-.05)]},{t Cos[2 Pi t],t , t Sin[2 Pi t]}},{t,0.7,2,.3}]},{RGBColor[0.5,0.5,0],Arrow[{{2,0.9,0},{2,.95,0}}],Arrow[{{2,1.7,0},{2,1.75,0}}]},
{PointSize[0.03],Red,Point[{0,0,0}],Point[{2,2,0}]},{Text[Style["A",FontSize->22,FontSlant->Italic],{0.1,0,-0.15}],Text[Style["B",FontSize->22,FontSlant->Italic],{2.1,2,-0.1}]}}],PlotRange->{{-2.1,2.1},{-0.1,2.1},{-2.1,2.1}},AxesLabel->{x,y,z},BaseStyle->{FontSize->18},ViewPoint->vp,AxesEdge->{{1,-1},{1,-1},{-1,1}},Ticks->{Automatic,{0,1,2},Automatic},ImageSize->600]

TwoCurvesP25=Rasterize[Show[
ParametricPlot3D[{t Cos[2 Pi t],0 , t Sin[2 Pi t]},{t,0,2},PlotStyle-> {{Thickness[0.007],RGBColor[0.5,0,0.5]}},PlotPoints->256],ParametricPlot3D[{t Cos[2 Pi t],t , t Sin[2 Pi t]},{t,0,2},PlotStyle-> {{Thickness[0.007],RGBColor[0,0,0.7]}},PlotPoints->256],ParametricPlot3D[{2, t,0 },{t,0,2},PlotStyle-> {{Thickness[0.007],RGBColor[0.5,0.5,0]}}],

Graphics3D[{{RGBColor[0.5,0,0.5],Arrow@Table[{{(t-.05) Cos[2 Pi (t-.05)],0 , (t-.05) Sin[2 Pi (t-.05)]},{t Cos[2 Pi t],0 , t Sin[2 Pi t]}},{t,0.7,2,.3}]},{RGBColor[0,0,0.5],Arrow@Table[{{(t-.05) Cos[2 Pi (t-.05)],t-0.05 , (t-.05) Sin[2 Pi (t-.05)]},{t Cos[2 Pi t],t , t Sin[2 Pi t]}},{t,0.7,2,.3}]},{RGBColor[0.5,0.5,0],Arrow[{{2,0.9,0},{2,.95,0}}],Arrow[{{2,1.7,0},{2,1.75,0}}]},
{PointSize[0.03],Red,Point[{0,0,0}],Point[{2,2,0}]},{Text[Style["A",FontSize->22,FontSlant->Italic],{0.1,0,-0.15}],Text[Style["B",FontSize->22,FontSlant->Italic],{2.1,2,-0.1}]}}],PlotRange->{{-2.1,2.1},{-0.1,2.1},{-2.1,2.1}},AxesLabel->{x,y,z},BaseStyle->{FontSize->18},ViewPoint->vp,AxesEdge->{{1,-1},{1,-1},{-1,1}},Ticks->{Automatic,{0,1,2},Automatic},ImageSize->600],ImageResolution->600]

Export["TwoCurvesP25.gif",TwoCurvesP25,"GIF","ImageSize"->600]

Export["TwoCurvesP25.png",TwoCurvesP25,"PNG","ImageSize"->600]

{t Cos[2 Pi t],t , t Sin[2 Pi t]}

{y z, z (x+1), x y + y +1}

Curl[{Yy Zz, Zz (Xx+1), Xx Yy + Yy +1}]

{y z, z (x+1), x y + y +1}/.{x-> t Cos[2 Pi t],y-> t , z-> t Sin[2 Pi t]}

Expand[Simplify[({y z, z (x+1), x y + y +1}/.{x-> t Cos[2 Pi t],y-> t , z-> t Sin[2 Pi t]}).D[{t Cos[2 Pi t],t , t Sin[2 Pi t]},t]]]

Integrate[2 π t Cos[2 π t]+2 π (t^2) Cos[2 π t]+2 π (t^3) Cos[4 π t]+Sin[2 π t]+2 t Sin[2 π t]+3 (t^2) Cos[2 π t] Sin[2 π t],{t,0,2}]

Expand[Simplify[({y z, z (x+1), x y + y +1}/.{x-> t Cos[2 Pi t],y-> 0 , z-> t Sin[2 Pi t]}).D[{t Cos[2 Pi t],0 , t Sin[2 Pi t]},t]]]

Expand[Simplify[({y z, z (x+1), x y + y +1}/.{x-> 2,y-> t, z->0}).D[{2,t ,0},t]]]

{y z, z (x+1), x y + y +1}

Integrate[y z,x]

D[x y z+z y,y]

D[x y z+z y,z]

D[x y z+z y+z,z]

Grad[Xx Yy Zz+Zz Yy+Zz]

(x y z+z y+z)/.{x->0,y-> 0,z->0}

(x y z+z y+z)/.{x->2,y-> 0,z->0}

(x y z+z y+z)/.{x->2,y-> 2,z->0}

Try another gradient field.

Grad[Xx Yy Zz+Yy+Zz]

Expand[({Yy Zz,1+Xx Zz,1+Xx Yy}/.{Xx-> t Cos[2 Pi t],Yy-> t , Zz-> t Sin[2 Pi t]}).D[{t Cos[2 Pi t],t , t Sin[2 Pi t]},t]]

Expand[({Yy Zz,1+Xx Zz,1+Xx Yy}/.{Xx-> t Cos[2 Pi t],Yy-> 0 , Zz-> t Sin[2 Pi t]}).D[{t Cos[2 Pi t],0 , t Sin[2 Pi t]},t]]

Integrate[1+2 π t Cos[2 π t]+2 π (t^3) Cos[4 π t]+Sin[2 π t]+(3/2) (t^2) Sin[4 π t],{t,0,2}]

Integrate[2 π t Cos[2 π t]+2 π (t^3) Cos[4 π t]+Sin[2 π t]+(3/2) (t^2) Sin[4 π t],{t,0,2}]

Integrate[2 π t Cos[2 π t]+2 π (t^3) Cos[4 π t]+Sin[2 π t]+(3/2) (t^2) Sin[4 π t],t]

Integrate[ t Cos[2 π t],t]

Integrate[ t Cos[2 π t],{t,0,2}]