Lösungen

1

a

f[x_,y_]:=Sin[x+y];
Plot3D[f[x,y],{x,0,Pi},{y,-Pi,Pi}];

Integrate[Evaluate[Sqrt[D[f[x,y],x]^2+D[f[x,y],y]^2+1] ],{x,0,Pi},{y,-Pi,Pi}]

NIntegrate[Evaluate[Sqrt[D[f[x,y],x]^2+D[f[x,y],y]^2+1] ],{x,0,Pi},{y,-Pi,Pi}]

b

f[x_,y_]:=Sin[x+y];
Plot3D[f[x,y],{x,-Sqrt[Pi/2-y^2],Sqrt[Pi/2-y^2]},{y,-Sqrt[Pi/2],Sqrt[Pi]}];

f1[φ_,r_]:={r Cos[φ],r Sin[φ],Sin[r Cos[φ]+r Sin[φ]]}

ParametricPlot3D[f1[φ,r],{r,0, Sqrt[Pi/2]}, {φ,0,2 Pi},ViewPoint->{1.495, -2.565, 0.329}];

Integrate[Evaluate[Sqrt[D[f[x,y],x]^2+D[f[x,y],y]^2+1] ], {x,-Sqrt[Pi/2-y^2],Sqrt[Pi/2-y^2]},{y,-Sqrt[Pi/2],Sqrt[Pi]}]

NIntegrate[Evaluate[Sqrt[D[f[x,y],x]^2+D[f[x,y],y]^2+1] ], {x,-Sqrt[Pi/2-y^2],Sqrt[Pi/2-y^2]},{y,-Sqrt[Pi/2],Sqrt[Pi]}]

NIntegrate[Evaluate[Norm[Cross[D[f1[φ,r],φ], D[f1[φ,r],r]]]],{φ,0,2 Pi},{r,0, Sqrt[Pi/2]}]

c

f[x_,y_]:=Sin[x y]+Sin[x];
Plot3D[f[x,y],{x,0,Pi},{y,-Pi,Pi}];

Integrate[Evaluate[Sqrt[D[f[x,y],x]^2+D[f[x,y],y]^2+1] ],{x,0,Pi},{y,-Pi,Pi}]

NIntegrate[Evaluate[Sqrt[D[f[x,y],x]^2+D[f[x,y],y]^2+1] ],{x,0,Pi},{y,-Pi,Pi}]

2

Remove["Global`*"]

h[x_,y_]:=x+x y +y^2;
f[x_,y_]:=1-x^2-y^2;

Integrate[Evaluate[h[x,y]* Sqrt[D[f[x,y],x]^2+D[f[x,y],y]^2+1] ],{x,0,y},{y,0,1}]

NIntegrate[Evaluate[h[x,y]* Sqrt[D[f[x,y],x]^2+D[f[x,y],y]^2+1]], {x,0,y},{y,0,1}]

NIntegrate[
      Evaluate[h[x, y]*Sqrt[D[f[x, y], x]^2 + D[f[x, y], y]^2 + 1]], {x, 0, 
        y}, {y, 0, 1}, Method -> QuasiMonteCarlo, 
      MaxPoints -> 5000]

Plot[UnitStep[x],{x,-2,2}];

v[x_,y_]:=Evaluate[(h[x,y]*Sqrt[D[f[x,y],x]^2+D[f[x,y],y]^2+1]) UnitStep[x-y]] ;
Plot3D[v[x,y],{x,0,1},{y,0,1}];

NIntegrate[v[x,y],{x,0,1},{y,0,1}]

3

Remove["Global`*"]

a

f[α_,r_]:= Sin[α] E^r;
v[α_,r_]:= {r Cos[α], r Sin[α], f[α,r]};
ParametricPlot3D[{r Cos[α], r Sin[α], f[α,r]},{α,0,Pi/4},{r,0.5,1}];
NIntegrate[Evaluate[Norm[Cross[D[v[α,r],α], D[v[α,r],r]]]],{α,0.5,2},{r,0.5, 1}]

b

v[α_,r_]:= {r Cos[α], r Sin[α], f[α,r]} UnitStep[α-r] UnitStep[2r-α];
ParametricPlot3D[v[α,r],{α,0,Pi/4},{r,0.5,1}];
NIntegrate[Evaluate[Norm[Cross[D[v[α,r],α], D[v[α,r],r]]]],{α,0.5,2},{r,0.5, 1}]

c

v[α_,r_]:= {r Cos[α], r Sin[α], f[α,r]} UnitStep[α-r] UnitStep[2r-α];
ParametricPlot3D[v[α,r],{α,0,Pi/4},{r,0.5,1}];
NIntegrate[Evaluate[Norm[Cross[D[v[α,r],α], D[v[α,r],r]]]],{r,0,2},{α,0,1}]

4

Remove["Global`*"]

a

f[x_,y_]:= Cos[x+y] ;
v[α_,r_]:= {r Cos[α], r Sin[α], f[r Cos[α],r Sin[α]]};
ParametricPlot3D[{r Cos[α], r Sin[α], f[r Cos[α],r Sin[α]]},{α,0,2 Pi},{r,0, Pi/2}];
NIntegrate[Evaluate[Norm[Cross[D[v[α,r],α], D[v[α,r],r]]]],{α,0,2 Pi},{r,0, Pi/2}]

b

f[x_,y_]:= y UnitStep[y-x^2]  UnitStep[1-y];
Plot3D[f[x,y],{x,-1,1},{y,0,1},PlotPoints->50];
f[x_,y_]:= y;
Integrate[Evaluate[(Sqrt[D[f[x,y],x]^2+D[f[x,y],y]^2+1]) UnitStep[y-x^2]  UnitStep[1-y] ],{x,-1,1},{y,0,1}]

N[%]

5

Remove["Global`*"]

a

x[r_,α_]:= Cos[α]+r;
y[r_,α_]:= Sin[α]+r^2;
z[r_,α_]:= Cos[α]-r+Sin[α]^2;
v[r_,α_]:= {x[r,α],y[r,α],z[r,α]};
ParametricPlot3D[v[r,α],{α,0,Pi/2},{r,0,2}];
NIntegrate[Evaluate[Norm[Cross[D[v[r,α],α], D[v[r,α],r]]]],{α,0,Pi/2},{r,0, 2}]

b

x[α_,β_]:= 4 Cos[α](1+Cos[β]);
y[α_,β_]:= 4 Sin[α](1+Cos[β]);
z[α_,β_]:= Sin[β];
v[α_,β_]:= {x[α,β],y[α,β],z[α,β]};
ParametricPlot3D[v[α,β],{α,0,Pi/2},{β,0,2 Pi}];
NIntegrate[Evaluate[Norm[Cross[D[v[r,α],α], D[v[r,α],r]]]],{α,0,Pi/2},{r,0, 2 Pi}]

c

x[α_,β_]:= Sin[α] Cos[β] +Sin[2 β]/4 ;
y[α_,β_]:= Sin[α] Sin[β];
z[α_,β_]:= Cos[α];
v[α_,β_]:= {x[α,β],y[α,β],z[α,β]};
ParametricPlot3D[v[α,β],{α,0,Pi/2},{β,0,2 Pi}];
NIntegrate[Evaluate[Norm[Cross[D[v[r,α],α], D[v[r,α],r]]]],{α,0,Pi/2},{r,0, 2 Pi}]

x[r_,α_]:= Sin[α] Cos[β] +Sin[2 β]/4 ;
y[r_,α_]:= Sin[α] Sin[β];
z[r_,α_]:= Cos[α];
v[r_,α_]:= {x[r,α],y[r,α],z[r,α]};
ParametricPlot3D[v[r,α],{α,0,Pi},{β,0,2 Pi}];

6

Remove["Global`*"]

f[u_,v_]:= E^(Sin[u]-Cos[v]);
Plot3D[f[u,v],{u,0,2 Pi},{v,0,2 Pi}];
NIntegrate[Evaluate[Sqrt[D[f[u,v],u]^2+D[f[u,v],v]^2+1] ],{u,0,2 Pi},{v,0,2 Pi}]

7

Krümmung und Anschmiegekreis

Krümmungskreis, 2-dimensional

1

Remove["Global`*"]

r[x0_]:=
(Abs[((D[f[x],x]^2+1)^(3/2))/D[f[x],{x,2}]]/.x->x0)

x0=0;
f[x_]:= Cos[x];
Evaluate[r[x0]]

p1=Plot[f[t],{t,-3,3},AspectRatio->Automatic,DisplayFunction->Identity];
p2=ParametricPlot[Evaluate[r[x0]]{Cos[t],Sin[t]},{t,0,2Pi},AspectRatio->Automatic,DisplayFunction->Identity];
Show[Graphics[{PointSize[0.03],Point[{x0,f[x0]}]}],p1,p2, DisplayFunction->$DisplayFunction,AspectRatio->Automatic];

2

Remove["Global`*"]

r[x0_]:=
(Abs[((D[f[x],x]^2+1)^(3/2))/D[f[x],{x,2}]]/.x->x0)

x0=0;
f[x_]:= x^2;
Evaluate[r[x0]]

p1=Plot[f[t],{t,-3,3},AspectRatio->Automatic,DisplayFunction->Identity];
p2=ParametricPlot[Evaluate[r[x0]]{Cos[t],Sin[t]+1},{t,0,2Pi},AspectRatio->Automatic,DisplayFunction->Identity];
Show[Graphics[{PointSize[0.03],Point[{x0,f[x0]}]}],p1,p2,Graphics[PointSize[0.03],Point[{x0,f[x0]}]],DisplayFunction->$DisplayFunction,AspectRatio->Automatic];

3

Remove["Global`*"]

r[x0_]:=
(Abs[((D[f[x],x]^2+1)^(3/2))/D[f[x],{x,2}]]/.x->x0)

x0=0;
f[x_]:= E^x;
Evaluate[r[x0]]

%//N

m[x0_]:= Evaluate[{x0,f[x0]}-{1,(-1/D[f[x],x]/.x->x0)}/Abs[Sqrt[{1,(-1/D[f[x],x]/.x->x0)}.{1,(-1/D[f[x],x]/.x->x0)}]] r[x0]];
m[u]

m[x0]

p1=Plot[f[t],{t,-3,3},AspectRatio->Automatic,DisplayFunction->Identity];
p2=ParametricPlot[Evaluate[r[x0]]{Cos[t],Sin[t]}+m[x0],{t,0,2Pi},AspectRatio->Automatic,DisplayFunction->Identity];
Show[Graphics[{PointSize[0.05],Point[{x0,f[x0]}]}],p1,p2,DisplayFunction->$DisplayFunction,AspectRatio->Automatic];

4

Remove["Global`*"]

r[x0_]:=
(Abs[((D[f[x],x]^2+1)^(3/2))/D[f[x],{x,2}]]/.x->x0)

x0=1;
f[x_]:= E^x;
Evaluate[r[1]]

%//N

m[x0_]:= Evaluate[{x0,f[x0]}-{1,(-1/D[f[x],x]/.x->x0)}/Abs[Sqrt[{1,(-1/D[f[x],x]/.x->x0)}.{1,(-1/D[f[x],x]/.x->x0)}]] r[x0]];
m[u]

p1=Plot[f[t],{t,-16,3},AspectRatio->Automatic,DisplayFunction->Identity];
p2=ParametricPlot[Evaluate[r[x0]]{Cos[t],Sin[t]}+m[x0],{t,0,2Pi},AspectRatio->Automatic,DisplayFunction->Identity];
Show[Graphics[{PointSize[0.05],Point[{x0,f[x0]}]}],p1,p2,DisplayFunction->$DisplayFunction,AspectRatio->Automatic];

Berechnung von Krümmung und Anschmiegekreis bei Raumkurven

Definition der Raumkurve

Remove["Global`*"]

x[t_] := 0.5 t Cos[t];   y[t_] :=  t Sin[t];   z[t_] := 
      t;    v[t_] := {x[t], y[t], z[t]};

Definition des Bereiches der Variablen

tmin = 0.1;   tmax =  10;

Plot Ausgangsfunktion

origPlot = 
        ParametricPlot3D[
          Evaluate[Join[v[t],  {RGBColor[1, 0, 0]}]], {t, tmin, tmax}, 
          PlotRange -> All, AxesLabel -> {x, y, z}, 
          AspectRatio -> Automatic];

Begleitendes Dreibein

vLen[v_] := Sqrt[v . v]; 
normVec[v_] := v/vLen[v];
tg[t_] = normVec[v'[t]];
n[t_] = 
        normVec[Cross[
            Cross[v'[t], v''[t]], v'[
              t]]];
b[t_] := Cross[tg[t], n [t]];

{tg[t], n[t], b[t]}

tVec[t_] := 
        Line[{v[t], v[t] + 3  tg[t]}];
nVec[t_] := 
        Line[{v[t], v[t] + 3  n[t]}];
bVec[t_] := Line[{v[t], v[t] + 3  b[t]}];

dreibein[t_] := 
      Graphics3D[{RGBColor[1, 0, 1], tVec[t], RGBColor[0.01, 0.6, 0.62], 
          nVec[t], RGBColor[0.54, 0.17, 0.89], bVec[t]}]

Show[origPlot, dreibein[5]];

Krümmung

ϰ[t_] = 
        Sqrt[(((v''[t] . v''[t])*(v'[t] . v'[
                          t]) - (v''[t] . v'[t])^2)/
              vLen[v'[t]]^6)];
ϰ[t]

Plot[Evaluate[ϰ[t]], {t, tmin, tmax}];

Krümmungsradius

ρ[t_] := Abs[1/ϰ[t]];   ρ[t]

Plot[Evaluate[ρ[t]], {t, tmin, tmax}];

Krümmungskreismittelpunkt (Evolute)

m[t_] := v[t] +  ρ[t]*n[t];   m[t]

evolute = 
        ParametricPlot3D[
          Evaluate[Join[m[t], {RGBColor[0, 0, 1]}]], {t, tmin, tmax}, 
          PlotRange -> All, AspectRatio -> Automatic];

Schmiegekreis

myCirc3d[t_, k_] :=  
      m[t] +  ρ[t] (n[t] Cos[k] + tg[t] Sin[k]);   myCirc3d[t, 
      k]

Plot

circPlot[t_] := 
        ParametricPlot3D[
          Evaluate[Join[myCirc3d[t, k], {RGBColor[0, 1, 0]}]], {k, 0, 2 Pi}, 
          DisplayFunction -> Identity];

Show[origPlot, circPlot[6]];

Alles zusammen

Show[origPlot, circPlot[6], dreibein[6]];

Möbiusband

Anwendung bei Riementrieben

Remove["Global`*"]

x[r_, α_] := 
        Cos[α] (1 + 
              r/2 Cos[α/2]);
y[r_, α_] := 
        Sin[α] (1 + 
              r/2 Cos[α/2]);
z[r_, α_] := 
        r/2 Sin[α/2];
ParametricPlot3D[{x[r, α], y[r, α], 
          z[r, α]}, {α, 0, 2 Pi}, {r, -1, 
          1}];

ParametricPlot3D[{x[r, α], y[r, α], 
          z[r, α]}, {α, 0, 2 Pi}, {r, -0.5, 
          0.5}];

ParametricPlot3D[{x[r, α], y[r, α], 
          z[r, α]}, {α, 0, 2 Pi}, {r, 0, 1}];

x[r_,α_]:= Cos[α] (1+r/2 Sin[α/2]);
y[r_,α_]:= Sin[α] (1+r/2 Sin[α/2]);
z[r_,α_]:= r/2 Sin[α/2];
v[r_,α_]:= {x[r,α],y[r,α],z[r,α]};
ParametricPlot3D[v[r,α],{α,0,2 Pi},{r,-1/2,1/2},
ViewPoint->{-0.078, -2.799, 1.082},Boxed->False, Axes->None,Shading->False];

ParametricPlot3D[v[r,α],{α,0,2 Pi},{r,-1/2,1/2},
ViewPoint->{-0.078, -2.799, 1.082}];

Evaluate[Norm[Cross[D[v[r,α],α], D[v[r,α],r]]]]

2 NIntegrate[Evaluate[Norm[Cross[D[v[r,α],α], D[v[r,α],r]]]],{α,0,2 Pi},{r,-1/2,1/2}]

Remove["Global`*"]

Mobius[s_, R_:  4, 
          t_] := {(R + s Cos[t/2]) Cos[t], (R + s Cos[t/2]) Sin[t], 
          s Sin[t/2]};
\(\(MobiusStrip[n_:  2, R_:  4, w : _  1, 
          opts___] := 
\ \ ParametricPlot3D[
          Evaluate[Mobius[s, R, t]], {s, \(-w\), w}, {t, 0, 
            2  Pi}, 
\ \ PlotPoints -> {5, 30}, 
          opts];\)\)
Show[{Graphics3D[{Thickness[ .02], 
              RGBColor[1, 0, 
                0], ParametricPlot3D[
                  Evaluate[Mobius[1, 4, t]], {t, 0, 2  Pi}, 
                  DisplayFunction -> 
                    Identity][[1]]}], 
MobiusStrip[2, 
            4, 1, DisplayFunction -> Identity]}];