Lösungen

1

Remove["Global`*"]

a

x[r_,u_,t_]:= (r Cos[u]+5) Sin[t];
y[r_,u_,t_]:= (r Cos[u]+5) Cos[t];
z[r_,u_,t_]:= r Sin[u];

w[u_,t_]:={(Cos[u]+5) Sin[t], (Cos[u]+5) Cos[t],Sin[u]};
ParametricPlot3D[w[u,t],{u,0,2Pi},{t,0,Pi}];

Halber Ring

b

j[r_,u_,t_]:= Det[{
{D[x[r,u,t],{r}],D[x[r,u,t],{u}],D[x[r,u,t],{t}]},
{D[y[r,u,t],{r}],D[y[r,u,t],{u}],D[y[r,u,t],{t}]},
{D[z[r,u,t],{r}],D[z[r,u,t],{u}],D[z[r,u,t],{t}]}
}];
j[r,u,t]//Simplify

j[r_,u_,t_]:= r (5+r Cos[u]);

c

V1 = Integrate[j[r,u,t],{r,0,1},{u,0,2Pi},{t,0,Pi}]

N[%]

d

V2 = Integrate[j[r,u,t],{r,0,2},{u,0,2Pi},{t,0,Pi}]

N[%]

V2/V1

2 Rohr mit Schauglas

Programm

Programm laufen lassen

Remove["Global`*"];

\(\(volProgr[r_, α_, schritt_] := 
        Module[{}, 
h0 = 
            2  r/Cos[α]; 
xh[h_, 
              rR_, αα_] := 
            rR - h\ Cos[αα]; 
z1[x_, 
              rR_, αα_] := \(-x\)\ \ Tan[αα]\  + \ rR\ Tan[αα]; 
z2b[x_, 
              rR_, αα_] := 
            x\ \ Tan[Pi/2 - αα]\  + \ 
              rR\ Tan[αα] + b; 
solv = 
            Solve[z1[xh[h, r, α], r, α] == 
                  z2b[xh[h, r, α], r, α], {b}] // 
              Flatten; 
b = 
            b /. solv; 
z2[x_, rR_, αα_, 
              h_] = z2b[x, rR, αα] /. 
              solv; 
IntZ[x_, 
              h_] := 
Integrate[
              1, {z, z1[x, r, α], 
                z2[x, r, α, h]}
, 
              GenerateConditions -> False]; 
IntZY[x_, 
              h_] := 
Integrate[
              IntZ[x, h], {y, \(-Sqrt[r^2 - x^2]\), 
                Sqrt[r^2 - x^2]}]; 
IntZYX[
              h_] := \((Integrate[IntZY[x, h], x] /. 
                  x -> r)\) - \((Integrate[IntZY[x, h], x] /. 
                  x -> r - 
                      h\ Cos[α])\); 
Print["Plot der Kurve 'Volumen als Funktion von h' "]; 
Plot[
            Evaluate[IntZYX[h]], {h, 0, 
              h0}]; 
Print["Tabelle zur Kurve 'Volumen als Funktion von h' "]; 
Prepend[
              Table[{h, IntZYX[h] // N}, {h, 0, h0, schritt}], {h, V}] // 
            MatrixForm
];\)\)

Eingaben

r = 5;   schritt = 1;    
α = 40 Degree;   α // N ;

volProgr[r, α, schritt]

Evaluate[IntZYX[8]]//N

3

Remove["Global`*"];

a

LaplaceTransform[Cosh[t]+Sinh[2 t],t,s]

%//Together

b

LaplaceTransform[(E^t) - 2  (E^t) Sin[t], t, 
      s]

% // Together

c

LaplaceTransform[(t/(1 + (t^2))), t, s]

?CosIntegral

"CosIntegral[z]"=-(∫_z^∞)cos(t)/tdt

d

?Dirac*

LaplaceTransform[DiracDelta[t] + 1 + t + t^2 + t^3, t, 
      s]

%//Together

e

InverseLaplaceTransform[(s/((s^2) - 1)), s, t]

%//Expand

f

InverseLaplaceTransform[((10 s)/((((2  s))^2) - 4)) + 
        (5/(1 + s + (s^2))), s, t]

%//Expand

g

InverseLaplaceTransform[((4 s)/(8 + (((2  s))^3))), s, 
      t]

%//TrigExpand

%//TrigFactor

h

InverseLaplaceTransform[(3/(1 + (s^2) + (s^4))), s, t]

%//TrigExpand

%//TrigFactor

4

Remove["Global`*"]

links = 
      LaplaceTransform[4 y''[t] - 2 y'[t] + y[t], t, 
          s] /. {LaplaceTransform[y[t], t, s] -> Y[s], 
          y[0] -> 1, y'[0] -> 1}

LaplaceTransform[DiracDelta[t-1],t,s]

rechts = 
      LaplaceTransform[
        DiracDelta[t] + DiracDelta[t - 1] + DiracDelta[t - 2] , t, 
        s]

solv = Solve[links == rechts, {Y[s]}]  //  Flatten

%//ExpandAll

U[s] := Y[s] /.  solv;   U[s]

%//ExpandAll

U[s] // Apart

u0[t_] := InverseLaplaceTransform[U[s], s, t] // Simplify;  
    u0[t]

u0[t]//Expand

Plot[Evaluate[{u0[t]}],{t,0,3Pi}];

Plot[Evaluate[{u0[t]}], {t, 0, 6  Pi}];

Plot[Evaluate[{u0[t]}],{t,0,12Pi}];

Plot[Evaluate[{u0[t]}],{t,0,24Pi},PlotRange->{-10^8,10^8}];

Plot[ UnitStep[-1+t],{t,0,2}];

5

a   RBD y1[0]->0, y1'[0]->0, y2[0]->1, y2'[0]->0

Remove["Global`*"];

links1 = LaplaceTransform[y2''[t] + a y2[t] - b y2[t] + b y1[t], t,s] /. 
{LaplaceTransform[y1[t],t,s]->Y1[s], 
LaplaceTransform[y2[t],t,s]->Y2[s], y1[0]->0, y1'[0]->0, y2[0]->1, y2'[0]->0}

links2 = LaplaceTransform[y1''[t] + a y1[t] - b y1[t] + b y2[t], t,s] /. 
{LaplaceTransform[y1[t],t,s]->Y1[s], 
LaplaceTransform[y2[t],t,s]->Y2[s], y1[0]->0, y1'[0]->0, y2[0]->1, y2'[0]->0}

solv=Solve[{links1 == 0, links2 == 0},{Y1[s],Y2[s]}] // Flatten

%//ExpandAll

U1[s]:=Y1[s]/. solv[[1]];
U2[s]:=Y2[s]/. solv[[2]];

u1[t_]:=InverseLaplaceTransform[U1[s],s,t]//Simplify; Print["u1(t) = ",u1[t]];

u2[t_]:=InverseLaplaceTransform[U2[s],s,t]//Simplify; Print["u2(t) = ",u2[t]];

u1P[t]:=u1[t]/. {a -> 10, b->1};
u1P[t]

u2P[t]:=u2[t]/. {a -> 10, b->1};
u2P[t]

Plot[Evaluate[u1P[t]],{t,0,10}];
Plot[Evaluate[u2P[t]],{t,0,10}];

u1P[t]:=u1[t]/. {a -> 5, b->1};
u1P[t]

u2P[t]:=u2[t]/. {a -> 5, b->1};
u2P[t]

Plot[Evaluate[u1P[t]],{t,0,25}];
Plot[Evaluate[u2P[t]],{t,0,25}];

u1P[t]:=u1[t]/. {a -> 5, b->2};
u1P[t]

u2P[t]:=u2[t]/. {a -> 5, b->2};
u2P[t]

Plot[Evaluate[u1P[t]],{t,0,25}];
Plot[Evaluate[u2P[t]],{t,0,25}];

b   RBD y1[0]->0, y1'[0]->0, y2[0]->1, y2'[0]->0

Remove["Global`*"];

links1 = LaplaceTransform[y2''[t] + a y2[t] - b y2[t] + b y1[t], t,s] /. 
{LaplaceTransform[y1[t],t,s]->Y1[s], 
LaplaceTransform[y2[t],t,s]->Y2[s], y1[0]->0, y1'[0]->1, y2[0]->1, y2'[0]->0}

links2 = LaplaceTransform[y1''[t] + a y1[t] - b y1[t] + b y2[t], t,s] /. 
{LaplaceTransform[y1[t],t,s]->Y1[s], 
LaplaceTransform[y2[t],t,s]->Y2[s], y1[0]->0, y1'[0]->1, y2[0]->1, y2'[0]->0}

solv=Solve[{links1 == 0, links2 == 0},{Y1[s],Y2[s]}] // Flatten

%//ExpandAll

U1[s]:=Y1[s]/. solv[[1]];
U2[s]:=Y2[s]/. solv[[2]];

u1[t_]:=InverseLaplaceTransform[U1[s],s,t]//Simplify; Print["u1(t) = ",u1[t]];

u2[t_]:=InverseLaplaceTransform[U2[s],s,t]//Simplify; Print["u2(t) = ",u2[t]];

u1P[t]:=u1[t]/. {a -> 10, b->1};
u1P[t]

u2P[t]:=u2[t]/. {a -> 10, b->1};
u2P[t]

pa1=Plot[Evaluate[u1P[t]],{t,0,10}];
pb1=Plot[Evaluate[u2P[t]],{t,0,10}];

u1P[t]:=u1[t]/. {a -> 5, b->1};
u1P[t]

u2P[t]:=u2[t]/. {a -> 5, b->1};
u2P[t]

pa2=Plot[Evaluate[u1P[t]],{t,0,25}];
pb2=Plot[Evaluate[u2P[t]],{t,0,25}];

u1P[t]:=u1[t]/. {a -> 5, b->2};
u1P[t]

u2P[t]:=u2[t]/. {a -> 5, b->2};
u2P[t]

pa3=Plot[Evaluate[u1P[t]],{t,0,25}];
pb3=Plot[Evaluate[u2P[t]],{t,0,25}];

Show[pa1,pa2,pa3];
Show[pb1,pb2,pb3];

c   RBD y1[0]->1, y1'[0]->0, y2[0]->0, y2'[0]->0

Remove["Global`*"];

links1 = LaplaceTransform[y2''[t] + g/L y2[t] - k/m y2[t] + k/m y1[t], t,s] /. 
{LaplaceTransform[y1[t],t,s]->Y1[s], 
LaplaceTransform[y2[t],t,s]->Y2[s], y1[0]->a, y1'[0]->0, y2[0]->0, y2'[0]->0}

links2 = LaplaceTransform[y1''[t] + g/L y1[t] - k/m y1[t] + k/m y2[t], t,s] /. 
{LaplaceTransform[y1[t],t,s]->Y1[s], 
LaplaceTransform[y2[t],t,s]->Y2[s], y1[0]->a, y1'[0]->0, y2[0]->0, y2'[0]->0}

solv=Solve[{links1 == 0, links2 == 0},{Y1[s],Y2[s]}] // Flatten

U1[s]:=Y1[s]/. solv[[1]];
U2[s]:=Y2[s]/. solv[[2]];

u1[t_]:=InverseLaplaceTransform[U1[s],s,t]//Simplify; Print["u1(t) = ",u1[t]];

u2[t_]:=InverseLaplaceTransform[U2[s],s,t]//Simplify; Print["u2(t) = ",u2[t]];

u1P[t]:=u1[t]/. {g -> 10, m->1, L->1, k->1, a->1};
u1P[t]

u2P[t]:=u2[t]/. {g -> 10, m->1, L->1, k->1, a->1};
u2P[t]

Plot[Evaluate[u1P[t]],{t,0,25}];
Plot[Evaluate[u2P[t]],{t,0,25}];

u1P[t]:=u1[t]/. {g -> 10, m->1, L->2, k->1, a->1};
u1P[t]

u2P[t]:=u2[t]/. {g -> 10, m->1, L->2, k->1, a->1};
u2P[t]

Plot[Evaluate[u1P[t]],{t,0,25}];
Plot[Evaluate[u2P[t]],{t,0,25}];

u1P[t]:=u1[t]/. {g -> 10, m->1, L->2, k->2, a->1};
u1P[t]

u2P[t]:=u2[t]/. {g -> 10, m->1, L->2, k->2, a->1};
u2P[t]

Plot[Evaluate[u1P[t]],{t,0,25}];
Plot[Evaluate[u2P[t]],{t,0,25}];

6

Remove["Global`*"];

a

i

f[x_,y_]:= E^(3 x+ 7 y)+c;

1/3 D[f[x,y],x]==1/7 D[f[x,y],y]

ii

E^(0)+c==2

f[1,1]

E^(3 1+ 7 1)+1

E^(7 y)+1

b

f[x_]:= a Cosh[x/a]

κ = f''[x]/(1+f'[x]^2)^(3/2)/.x->0

==> Krümmungsradius ϱ = a