Lösungen/Solutions/E1S2AnUe/Ex9

L=Lektion/Leçon-- Sprechstunde/heures de consultation
W=Work/Arbeit/Travail
A  =Anhang/  Annexe / Appendice

Ueb/Ex 1      W

f1[x_, y_] := 
        3  x^4 - 2  x^3 y + 2  x y^2 - 4  x + y + 
          1; 
f1[{x_, y_}] := 
        3  x^4 - 2  x^3 y + 2  x y^2 - 4  x + y + 
          1; 
f1[x, y]

Plot3D[f1[x, y], {x, -10, 10}, {y, -10, 10}];

Plot3D[f1[x, y], {x, -5, 5}, {y, -10, 7}];

Plot3D[f1[x, y], {x, -0.5, 1}, {y, -2, 2}];

c1 = 
        ContourPlot[f1[x, y], {x, -5, 8}, {y, -15, 15}];

c2 = 
        ContourPlot[f1[x, y], {x, -0.5, 1}, {y, -2, 2}];

Remove[grad]

grad[f_] := {D[f, x], D[f, y]}; res =  grad[f1[x, y]]

{res[[1]] == 0, res[[2]] == 0}

solv = 
      NSolve[{res[[1]] == 0, res[[2]] == 0}, {x, 
          y}]

p1 = {x, y} /. solv[[1]]; p4 = {x, y} /. solv[[4]]; 
    p5 = {x, y} /. solv[[5]]; 
    p6 = {x, y} /. solv[[6]];  {p1, p4, p5, p6}

Show[c1, 
        Graphics[{PointSize[0.02], Point[p1], Point[p4], Point[p5], 
            Point[p6]}]];

Show[c2, 
        Graphics[{PointSize[0.02], Point[p4], Point[p5], 
            Point[p6]}]];

Solve[1 - 2 (x^3) + 4 x y == 0, {y}]

a = (-4 + 12 (x^3) - 6 (x^2) y + 
              2 (y^2) /. {y -> -((1 - 2 (x^3))/(4 x))}) // 
        Expand; a x^2 // Expand

Solve[a x^2 == 0, {x}]

Gleichung Grad grösser 4, kein Lösungsalgorithmus

Plot[-4 + (1/(8 (x^2))) + x + 
          12 (x^3) - ((5 (x^4))/2), {x, -1, 5}];

Plot[-4 + (1/(8 (x^2))) + x + 
          12 (x^3) - ((5 (x^4))/2), {x, -1, -0.1}];

Plot[-4 + (1/(8 (x^2))) + x + 12 (x^3) - ((5 (x^4))/2), {x, 
          0.1, 5}];

a

hess[f_] := {{D[f, x, x], D[f, y, x]}, {D[f, x, y], D[f, y, y]}}; 
    res1 =  hess[f1[x, y]] // MatrixForm

Det[hess[f1[x, y]]]

Det[hess[f1[x, y]]] /. solv[[1]]

--->  Sattel

Det[hess[f1[x, y]]] /. solv[[4]]

D[f1[x, y], x, x] /. solv[[4]]

--->  Minimum

Det[hess[f1[x, y]]] /. solv[[5]]

--->  Sattel

Det[hess[f1[x, y]]] /. solv[[6]]

--->  Sattel

b

v[t_] := {Cos[t], Sin[t]} ; v[t]

f1[v[t]]

D[f1[v[t]], t] == 0

Plot[Evaluate[D[f1[v[t]], t]], {t, -10, 10}];

Plot[Evaluate[D[f1[v[t]], t]], {t, -1, 5}];

3 periodische Lösungen

c

w[r_, t_] := {r E^t, r^3 - t} ; w[r, t]

F[r_, t_] := f1[w[r, t]]; F[r, t]

D[F[r, t], r] // Simplify

D[F[r, t], t] // Simplify

Ueb/Ex 2      W

f2[x_, y_, z_] := E^(x^2 - y^2 + z^2) (x^2 + y + z);  
    f2[x, y, z]

grad3[f_] := {D[f, x], D[f, y], D[f, y]}; 
    res =  grad3[f2[x, y, z]]

res2 = res/E^(x^2 - y^2 + z^2)  // Expand

Solve[{2 x + 2 (x^3) + 2 x y + 2 x z, 
          1 - 2 (x^2) y - 2 (y^2) - 2 y z, 
          1 - 2 (x^2) y - 2 (y^2) - 2 y z} == {0, 0, 0}, {x, y, 
        z}]

Plot3D[f2[0, y, z], {y, -5, 5}, {z, -5, 5}, 
        PlotPoints -> 100];

Plot3D[f2[0, y, z], {y, -5, 5}, {z, -5, 5}];

Plot3D[f2[0, y, z], {y, -2, 2}, {z, -2, 2}, 
        PlotPoints -> 100];

Plot3D[f2[x, -1/2, z], {x, -5, 5}, {z, -5, 5}, 
        PlotPoints -> 100];

Plot3D[
        f2[x, -1/2, z], {x, -5, 5}, {z, -5, 5}];