Lösungen/Solutions/E1S2AnUe/Ex8

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

Hinweis: Zur Konvergenzentscheidung werden hier auch noch gleich die Berechnungen druchgeführt.

Ueb/Ex 1      W

f1[x_, y_] := E^Sin[x + y];  f1[x, y]

f1[{x_, y_}] := E^Sin[x + y];  f1[{x, y}]

Plot3D[f1[x, y], {x, -4, 4}, {y, -4, 4}];

f2[x_, y_] := x^2 + x y  - 3  y;  f2[x, y]

f2[{x_, y_}] := x^2 + x y  - 3  y;  f2[{x, y}]

Plot3D[f2[x, y], {x, -20, 20}, {y, -20, 20}];

P1 = {0, 0};  P2 = {2, 1};

a

Dt[x]= dx     etc.!!!!!!

Dt[f1[x, y]]

Dt[f1[x, y]] /.  {Dt[x] -> dx, Dt[y] -> dy}

Dt[f1[x, y]] /.  {Dt[x] -> dx, 
          Dt[y] -> dy} /.  {x -> 0, y -> 0}

Dt[f1[x, y]] /.  {Dt[x] -> dx, 
          Dt[y] -> dy}  /. {x -> 2, y -> 1}

% // N

Dt[f2[x, y]]

Dt[f2[x, y]] /.  {Dt[x] -> dx, Dt[y] -> dy}

Dt[f2[x, y]] /.  {Dt[x] -> dx, 
          Dt[y] -> dy} /.  {x -> 0, y -> 0}

Dt[f2[x, y]] /.  {Dt[x] -> dx, 
          Dt[y] -> dy} /.  {x -> 2, y -> 1}

b

Φ1[x_, y_, x0_, y0_] := 
      f1[P1] + (x - 2) D[f1[x0, y0], x0] + (y - 1) 
          D[f1[x0, y0], y0] ;  Φ1[x, y, x0, y0]

Φ1[x, y, x0, y0] /.  {x0 -> 2, 
        y0 -> 1}

% // N // Expand

Φ2[x_, y_, x0_, y0_] := 
      f2[P1] + (x - 2) D[f2[x0, y0], x0] + (y - 2) 
          D[f2[x0, y0], y0] ;  Φ2[x, y, x0, y0]

Φ2[x, y, x0, y0] /.  {x0 -> 2, 
        y0 -> 1}

% // N // Expand

c

Package laden

<< Calculus`VectorAnalysis`

?Grad

Coordinates[]

3. Koordinate anfügen, fallen lassen

?Drop

grad1  =  Drop[Grad[f1[Xx, Yy]], -1]

grad11  =  
      Drop[Grad[f1[Xx, Yy]], -1] /. {Xx -> 0, 
          Yy -> 0}

grad12  =  
      Drop[Grad[f1[Xx, Yy]], -1] /. {Xx -> 2, 
          Yy -> 1}

% // N

grad2  =  Drop[Grad[f2[Xx, Yy]], -1]

grad21  =  
      Drop[Grad[f2[Xx, Yy]], -1] /. {Xx -> 0, 
          Yy -> 0}

grad22 =  
      Drop[Grad[f2[Xx, Yy]], -1] /. {Xx -> 2, 
          Yy -> 1}

% // N

d

<< Graphics`Arrow`

p[P_] := Point[P];

s1 = 
        Show[Graphics[
{PointSize[
                0.03], 
p[P1], p[P1 + grad11], 
              Arrow[P1, P1 + grad11], 
p[P2], 
              p[P2 + grad12], Arrow[P2, P2 + grad12]
}], 
          AspectRatio -> 01];

s2 = 
        Show[Graphics[
{PointSize[
                0.03], 
p[P1], p[P1 + grad21], 
              Arrow[P1, P1 + grad21], 
p[P2], 
              p[P2 + grad22], Arrow[P2, P2 + grad22]
}], 
          AspectRatio -> 01];

c1 = 
        ContourPlot[f1[x, y], {x, -1, 3}, {y, -1, 3}];

Show[c1, s1];

c2 = 
        ContourPlot[f2[x, y], {x, -3, 8}, {y, -3.5, 4}];

Show[c2, s2];

e

α = Pi/4

dir = {Cos[α], Sin[α]}

{g1, g2} . {r, s}

Dα12 = grad12 . dir

% // N

Dα22 = grad22 . dir

% // N

f

grad1 == {0, 0}

Solve[grad1 == {0, 0}, {Xx, Yy}] // Flatten

Plot3D[
        f1[x, y], {x, -Pi/2 - 4, -Pi/2 + 4}, {y, -4, 
          4}];

grad2 == {0, 0}

Solve[grad2 == {0, 0}, {Xx, Yy}] // Flatten

Plot3D[
        f2[x, y], {x, 3 - 4, 
          3 + 4}, {y, -6 - 4, -6 + 4}];

ContourPlot[
        f2[x, y], {x, 3 - 4, 
          3 + 4}, {y, -6 - 4, -6 + 4}];