Lösungen/Solutions/E1S2AnUe/Ex15

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

Remove["Global`@*"]

Ueb/Ex 1      W

mF[x_] := MatrixForm[x]

x[r_, φ_, ϑ_] := 
        r Cos[φ] Sin[ϑ];
y[r_, φ_, ϑ_] := 
        r Sin[φ] Sin[ϑ];
z[r_, φ_, ϑ_] := 
        r Cos[ϑ];

m[r_, φ_, ϑ_] := {
{D[
            x[r, φ, ϑ], φ], 
          D[y[r, φ, ϑ], φ], 
          D[z[r, φ, ϑ], φ]}, 
{D[x[r, φ, ϑ], ϑ], 
          D[y[r, φ, ϑ], ϑ], 
          D[z[r, φ, ϑ], ϑ]}, 
{D[x[r, φ, ϑ], r], 
          D[y[r, φ, ϑ], r], 
          D[z[r, φ, ϑ], r]}
};  
    m[r, φ, ϑ] // mF

Det[m[r, φ, ϑ]] // Simplify

dV = 
      Simplify[Det[
            Evaluate[
              m[r, φ, ϑ]]]]  /. { ϑ -> -ϑ1}

Ueb/Ex 2      W

a

Remove["Global`@*"]

x[r_, φ_, ϑ_] := 
        a r Cos[φ] Sin[ϑ];
y[r_, φ_, ϑ_] := 
        b r Sin[φ] Sin[ϑ];
z[r_, φ_, ϑ_] := 
        c r Cos[ϑ];

m[r_, φ_, ϑ_] := {
{D[
            x[r, φ, ϑ], φ], 
          D[y[r, φ, ϑ], φ], 
          D[z[r, φ, ϑ], φ]}, 
{D[x[r, φ, ϑ], ϑ], 
          D[y[r, φ, ϑ], ϑ], 
          D[z[r, φ, ϑ], ϑ]}, 
{D[x[r, φ, ϑ], r], 
          D[y[r, φ, ϑ], r], 
          D[z[r, φ, ϑ], r]}
};  
    m[r, φ, ϑ] // mF

dV = 
      Simplify[Det[Evaluate[m[r, φ, ϑ]]]]

dV1 = 
      Simplify[Det[
            Evaluate[
              m[r, φ, ϑ]]]]  /. { ϑ -> -ϑ1}

b

V = 
      Integrate[
Integrate[
Integrate[  dV , {r, 0, R}]
, {φ, 0, 
            2  Pi}]
, {ϑ, 0, Pi}]

V1 = 
      Integrate[
Integrate[
Integrate[  dV1 , {r, 0, R}]
, {φ, 0, 
            2  Pi}]
, {ϑ1, 0, Pi}]

Ueb/Ex 3      W

Integrate[
Integrate[
Integrate[  Sin[φ] Sin[ϑ] r^2 , {r, 0, 
            1}]
, {φ, 0, 
          2  Pi}]
, {ϑ, 0, Pi}]

Ueb/Ex 4      W

res1 = 
      Integrate[
        Integrate[
Sin[Log[x] + E^y] - 
            Cos[Log[x] - E^y],  {x, 1, 2}] , {y, 0, 1}]

N[res1]

x = E^u;  y = Log[v];
expr = 
      Evaluate[
Sin[Log[x] + E^y] - 
            Cos[Log[x] - E^y]] /. {Log[E^u] -> u}

D[x, u]

j = {{D[x, u], D[y, u]}, {D[x, v], D[y, v]}};  
    j // MatrixForm

dG=Det[j]

res2 = 
      Integrate[
        Integrate[expr dG, 
 {u, 0, Log[2]}] , {v, 1, 
          E}]

N[res2]