Lösungen mit Exkursen

1

a

1

Remove[a]

A = {{0,a,-1},{0,2,2},{3,1,2}};A//MatrixForm

A//TeXForm;

Det[A]

B = {{1,0,1},{1,1,0},{0,1,1}};B//MatrixForm

B//TeXForm;

Det[B]

M = {{3,2,2},{1,1,2},{4,3,4}};M//MatrixForm

M//TeXForm;

Det[M]

2

Det[X] = 0

3

Det[A]

Det[A]/.a->-1

b

A.X.B == M

X = Inverse[A].M.Inverse[B]//Simplify//MatrixForm

X = ( Inverse[A].M.Inverse[B]/.a->0)//Simplify//MatrixForm

X = ( Inverse[A].M.Inverse[B]/.a->3)//Simplify//MatrixForm

%//N

c

A = A/.a->-1; A//MatrixForm

X = {{x11,x12,x13},{x21,x22,x23},{x31,x32,x33}}; X//MatrixForm

A.X//MatrixForm

M.Inverse[B]//MatrixForm

(A.X//MatrixForm) == (M.Inverse[B]//MatrixForm)

Anhang

Transpose[X][[1]]

RowReduce[A.X]//MatrixForm

RowReduce[M.Inverse[B]]//MatrixForm

LinearSolve[{{1,0,0},{0,1,0},{0,0,1}},{1,2,3}]

LinearSolve[{{1,0,0},{0,1,0},{0,0,0}},{1,2,3}]

LinearSolve[A,Transpose[M.Inverse[B]][[1]]]

LinearSolve[A,Transpose[M.Inverse[B]][[2]]]

LinearSolve[A,Transpose[M.Inverse[B]][[3]]]

2

Remove["Global`*"]

a Gleichmässig verteilte Streckenlast plus Gegenkraft am Balkenende, allgemeineGleichung

m[x_]:= -q1 (zL-x)^2 + F1 (zL-x); m[x]

s[x_] := 
      DSolve[y''[x] == -m[x]/zE/zI, y[x], x] // 
        Flatten;  s[x]

s[z]

s[z][[1]][[2]] //FullForm

s[z][[1]][[2]] /. {C[1]->C1, C[2]->C2}

h[z_]:=s[z][[1]][[2]] /. {C[1]->C1, C[2]->C2} ; h[z]

h[3]

h[z]/.z->3

h1[u_]:=h[z]/.z->u;  h1[3]

solv1=Solve[{h1[0]==0,Evaluate[h1'[u]==0]/.u->0},{C1,C2}]//Simplify//Flatten

h2[u_]:=(h1[u]/.solv1)//Simplify;  h2[t]

Collect[h2[t],{t,t^2,t^3,t^4}]

Collect[h2[t],{t,t^2,t^3,t^4}]//Together

(Collect[h2[t],{t,t^2,t^3,t^4}] /. q1-> q/2) //Together

Anhang

h2[u_]:=(h1[u]/.solv1)//Simplify; 
h3[t_]=InputForm[h2[u]]/.u->t; {h2[w],h3[w]}

h3[t_]=h2[u]/.u->t; {h2[w],h3[w]}

FullForm[h3[t]]

b

q1=q/2

Masse in Meter und kg

F1 = 500 9.81 Newton

hoehe= 6 Zentimeter; A=hoehe^2; zI = A hoehe^2 /12

h3[u]

A

zE=210000 Newton /Mimimeter^2

zL = 4 Meter

q = 800  9.81 Newton /zL

h3[t]/.{Zentimeter-> 0.01 Meter, Mimimeter-> 0.001 Meter}

h3[zL]/.{Zentimeter-> 0.01 Meter, Mimimeter-> 0.001 Meter}

h3[zL]/.{Meter-> 100 Zentimeter, Mimimeter-> 0.1 Zentimeter}

Anhang

q zL^4/zE/zI (* Einheitenvergleich *)

c

h3[t]//Expand//Simplify

hh=h3[t]//Expand//Simplify

h3[t]/.t->u

hh[[1]]/.t->u

h4[s_]:=hh[[1]]/.t->s ; h4[u]

h4[s_]=Evaluate[h3[t]]/.t->s

D[h4[v],v]

D[h4[v],v]/.{v->zL}

%/. {Meter-> 100 Zentimeter, Mimimeter-> 0.1 Zentimeter}

ArcTan[%]

ArcTan[%%]/(1. Degree)

Remove["Global`*"]

3  Eigenwerte

a

a1

v1 = {1,2,1}; v2={3,8,4}; v3={-4,0,3};
k1=1; k2=4; k3=-2;

Det[{v1,v2,v3}]

M = Transpose[{k1 v1, k2 v2, k3 v3}]. Inverse[Transpose[{v1,v2,v3}]]; M//MatrixForm

M//InputForm

a2

Eigensystem[M]

b

b1

v1 = {1,2}; v2={3,8};
k1=1; k2=4;

M = Transpose[{k1 v1, k2 v2}]. Inverse[Transpose[{v1,v2}]]; M//MatrixForm

M//TeXForm;

Eigensystem[M]

M.{3/8, 1}

M.{1/2, 1}

b2

a[t_]:= +t v1;

a[t]

g[s_]:={6,3}+s {1,1};

M.g[s]//Simplify

Solve[g[s]==M.g[s],{t}]

b3

(M.g[1]-g[1])/(-120)

(M.g[0]-g[0])/(-108)

Solve[a[t]==g[s],{t,s}]

Solve[a[t]==M.g[s],{t,s}]

a[-3]

(M.g[s]-g[s])/(9+s)/(-12)//Simplify

4  Bewegungen von Punkten, Abstände

a Projiziere die Gerade: g (0), g(1)

u = {1,2,1}; a={3,8,4}; b={-4,0,3}; n={0,0,0};

g[s_]:={6,3,0}+s{1,1,1};

g10=g[0]

g20=g[1]

mP = Transpose[{a,b,n}].Inverse[ Transpose[{a,b,u}]]; mP//MatrixForm

g11=mP.g[0]

%//N

g21=mP.g[1]

%//N

b Drehe die projizierte Gerade (Punkte) um 30 Grad um die z-Achse

Remove[d,x]

d[x_]:= {{Cos[x],-Sin[x],0},{Sin[x],Cos[x],0},{0,0,1}}

d[30 Degree]//MatrixForm

%//N

g12=d[30 Degree].(mP.g[0])

%//N

g22=d[30 Degree].(mP.g[1])

g22=d[30 Degree].mP.g[1]//Simplify

%//N

c Berechne die minimalen Abstände der ursprünglichen Geraden und der bewegten Geraden von 0

abst[r_,s_]:=Norm[Cross[r,s]]/Norm[r-s]

abst[g10,g20]

%//N

abst[g11,g21]

%//N

abst[g12,g22]

%//N

d

M={g20-g10,g12-g10,g22-g10}; Det[M]/6

%//N

5 a

M=2 {{-8, 17/2, -8}, {-24, 29, -32}, {-12, 31/2, -18}}; M//MatrixForm

M//TeXForm;

IdentityMatrix[3]//TeXForm;

Det[M- t IdentityMatrix[3]]

-64 IdentityMatrix[3] + 24 M + 6 M . M - M . M . M

%//MatrixForm