Zamieszczam plik  rozwiązanego:



In[1]:= r = {Cos[u]*Sin[v], Cos[u]*Cos[v], u^2}

Out[1]= {Cos[u] Sin[v], Cos[u] Cos[v], u^2}

In[2]:= ParametricPlot3D[r, {u, 0, 2*Pi}, {v, 0, 2*Pi}]

Out[4]= {-Sin[u] Sin[v], -Cos[v] Sin[u], 2 u}

In[5]:= r2 = D[r, v]

Out[5]= {Cos[u] Cos[v], -Cos[u] Sin[v], 0}

In[6]:= wn = FullSimplify[Cross[r1, r2]]

Out[6]= {2 u Cos[u] Sin[v], 2 u Cos[u] Cos[v], Cos[u] Sin[u]}

In[7]:= m = FullSimplify[wn/Sqrt[wn.wn]]

In[8]:= g11 = FullSimplify[r1.r1]

Out[8]= 1/2 (1 + 8 u^2 - Cos[2 u])

In[9]:= g12 = FullSimplify[r1.r2]

Out[9]= 0

In[10]:= g22 = FullSimplify[r2.r2]

Out[10]= Cos[u]^2

In[11]:= g = g11*g22 - (g12)^2

Out[11]= 1/2 Cos[u]^2 (1 + 8 u^2 - Cos[2 u])

In[12]:= r11 = D[r1, u]

Out[12]= {-Cos[u] Sin[v], -Cos[u] Cos[v], 2}

In[13]:= r12 = D[r1, v]

Out[13]= {-Cos[v] Sin[u], Sin[u] Sin[v], 0}

In[14]:= r22 = D[r2, v]

Out[14]= {-Cos[u] Sin[v], -Cos[u] Cos[v], 0}

In[15]:= b11 = FullSimplify[m.r11, 3 + Cos[v] > 0]

Out[15]= (2 Sqrt[2] Cos[u] (-u Cos[u] + Sin[u]))/Sqrt[
Cos[u]^2 (1 + 8 u^2 - Cos[2 u])]

In[16]:= b12 = FullSimplify[m.r12]

Out[16]= 0

In[17]:= b22 = FullSimplify[m.r22, 3 + Cos[v] > 0]

Out[17]= -((2 Sqrt[2] u Cos[u]^2)/Sqrt[
Cos[u]^2 (1 + 8 u^2 - Cos[2 u])])

In[18]:= b = FullSipmlify[b11*b22 - (b12)^2]

Out[18]= FullSipmlify[-((8 u Cos[u] (-u Cos[u] + Sin[u]))/(
  1 + 8 u^2 - Cos[2 u]))]


Na koniec na podstawie tych wyliczen trzeba jeszcze zapisac 1 i 2 forme :
Ponizej podaje wzor ogolny:

http://zapodaj.net/d7e264a37f92.bmp.html


Ogolnie to jest zadanie 3 z zadan nr 3 w pdf

Ostatnio edytowany przez nitram (2011-06-14 09:06:40)