Example
Example
In[63]:=
Clear[v,S,g,n]
v[x_,y_,z_]:={x,y,z}
S[x_,y_]:=1-x^2-y^2
g[x_,y_,z_]:=(1-x^2-y^2)-z
In[64]:=
vf=Plot3D[S[x,y],{x,-1,1},{y,-1,1}];
In[65]:=
grad3[g][x,y,z]
Out[65]=
{-2 x, -2 y, -1}
In[66]:=
n=grad3[g][x,y,z]/norm[grad3[g][x,y,z]]
Out[66]=
-2 x -2 y
{---------------------, ---------------------,
2 2 2 2
Sqrt[1 + 4 x + 4 y ] Sqrt[1 + 4 x + 4 y ]
1
-(---------------------)}
2 2
Sqrt[1 + 4 x + 4 y ]
In[67]:=
vp=vectorPlot[{{0,0,S[0,0]}+{0,0,0}},{{0,0,S[0,0]}+n/.{x->0,y->0,z->0}}];
In[68]:=
Show[vf,vp];
Where's the normal?
In[69]:=
Show[vf,vp,ViewPoint->{0,-1,-1}];
It is an inward pointing normal. We will get the flux from the outside to the inside.
In[70]:=
Simplify[v[x,y,S[x,y]].n]
Out[70]=
2 2
1 + x + y
-(---------------------)
2 2
Sqrt[1 + 4 x + 4 y ]
In[71]:=
Dx=D[S[x,y],x]
Dy=D[S[x,y],y]
Out[71]=
-2 x
Out[72]=
-2 y
In[73]:=
dS=Sqrt[1+Dx^2+Dy^2]
Out[73]=
2 2
Sqrt[1 + 4 x + 4 y ]
In[74]:=
Simplify[(v[x,y,S[x,y]].n)*dS]
Out[74]=
2 2
-1 - x - y
In[75]:=
Integrate[1+x^2+y^2,{x,-1,1},
{y,-Sqrt[1-x^2],Sqrt[1-x^2]}]
Out[75]=
3 Pi ---- 2