Functions of Several Variables
Functions of Several Variables
In[117]:=
p3=Plot3D[Sin[x y],{x,0,3},{y,0,3}];
In[118]:=
Show[p3,ViewPoint->{0,-2,0}];
In[119]:=
Show[p3,Shading->False];
Partial derivatives hold one or more of the variables constant and differentiate with respect to the variable that is not fixed.
Here are some first partial derivatives.
In[120]:=
D[x^2 y + 2 Sin[y],x]
Out[120]=
2 x y
In[121]:=
D[x^2 y + 2 Sin[y],y]
Out[121]=
2 x + 2 Cos[y]
Here are some second partial derivatives.
First Dxy, that is, partial with respect to x first, then with
respect to y.
In[122]:=
D[x^2 y + 2 Sin[y],y,x]
Out[122]=
2 x
Next Dyx
In[123]:=
D[x^2 y + 2 Sin[y],x,y]
Out[123]=
2 x
Now, D^3 y, the third partial with respect to y.
In[124]:=
D[x^2 y + 2 Sin[y],{y,3}]
Out[124]=
-2 Cos[y]
You will recall that for "nice" functions, that is, functions whose first and second partials, along with f itself, are continuous, we always have the mixed second partials being equal.
In[125]:=
Clear[x,y,z,g,h]
g[x_,y_]:=Sin[x^2 y]
h[x_,y_,z_]:=x Exp[y] Sin[Pi z]
The following (double equal signs, ==) is Mathematica's way of asking if something is true or false.
In[126]:=
D[g[x,y],x,y]==D[g[x,y],y,x]
Out[126]=
True
In[127]:=
D[h[x,y,z],x,y]==D[h[x,y,z],y,x]
Out[127]=
True
In[128]:=
D[h[x,y,z],x,z]==D[h[x,y,z],z,x]
Out[128]=
True
In[129]:=
D[h[x,y,z],z,y]==D[h[x,y,z],y,z]
Out[129]=
True