Example Of Unequal Mixed Partails.

In[60]:=

f[x_, y_] := x y (y^2 - x^2)/(x^2 + y^2)

In[61]:=

Plot3D[f[x, y], {x, -1, 1}, {y, -1, 1}]

[Graphics:HTMLFiles/index_3.gif]

Out[61]=

⁃SurfaceGraphics⁃

In[62]:=

dfx = D[f[x, y], x]

Out[62]=

-(2 x^2 y (-x^2 + y^2))/(x^2 + y^2)^2 - (2 x^2 y)/(x^2 + y^2) + (y (-x^2 + y^2))/(x^2 + y^2)

In[63]:=

Plot3D[dfx , {x, -1, 1}, {y, -1, 1}]

[Graphics:HTMLFiles/index_8.gif]

Out[63]=

⁃SurfaceGraphics⁃

In[64]:=

dfy = D[f[x, y], y]

Out[64]=

-(2 x y^2 (-x^2 + y^2))/(x^2 + y^2)^2 + (2 x y^2)/(x^2 + y^2) + (x (-x^2 + y^2))/(x^2 + y^2)

In[65]:=

Plot3D[dfy, {x, -1, 1}, {y, -1, 1}]

[Graphics:HTMLFiles/index_13.gif]

Out[65]=

⁃SurfaceGraphics⁃

In[66]:=

dfxy = D[dfx, y]

Out[66]=

(8 x^2 y^2 (-x^2 + y^2))/(x^2 + y^2)^3 - (2 x^2 (-x^2 + y^2))/(x^2 + y^2)^2 - (2 y^2 (-x^2 + y^2))/(x^2 + y^2)^2 - (2 x^2)/(x^2 + y^2) + (2 y^2)/(x^2 + y^2) + (-x^2 + y^2)/(x^2 + y^2)

In[67]:=

dfyx = D[dfy, x]

Out[67]=

(8 x^2 y^2 (-x^2 + y^2))/(x^2 + y^2)^3 - (2 x^2 (-x^2 + y^2))/(x^2 + y^2)^2 - (2 y^2 (-x^2 + y^2))/(x^2 + y^2)^2 - (2 x^2)/(x^2 + y^2) + (2 y^2)/(x^2 + y^2) + (-x^2 + y^2)/(x^2 + y^2)

In[69]:=

Plot3D[dfxy, {x, -1, 1}, {y, -1, 1}]

[Graphics:HTMLFiles/index_20.gif]

Out[69]=

⁃SurfaceGraphics⁃

In[70]:=

Plot3D[dfyx, {x, -1, 1}, {y, -1, 1}]

[Graphics:HTMLFiles/index_23.gif]

Out[70]=

⁃SurfaceGraphics⁃

Created by Mathematica  (February 10, 2006)