The Gradient

In[48]:=

  grad2[g][x,y]

Out[48]=

              2      2      2
  {2 x y Cos[x  y], x  Cos[x  y]}

Note about notation: The first of the above output means the partial derivative of g with respect to x, the second means the partial derivative of g with respect to y.

In[49]:=

  g[x_,y_]:=Exp[- y]Sin[x]

In[50]:=

  grad2[g][x,y]

Out[50]=

   Cos[x]    Sin[x]
  {------, -(------)}
      y         y
     E         E

In[51]:=

  grad2[h][x,y]

Out[51]=

    (1,0)         (0,1)
  {h     [x, y], h     [x, y]}

In[52]:=

  h[x_,y_,z_]:=x y^2 + y^2 z^3 + z^3 x

In[53]:=

  grad3[h][x,y,z]

Out[53]=

    2    3               3       2      2  2
  {y  + z , 2 x y + 2 y z , 3 x z  + 3 y  z }

Chain Rule

Directional Derivatives

The Gradient Vector is Perpendicular to the Level Curve

Rate of Change and Gradient