Example

The temperature at each point of a metal plate is given by T(x,y)=10+x^2-y^2. Find the path of a heat seeking particle that originates at (-2,1).

We need the curve r(t)=x(t) i + y(t) k which begins at (-2,1) and at each point has direction (direction of r'(t)) in the direction of the gradient vector.

In[87]:=

  T[x_,y_]:=10+x^2-y^2

In[88]:=

  grad2[T][x,y]

Out[88]=

  {2 x, -2 y}

This is easy enough to solve by hand, but let's practice our commands.

In[89]:=

  Clear[x,y,t]
  DSolve[{x'[t]==2 x[t],y'[t]==-2 y[t],x[0]==-2, y[0]==1},
  {x[t],y[t]},t]

Out[89]=

                2 t           -2 t
  {{x[t] -> -2 E   , y[t] -> E    }}

Do you see why this is just the path xy=-2?

In[90]:=

  PlotGradientField[10+x^2-y^2,{x,-2,2},{y,-2,2}]

Out[91]=

  -Graphics-

Up to Rate of Change and Gradient