Gradient vector and level curves

The gradient vector of a function of two variables, evaluated at a point (a,b), points in the direction of maximum increase in the function at (a,b). The gradient vector is also perpendicular to the level curve of the function passing through (a,b).

Below is the graph of the level curve

 2     2
x + 2y  = 1
of the function
          2     2
f|\x,y/| = x +  2y ,
whose gradient vector is
      @f     @f
 \~/ f = ---i+  ---j = 2x i+ 4yj.
       @x    @y
At each point on this level curve the gradient vector is perpendicular to the level curve.