This applet is designed to build intuition about finding critical
points of functions of two variables, using the gradient vector field
- The applet has a control window, a FieldMap window, and a
ThetaSliceCurve window. The FieldMap window initially shows the
gradient field in white, the point P0 and the gradient at P0 in red, and a line through P0 at angle theta in green.
The ThetaSliceCurve gives a green cross section of the surface
in direction theta, a thin green tangent line to that slice
curve, and red and pink lines corresponding to the slope of
the gradient and its negative respectively.
gradient vector is (f_x, f_y, -1). It is the gradient to the
function h(x,y,z)=f(x,y-z) which has the graph of z=f(x,y) as a level
- Local extrema are either sources of sinks for the gradient vector
- Extrema are also places where the gradient goes to
zero, making the corresponding tangent lines flat.
- A point that is a local extreme point (maximum or minimum) will
be an extreme point of the same kind (maximum or minimum) for slices in
- We expect the contours to form a loop around extreme points.
Whenever the contours form a loop for a continuous function,
there is an extreme point inside the loop.