This applet is designed to build intuition about finding local extrema
of functions of two variables when constrained to a parameterized curve
- We are trying to find the extrema of f(x,y), constrained to
the parametric curve (gx(t), gy(t)).
- The applet has a control window, a FieldMap
window and a ConstraintCurveUnwound window. The FieldMap window
initially show the gradient field as white arrows, the constraining
curve in green with a perpendicular arrow, a point on th curve in red
with a gradient arrow, and a contour at height z0 in blue.
The ConstraintCurveUnwound graphs t against f(gx(t),gy(t)) with
the point, a tangent vector to the point, and the contour in
the FieldMap window, local extrema on a constraint curve can be
either by having the gradient of f perpendicular to the curve (the red
and green arrow line up), or by
having the a level curve tangent to the constraint curve (the glue and
green curves just touch). In the ConstraintCurveUnwound window,
local extrema are old fashioned local extrema. The purple tangent
vector is flat at critical points.