(rename! sounds like differential forms!)
Dealing with curves (manifolds!) presented 3 different ways:
- graph of function
- parametrized
- level set
parametrized = image of map *in* = explicit
level set = fibre of map *out* = implicit
OIC!!
A function R -> R
can be seen *either* as parametrization *or* a functional
max/min (optimization) is looking in terms of *functional*
(or as parametrization: when you're turning around/stopping)
graph of function = computational, "graph fetishism"
(solved for one in terms of other)
Computing the tangent line to the circle (at (\sqrt{2}/2, \sqrt{2}/2))
several different ways:
- as function
- parametrized
- level set
Level set pretty easy b/c circle is *given* implicitly.
Parametric would be easy in different context:
if going in circle, what's your velocity at time t=\pi/6?
BTW, given a parametrized curve/surface/etc.,
get a basis for the tangent space (push forward the coordinates).
Given a functional, get a vector (gradient),
but the tangent space is the perp
and has no basis.