Explicit/Implicit

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Actually, there's -3- ways that sets are presented
...and these each correspond to different -computational- ways of
discussing the same -geometric- situation.
Keep 'em straight, and present 'em individually.

- parametrically/explicit
- level set/implicitly
- graph of function

EG, for the unit circle, these are:
- $(\cos \theta, \sin \theta)$
- $x^2 + y^2 = 1$
- $y = \sqrt{1-x^2}$, $y=-\sqrt{1-x^2}$
...and this in the only example you need.

Other EG: the cuspital cubic:
- $(t^2,t^3)$
- $y^2=x^3$
- $y=\pm x^{3/2}$
[this one is confusing b/c the first two are so close]

The nodal cubic is another good EG, and is sufficiently complicated
that one can keep all of 'em straight.

The form the derivative takes here is
- explicit: point-vector form
     (as -image-)
     geometrically, -velocity-
- implicit: perp to grad
     (as -kernel-)
     geometrically, -steepest descent- (or rather, its perp)
- graph: point-slope form

These -all follow- from the POV of "local linear approximation",
and should -all- be seen as -illustrations- and -consequences-.

The connections between them:
- graph is a special case of parametric:
  $(x,f(x))$
- parametric can be graphed:
  $(t,x(t),y(t))$

- graph is a special case of implicit:
  $y=f(x)$ goes to $y-f(x)=0$
- implicit -> graph:
  solve for $y$ (not always possible!)

- explicit -> implicit
  find -defining equations-;
  hard question in general (and careful of -constructible sets-:
  EG, in the rational parametrization of the circle, you miss one
  point: so it satisfies $x^2+y^2=1$, but doesn't hit -all- of 'em)

- implicit -> explicit:
  parametrization (leads to issues of rationality of curves!)
  (EG of conics and singular cubics being rational,
   via stereographic projection; also interpret via tan)
  ...and this also leads -immediately- to the implicit function
  theorem, namely that:
  - ``this is always -locally- possible''
  - and in fact you can just use some of the coordinates
    as the independent variables
    (you can project to the tangent hyperplane, obviously,
     and some coordinates give a good basis for this)

The "function" POV is $\bR^1 \to \bR^1$,
  and is most fundamental,
  but is hard to visualize.
The parametric POV is much easier to visualize,
  and really just illustrates the function POV.
However, students are not familiar with parametrics
  and their heads start to hurt.

The level set POV is the geometry behind -equations-
  (and leads to varieties and algebraic geometry),
  and is thus the -most- computationally familiar.
Unfortunately, it's also geometrically the most complicated
  (though it's not really that awful).

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A -key- organizing idea of multivariable calculus is -explicit- vs.
-implicit- descriptions of subset.
IE, describing a subset as an -image- vs. a -fibre-.
The -key- example is the unit circle:
$(\cos \theta, \sin \theta)$ is the
  -explicit-, -parametrized-, -image- example;
$x^2 + y^2 = 1$ is the
  -implicit-, -level set-, -fibre- example.

The point is that sometimes, it's easier to parametrize a set;
other times, it's easier to give -defining equations-.

the linear version is:
a subspace of $V$ being described as
$\im(U \to V)$
or as
$\ker(V \to W)$

Passing between these is -exactly- Linear Algebra 101:
going from kernel to image is -exactly- solving a (homogeneous) system
of linear equations;
going from image to kernel is exactly -dual-.

Indeed, this whole topic can be seen as the two faces of duality.

Basics ---------------------------------
One should wallow in this for a little while, to appreciate it.
In general, it's -hard- to pass between these descriptions:
given a parametrization, you want a set of equations that define it.
Conversely, given an implicit set, you want a parametrization.

Indeed, in algebraic geometry, varieties are defined as -implicitly
defined- subsets; if you include explicitly defined ones (images), you
get the messy collection of ``constructible sets''.

Further, trying to give -rational- parametrizations of various curves

Rational parametrizations of $x^2+y^2=1$ (quadratic -- or other
quadratics) and $y^2=x^3$ (singular cubic) (or $y^2=x^3-x$)
is a useful exercise;
OTOH, elliptic curves (smooth cubics) -do not- admit rational
parametrizations, and this is the beginings of algebraic geometry.

Differential ---------------------------
For tangent (hyper)planes:
- given a -parametrized- subset, $\im (U\from \bR^m \to \bR^n)$,
  the tangent plane is $U(p) + dU d\vect x$
  This is an -explicit- description;
    it's the derivative of an explicit description
- given an -implicit- subset,
  $\ker (f \from \bR^n \to \bR)$
  the tangent plane at $x$ is
  $x + \set{v \st df(v) = 0}$
     ...which is an -implicit- description.
  ...obviously if it's the kernel of a map $\to \bR^p$,
  it's the intersection of several hypersurfaces,
  and the tangent plane is
  $x + \set{v \st \forall i, df_i(v) = 0}$
  (it's the intersection of all the tangent planes)
Giving an -explicit- parametrization of the tangent plane to a set
given -implicitly-, or conversely, is exactly linear algebra.

For optimization:
- optimizing a function on a -parametrized- subset is trivial:
  just pull it back to the original set,
  optimize there, and then push the optima forward

- optimizing a function an a -level set-
  is exactly what Lagrange multipliers are for

Integral -------------------------------
I don't know how to integrate over an implicitly defined set.
The only way I know how to integrate is by -parametrizing- the set,
as
$x_1=[a,b], x_2 = [a(x_1),b(x_1)],
 x_3=[a(x_1,x_2),b(x_1,x_2)], \dotsc$
[this is simplest if it's over a rectangle.]
