the other file, "Where Now", is for students looking -forward- to what
math to do;

this file is about how to recognize calc when you see it again
...which is a bit different;
this is supposed to be read -while- taking a course,
rather than to decide what to take

complex analysis:
- power series = analytic/holomorphic functions
- ramified covering spaces: explain how integration tricks worked
- elementary functions (exp/log, trig/arc-trig) over complex plane
  residues as deck transforms...
- covering spaces as graphs of multi-valued functions,
  or rather, as subsets of $\bC^2$, together with a projection to
  $\bC$;
  log (and arcsin etc.) is a section, not a global inverse function
- infinite products (not usually covered in calc): like infinite sums

topology:
- metric spaces: the abstraction of epsilon-deltas
- covering spaces (see also Complex analysis):
    differentiation is local, as is anti-differentiation;
    first source of manifolds
- connectedness/compactness:
  the topological context of the IVT and maximum principle
  (image of connected/compact is connected/compact)

algebra:
- rings/algebras/graded/DGAs (Differential Graded Algebras):
  think of various algebras of functions, and polynomials
- local rings: think of power series and taylor series:
  power series in $(x-a)$ are the local ring at $a$
  (and are the local ring par excellence);
  Taylor series are the image of the germ in the local ring

analysis:
everything...
- compare Lebesgue integral (measure) vs. Darboux/Riemann (intervals)
- differentiating distributions: integration by parts is now a
  defn/axiom
- topologies on function spaces: read my notes on calc from point of
  view of topological vector spaces (uniform convergence of
  derivative, etc.)

differential geometry:
everything...
- calculus of variations as minimization in weird space;
  geodesics
- curvature: think of second derivative test for $\bR^2 \to \bR$
- parametrized curves (familiar for my students ;-)

algebraic geometry:
- rational parametrization of curves:
  think of $u=\arctan (x/2)$ substitution in rational parametrization
  of circle, and why we care (it makes life -nice-!)
  (also enumeration of Pythagorean triples)
- classification of degree 2 quadratic forms over the reals
  = classification of conics (esp. rotation of axes)

linear algebra:
- linear maps are the generalization of lines needed to define
  derivatives for maps $\bR^m \to \bR^n$
- derivative is a linear operator
- the "+C" in integration qua anti-differentiation is first EG of kernel
- constant coefficient (homogenous!) ODEs are easy linear algebra

representation theory:
- Fourier analysis (done a little in my class) is representation
  theory of the circle

dynamics:
- fixed point theorems, derivatives -- these are all basics of
  understanding dynamics
