I'd like to summarize my notes into a calculus book.

Why?:
There's countless calculus books out there; why another one?

(1) Freedom
 - The era of free content is beginning, and the threats of shackled
   electronic textbooks is pressing.
 - Also, by making a free text, hopefully others will be able to work
   with it to be customizable to their teaching styles.
 - also, less waste (no need for hundreds of books if there's a few
   free ones)
 
(2) Most calc books suck
 - inappropriate rigor
   Often results are proven by laborious elementary methods
   in the text, yet no such sophistication is required in the
   exercises: you can't read math if you don't write math.
   
   Further, often definitions are introduced based not on pedagogy,
   but on amenability to proofs. E.g., introducing log as an integral,
   or sin as (Spivak's way).
   
   As a result, students skip proofs or learn that they are pedantry.
   This wastes time and space, contributing to the obessity of these
   books.
   (and anyway, these proofs don't stick)
   
   These proofs are there only to soothe the guilt of textbook
   authors and to give instructors something to copy on the board,
   rather than engage the material: they serve no purpose for the
   student.

 - epsilon/deltas too early
   epsilon-delta proofs are -hard-!
     they involve convoluted logical backflips, and rely on an opaque
     definition.
     ultimately, they are pointless: other than pathological
     functions, there's only 5 such proofs you need to do:
     1 is continuous, x is continuous,
     sum, product, uniform limit are continuous.
   continuity as "continuous at a
   point for every point" is -very- unintuitive.
   this should -not- be done in the freshman fall.

 - overuse of elementary methods
   eg, defining e as lim (1+1/n)^n -- b/c you -can-!
     (no-one understands a definition that requires L'Hopital's
      rule to prove is well-defined!)
   (also present in treatment of log as integral, etc.)
   -- related to over-rigor
 
 - stuck in muddled past
   (what is a function? what's an integral?
    newton-era notions, 19th century rigor, 20th century rigor
    (e.g., def'n of function) all muddled)
 
 - lack of physics
   Math and physics have an intimate connection, one that is largely
   glossed over or ignored in calculus texts (and even more in
   introductory physics texts)

 - worthless computations
   Since calculus, I have never computed the volume or surface area of
   a solid/surface of revolution; whatever minor interest these
   objects have is eclipsed by their use as "motivated" sources of
   integrals to compute.

 - picky, picky, picky
   There are many dark corners and sneaky details in elementary
   mathematics, and many of these are glossed over or "accepted"
   by instructors. If they are flagged, then they can be avoided or
   addressed, rather than lurking underneath to trap the unwary.
   + pitfalls for students
   + stumbling blocks for instructors

(3) All same
  all these books are the same!!!
  (okay, there's some variation)
  Why do they have integration by parts in "integration techniques",
  but u-substitution in the definition of integral???

(4) Liberal art
  Math is one of the liberal arts (actually, it's 2 or 3 of them ;-),
  yet it is taught like a base calculation tool as it is so useful.
  In literature and history classes, students learn to examine several
  sides to a question, to argue a point, to express themselves.
  In math, they learn to punch buttons, apply algorithms.
  (movement to add intuition to young math...
   college version deals with woefully unprepared students)
  (state of math education is abysmal -- I can't fix this, but perhaps
   his can help a little...)

My model has been Spivak's "Calculus", the one book I know that falls
at the intro calc level but is not pablum

Goals:
1) mathematical maturity
   Non-mathematicians will frequently take no math classes after
   calculus, but will be asked to apply logical and quantitative
   reasoning later on;
   Mathematicians (and physicists) will generally take further math
   classes, and should be prepared.

   In either case, either because this will be their only opportunity
   to learn to think as mathematicians, or because this is the
   beginning of learning to think like that, it's valuable to learn
   not just computational skills, but thinking techniques.

   -- progression of maturity --

2) understand functions
  A fundamental idea of mathematics (particularly analysis, for our
  purposes) is that of -function-. That said, most students exiting
  calculus have only the vaguest understanding of what a function is,
  due to the obsession with functions R -> R.

3) intuitive presentation of transcendental functions
  (in particular, of their calculus properties)
  intuitive derivative of (cos,sin), exp, log

idea:
quarter by quarter progression, going forward in time and maturity
(start with vague derivatives, be more rigorous with integrals,
 finally have epsilons, deltas and N for sequences/series/continuity)
maybe use cos/sin and exp/log as recurring features: every time, we get more
understanding (vague at first, then more rigorous, then really rigorous at
end)

heavy emphasis on paths in plane
why?
1) b/c these are functions that aren't just R -> R
2) more geometric, and related to physics
   makes partical motion easier to work with -- emphasizes domain as t=time
	 (eg, when you graph f(x), you're graphing the path x,f(x), so high points
	  really look like maximums etc.)
3) many properties of graphs are really properties of paths in plane,
  viewed as x -> (x,f(x))

chapter 1: functions
(introduce wacky functions;
 1.1: paths in the plane
   use this to define cos,sin, and use cos,sin as illustration
 1.2: rational numbers and dirichlet function
 (bonus: that wack-ass function that's 1/q on rationals)
 1.3: functions R^n -> R^m
   (e.g., plus, times: R^2 -> R;
    and make some cryptic remarks about x^y -- more later ;-)

presentation of e:
derivative of 2^x is a bit less than 2^x, and derivative of 3^x
is a bit more, so choose a number in between, which is approx 2.7

re-parametrization of curves (to unit speed) as illustration of chain rule:
given a path in plane, sqrt(f_x(t)^2 + f_y(t)^2) is the speed is a continuous
function and is differentiable if speed stays away from 0 (you don't stop!).
let g(t)=integral_0^t 1/speed, and look at:
R -f-> R^2
`-g-> R
so both have same speed, so completing it to fg^{-1} gives a unit speed
parametrization.

defer to last quarter:
 - rigorous definition of 'e', proof of its properties
   (sooooo much easier/better with power series)
   
 - improper integrals
   (if introduced with integrals, devaluates definite
    integrals and encourages sloppiness;
    also can't understand subtleties until they see non-absolutely
    convergent series)
    
 - define continuity (don't do it earlier! it's hard!)

emphasize -cauchy- defn of convergence: so you can see -internally- what's
going on with the sequence

bonus sections:
 - construction of real numbers
   (by cauchy sequences, seen as canonicalization of decimal expansions)
	 (use indeterminacy of decimal expansion (0.999... = 1) to motivate
	 equivalence classes)
	 - extending functions from Q to R, eg exponential
	   see how the problem of extending a function continuously is intimately
		 related to what you mean by a real number
		 (to make it continuous, it must converge on cauchy sequences -- so cauchy
		  sequences are what we should think of as functions)
 - function spaces and limits of functions
   (in particular:
    pointwise limits
    uniform limit
    uniform on compact sets (compact-open)
      any local result that is true of uniform convergence
      is true in uniform on compact sets
    pf/note: thanks to taylor's thm, taylor polynomial converges
      uniformly on compact sets (well, sometimes ;-)
    pf: so differentiation term-wise makes sense, i.e.,
      we can differentiate power series.

-------------------------------------
Later volume:
multi-variable

key concept: linear approximation in several variables:
  derivative R^n -> R