> I will concede that once you have broken into whole > hearted formal analysis and its development, the notion > of anti-derivative disappears. Even if you had it to > begin with.
I wasn't very clear about (at least) one thing yesterday. The books that cover integration first only cover a chapter or two of mostly theoretical issues when integration is introduced. Later, after differentiation, the Fundamental Theorem of Calculus is covered and then one gets into all the various anti-derivative methods. Incidentally, the idea of anti-derivatives still shows up in higher level analysis (a classical term for it is "a primitive of the function"), but the concern shifts from finding explicit formulas for them to their existence for sufficiently nice functions and to its properties (such as being continuous).
> By the way, in what sort of conditions would a book like > Apostol's be the first calculus a students sees. I mean, > like quadratic formula and then boom Apostol?
Essentially every student entering Caltech (where Apostol had taught; "had", because he's retired now) knows some calculus, whether from a high school course, a college course, or from their own independent study. Thus, the book isn't really the first calculus a student sees. Also, discussions of tangent lines and the derivative and even power series expansions of functions used to be part of advanced algebra many years ago, so when Courant wrote his book (1930), the background one could assume was quite a bit higher than what one can assume today. You might find it amusing to look through the following books, which are admittedly very selectively chosen and probably at an even higher level than Courant could safely assume his readers knew in 1930:
> Also, do you have any thing in the way of a mathematical > description of the different nature between the mapping of > functions to their derivatives and the mapping to their > integrals. In other words why one direction is so much more > easily solvable (closed?) than the other. Or is it just the > nature sums of infinite series versus, well, not sums of > infinite series. I suppose that many solutions are so > transcendental in nature that they defy solved forms.
Not anything particularly enlightening, although I will mention that you're only focusing on the "algebraic and combinatorial complexity" aspect. In ways that are more important and more interesting, the process of integration is "nicer" than the process of differentiation.
For one thing, the derivative of a function is typically less "smooth" than the function. A function has to be continuous (and more) to be differentiable, but after you take the derivative of a differentiable function, the result can fail to be continuous (it can even be discontinuous on a dense set of points). Also, if you consider the differentiation operator as a function from the set of functions with a continuous derivative to the set of continuous functions (that is, d/dx is the function that, when you input a function f, the output is f') is discontinuous. In fact, it is not continuous at any point in its domain. (Usual C^1 sup norm on the domain space and the usual C^0 sup norm on the range space, for those wanting the i's dotted and t's crossed.)
On the other hand, the indefinite Riemann integral of a function (even some that are fairly badly behaved) will be continuous. Also, pretty much any kind of operator you define using integrals will be everywhere continuous as a function with domain an appropriate space of functions and range another appropriate space of functions.