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Topic: Integration Before Differentiation
Replies: 31   Last Post: Oct 7, 2009 10:38 AM

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 Dave L. Renfro Posts: 4,792 Registered: 12/3/04
Re: Integration Before Differentiation
Posted: Oct 6, 2009 4:13 PM

Robert Hansen wrote (in part):

http://mathforum.org/kb/message.jspa?messageID=6861769

> 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
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:

George Chrystal
"Algebra" (2 volumes)

Henry Sinclair Hall and Samuel Ratcliffe Knight
"Higher Algebra"

Elias Loomis
"A Treatise on Algebra"

Charles Smith
"A Treatise on Algebra"

Aldis William Steadman, "A Text Book of Algebra"

Isaac Todhunter
"Algebra"

> 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.

Dave L. Renfro