Mathematics help needed

[aneeshm]

There are a few things I've been trying to understand, but which I've been having a bit of difficulty with.

The first is the rigorous concept of a limit. Sure, the concept itself is simple enough in definition, but I'm having trouble seeing all its implications.

Secondly, the construction of the calculus. I want to know how the laws of calculus are derived from lower-level concepts. The problem, of course, is that the mathematics curriculum in an engineering course is more application-oriented (it's being updated this year, IIRC, taking into consideration that comp. sci. is, in a sense, mathematics).

The "naive" derivation of calculus has, of course, been covered in my school courses, but it wasn't rigorous, so this is a bit of a gap in my knowledge that I want to fill.

[Adalbertus]

Any attempt to answer will probably give you more questions than answers, so you might try to get a book on calculus.

Calculus as I learned it started off from a set of 13 axioms, which are based more or less on every day's language - it is impossible to have a foundation of mathematics without recurring to "weak" normal language.
From these axioms (and a "weakly" defined understanding of sets and mappings) you can derive everything with use of the laws of mathematical logic, including limits.

[aneeshm]

Which one would you recommend that is for pure self-study, assuming that I have no instructor but the book?

[Ramo]

Go to amazon.com and look up real analysis...

[aneeshm]

And one more problem, which is far less general than the others:

The definition of a measure of length on the real line. How do we know that the interval [0,1] is in some sense smaller than the interval [1,infinity] on the (extended) real line, given that the cardinalities of both are exactly the same?

[Lul Thyme]

Originally posted by aneeshm
And one more problem, which is far less general than the others:

The definition of a measure of length on the real line. How do we know that the interval [0,1] is in some sense smaller than the interval [1,infinity] on the (extended) real line, given that the cardinalities of both are exactly the same?

According to wiki :
a measure , by definition, must satisfy sigma_additivity that is that the measure of a union of sets is the sum of the measures of the sets. (For a countable number of sets).

With this definition, the measure of the interval [1,infinity] is larger than the measure of the interval [0,1] (excluding the degenerate scenario where the measure of the rest of the interval is not strictly positive).
This isn't "in some sense", it's just a consequence of the definition of the measure.

[Lul Thyme]

Re: Mathematics help needed

Originally posted by aneeshm
There are a few things I've been trying to understand, but which I've been having a bit of difficulty with.

The first is the rigorous concept of a limit. Sure, the concept itself is simple enough in definition, but I'm having trouble seeing all its implications.

Secondly, the construction of the calculus. I want to know how the laws of calculus are derived from lower-level concepts. The problem, of course, is that the mathematics curriculum in an engineering course is more application-oriented (it's being updated this year, IIRC, taking into consideration that comp. sci. is, in a sense, mathematics).

The "naive" derivation of calculus has, of course, been covered in my school courses, but it wasn't rigorous, so this is a bit of a gap in my knowledge that I want to fill.

When I first learned calculus in college, we did every much pretty rigorously except for the limit, where we used an "intuitive" definition. Now of course since everything else depends on that, it wasn't rigorous but it made it much easier the second time when we went through it again using the delta-epsilon definition for the limit.

[aneeshm]

Originally posted by Lul Thyme


According to wiki :
a measure , by definition, must satisfy sigma_additivity that is that the measure of a union of sets is the sum of the measures of the sets. (For a countable number of sets).

With this definition, the measure of the interval [1,infinity] is larger than the measure of the interval [0,1] (excluding the degenerate scenario where the measure of the rest of the interval is not strictly positive).
This isn't "in some sense", it's just a consequence of the definition of the measure.

OK, but how do we define the measure in the first place?

[aneeshm]

Re: Re: Mathematics help needed

Originally posted by Lul Thyme


When I first learned calculus in college, we did every much pretty rigorously except for the limit, where we used an "intuitive" definition. Now of course since everything else depends on that, it wasn't rigorous but it made it much easier the second time when we went through it again using the delta-epsilon definition for the limit.

We did precisely the same thing, we just did it in the last year of high school.

I'd like to know how calculus is derived from the delta-epsilon definition. That's what I've been getting at. Could you recommend something for that?

I found Rudin to be like the reals - dense.

[Lul Thyme]

Originally posted by aneeshm


OK, but how do we define the measure in the first place?

http://en.wikipedia.org/wiki/Measure_%28mathematics%29

See definition for example.

Technically, it isn't THE measure but A measure.

There are many different measures you can define on the real numbers.

The most common one on the real numbers (and the one you're thinking of I think) is called the Lebesgue measure.

http://en.wikipedia.org/wiki/Lebesgue_measure

[Lul Thyme]

Re: Re: Re: Mathematics help needed

Originally posted by aneeshm


We did precisely the same thing, we just did it in the last year of high school.

I'd like to know how calculus is derived from the delta-epsilon definition. That's what I've been getting at. Could you recommend something for that?

I found Rudin to be like the reals - dense.

My point is that in most calculus classes (in the one I took for example), most definitions were rather rigourous EXCEPT for the limit, and the fact that the other definitions used limits. Hence if you define limits properly (say using delta-epsilon), then the rest should follow by calculus.


For example, the definition of the derivative is the lim as h goes to 0 of the Newton difference quotient (f(a+h)-f(a))/h, which is the definition most people use in a calculus class anyway.
If you use the proper definition of the limit, then this is the proper definition for the derivative.



As Ramo said, this is usually called Real Analysis and there many books on this.

[Adalbertus]

So, I've got a little time now. Please take letters as their greek equivalent where appropriate. A _ means the following letter is a subscript.

Definition of the limit of a sequence:
Let (a_n) a sequence of numbers, i. e. for each positive integer, a_n is a real number (or complex number or an element of some metric space). If there is a real number a such that for each e>0 there is a positive integer n_0, such that for every n > n_0 the distance |a - a_n| < e, then the sequence converges, and its limit is a.

Looks a bit complicated, but I think you can get through it.

Limit of a function:
Let f be a (real) function, which is not necessarily continuous, defined on a subset D of R. Let x_0 be a cumulation point of D (meaning, x_0 is sufficiently "inside" D). If there is a real number f_0 such that for each e>0 there is a d>0 such that for every x in D with |x-x_0| < d holds that |f(x) - f_0| < e, then f_0 is the limit of f(x) for x -> x_0.
In perhaps somewhat more intuitive language: For every neighbourhood N_f of f_0 (neighbourhood is a set which contains f_0 in its "interior"), there is a neighbourhood N_x of x0 in D which is entirely mapped by f into N_f.

There exist (simple) extensions to multidimensional spaces and even topological spaces, IMO the best way to really understand the thing, but maybe a bit too far off for engineers.

[Ben Kenobi]

Well the way I was taught calculus is to understand the fundamental relationship between distance, velocity and acceleration.

velocity is the rate of change of distance over time, hence it's the first derivative of distance, and the area under the velocity over time curve is your distance.

Acceleration is the rate of change of velocity over time, hence it's the first derivative of velocity and the second derivative of distance. If you were to look at a graph of distance over time, the concavity would show the acceleration.

That's why calculus was developed, so they could calculate what happens when you don't have constant acceleration.

[aneeshm]

OK, thanks for all the responses.

Update on the situation: I went to the Bhaskaracharya Pratishthan (it's a huge-ass institution dedicated solely to pure mathematics, and a gathering point for math lovers of all sorts) to return a book I had issued from their library.

There, I asked the librarian for books on elementary or introductory analysis. It so happened that a Ph.D. student doing his thesis from the University of Pune (the one my college is affiliated to) in pure mathematics, and who was teaching at the uni's math department, had come there.

She (the librarian) told me to ask him. He gave me a list of books available in the library by means of which I could teach myself analysis. The list I reproduce below, for anyone who may have had the same question at some time or the other:

1) Elementary Analysis: The Theory of Calculus (Kenneth A. Ross)
2) The Elements of Real Analysis (Robert G. Bartle and Shepherd)
3) Real Analysis (R. Goldberg)
4) Mathematical Analysis (Apostol)

He also recommended I learn basic topology before reading the latter two books. For topology, he recommended:

1) Introduction to Topology and Modern Analysis (G. F. Simmons)
2) Topology (James R. Munkres)



The BP has probably the best (pure) mathematical library in this entire quadrant of the country, and they charge next to nothing for a year's membership (250 Rs. per year = $ 6.30 US). And they have multiple copies of every single book given in the list. This is a stroke of fantastic luck.

[aneeshm]

So now I'm learning from the (extremely) readable text by Ross.

It's really good.