Calculus

[rmjohn.arbis]

Please give me ideas about Calculus

Do not use histort but focus on math concepts, principes, laws, theorems etc.

[ajb]

You have asked a very wide question here. Why not look at wikipedia first? You will get more information quickly that way.

What I will say, is that (standard) calculus is broken into two peices.

i) Differential calculus - which is looking at the instantaneous rates of change of objects with respect to some variables. We have the notion of the derivative of a function.
ii) Integral calculus - which is calculating the area under curves, calculating volumes and so on. This is all given in terms if the (indefinite or definite) integral of a function.

The two notions are tied together via the fundamental theorem of calculus. This says that the derivative and indefinite integral are basically mutual inverses (but not quite).

 

I hope that gives you something to start with.

[Strange]

The two notions are tied together via the fundamental theorem of calculus. This says that the derivative and indefinite integral are basically mutual inverses (but not quite).

 

 

Do you know if, historically, integral and differential calculus were developed as separate things - and then later it was realised they were inverses? Or was that always known?

[ajb]

Do you know if, historically, integral and differential calculus were developed as separate things - and then later it was realised they were inverses? Or was that always known?

Isaac Barrow (1630–1677) is one of the first to publish on a version of this theorem, while his student Isaac Newton (1642–1727) really developed the mathematics here. So it seems the theorem was known right from the start of calculus.

[blue89]

 

 

Do you know if, historically, integral and differential calculus were developed as separate things - and then later it was realised they were inverses? Or was that always known?

 

I quite againts this idea and also I am sure that this expression should be incorrect. look please golden rate may told something in general ,then I think it is highly clear that at the top of science almost every subjects will be related each with them.

[ajb]

I quite againts this idea and also I am sure that this expression should be incorrect.

You do not think the fundamental theorem of calculus is correct?

[Strange]

Isaac Barrow (1630–1677) is one of the first to publish on a version of this theorem, while his student Isaac Newton (1642–1727) really developed the mathematics here. So it seems the theorem was known right from the start of calculus.

 

 

Thanks for this, BTW.

[ajb]

Thanks for this, BTW.

No problem... it seem that the idea of both parts of the fundamental theorem of calculus are older than calculus as we know it today! It took a lot of work to formalise all this, but the ideas are right there from the start of the subject.

[StringJunky]

No problem... it seem that the idea of both parts of the fundamental theorem of calculus are older than calculus as we know it today! It took a lot of work to formalise all this, but the ideas are right there from the start of the subject.

Newton didn't invent calculus but developed it? I read he invented it.

[ajb]

Newton didn't invent calculus but developed it? I read he invented it.

He did, but not 100% from scratch.

[StringJunky]

He did, but not 100% from scratch.

OK Ta. Nothing ever is, is it?

[imatfaal]

OK Ta. Nothing ever is, is it?

 

"If I have seen little farther..."

[StringJunky]

 

"If I have seen little farther..."

Of course.

[blue89]

You do not think the fundamental theorem of calculus is correct?

 

oh dear ,

 

apology ,I had understood incoorrectly , I have supposed that he implied studying partial and to continue so in progress&later would be better. probably one associate.proffessor at mimar sinan university caused thinking so.

 

it seems this is not relevant to this thrad my idea I shared above. but again I will write it.

 

*** * ***

I think studying interdisciplinary would be very effective and may be useful. of course it would require studying much more than particular. but I am sure we would be able to produce useful and so we would be able to be efficacious to humanity.

I think the last expression is already should be science's goal. and it is clear that scientists who created usefuld products (theorie,patent) are richy or beloved.

 

*** * ***

 

I would study like tesla, gauss,thomas edison now.

You do not think the fundamental theorem of calculus is correct?

 

then, really I am more interested in mathematical modelling and this just throughout to find useful cryptology.

but I would express this : it is clear that now some theories relevant to integral (and also other subjects,too) already do not contain universal information. for instance calculating riemann integral is approximation ,it is not certitude. but of course according to mathematical principles ,there is no contradiction. because this is the result of limit.

Edited June 25, 2016 by blue89

[ajb]

... for instance calculating riemann integral is approximation ,it is not certitude. but of course according to mathematical principles ,there is no contradiction. because this is the result of limit.

I am not sure I quite follow you, but you are right that there is a proof of the second fundamental theorem of calculus that uses the Riemann integral. That is enough for a huge class of functions, i.e., those that have a Riemann integral. For wider classes of functions you need ideas from measure theory - Lebesgue integrable functions etc.

[blue89]

I am not sure I quite follow you, but you are right that there is a proof of the second fundamental theorem of calculus that uses the Riemann integral. That is enough for a huge class of functions, i.e., those that have a Riemann integral. For wider classes of functions you need ideas from measure theory - Lebesgue integrable functions etc.

 

 

!!: read please the guidance below ,riemann integral might/does not provide us understanding fundamental theorem.there are some another strong criteria available!.

the message i sent you contain something about Lebesgue integral with relevancy of functional analysis.I recommend that you check surroundingly to all related statements [definitions ,theorems] for your question above.(there might be some other (additional) requirements e.g. : continuoum or regaular continuoum ( as you know continuoum and regular continuoum are not same! ,define separate things..(look please exxtra : lipschitz (Criterion) and cantor theorem [!!: cantor theorem says ; if any function is continuous on the closed interval then, this function will be regularly continuous on that interval!..)..

 

in addition ,i have sent to you some information in private. o.k. really because of I am not interested a lot to pure mathematics ,I am not willing to research anymore in traditional way (I am also different than others as I implied Before times). because I am sure and I see ,there are already lots of theorem ,but if we ask that what is the reason for creating all these?? in my opinion the LOGICAL answer is not visible!... (otherwise ,of course it does not mean that I am not able to study so, look please the sentence written in bold,it shows the reason why I do not want to study as traditional)

 

please do not hesitate asking me any question else if you have.(especially related analysis ,functional analysis, and basic algebra )

Edited June 26, 2016 by blue89

[ajb]

... for your question above.

What questions?

 

 

(there might be some other (additional) requirements e.g. : continuoum or regaular continuoum ( as you know continuoum and regular continuoum are not same! ,define separate things..

I forget details here also, but for sure with any theorem you need to be careful with details. For classes of functions that do not have a Riemann integral then one has to do something else - if possible.

 

Anyway, as far as rmjohn.arbis is worried, as an introduction to the subject the Riemann integral is enough and would cover most of the functions he is likely to meet in real life.

[blue89]

What questions?

 

didn't you ask this question below(??)

You do not think the fundamental theorem of calculus is correct?

 

for this question.riemann integral is strong definition. but may not be enough to understand everything about integral properly ..

. For classes of functions that do not have a Riemann integral then one has to do something else - if possible.

 

here is something about fundamental theorem and just as I predicted it mentions about being regular continuous. http://mathworld.wolfram.com/FundamentalTheoremsofCalculus.html

Edited June 26, 2016 by blue89

[ajb]

 

for this question ...riemann integral is strong definition. but may not be enough to understand properly ..

 

Okay, right. The Riemann integral is enough for a large class of functions, and for sure enough for the kinds of functions one meets in 'introduction to calculus'. In this context, for functions that have a well defined Riemann integral, the proof of the second fundamental theorem of calculus using the Riemann integral is fine.

 

If you want to think about other classes of functions then one has to do more. I am not sure what the most general setting is.

[blue89]

 

Okay, right. The Riemann integral is enough for a large class of functions, and for sure enough for the kinds of functions one meets in 'introduction to calculus'. In this context, for functions that have a well defined Riemann integral, the proof of the second fundamental theorem of calculus using the Riemann integral is fine.

 

If you want to think about other classes of functions then one has to do more. I am not sure what the most general setting is.

 

don't be confused please , it is not so difficult to understand fundamental theorem. (I looked one of your paper and I thought this was more difficult than fundamental , really ,I do not believe that something is difficult already,I only would express that it will not be so worth spending our long times ,but it is generally required.)then , make it simple please , I remember omething for the second part of fundamental ,and I remember my hodja given this is relevant to continuoum too. care please that if any function is continuous on closed interval , then it will be regularly continuous.

look please this document at wikipedia is more comprehensive to understand : https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus .

may contain some faults in my opinion. or it is telling in different way.

but again really I request you to inform me ;

 

"are you asking something about a question that every details of its have not been understood (yet) ? " or

"would you like to speak something scientific only, relevant pure mathematics (having no question but being willing to speak scientific conversation) ?"

Edited June 26, 2016 by blue89

[ajb]

I am just saying that a proof of the second theorem exists based on Riemann integrals, at least for functions that have a well defined Riemann integral. I say no more and no less.

 

If I misunderstood you and thought that you were arguing against the theorem, then I am sorry.

Edited June 26, 2016 by ajb

[blue89]

I am just saying that a proof of the second theorem exists based on Riemann integrals, at least for functions that have a well defined Riemann integral. I say no more and no less.

 

If I misunderstood you and thought that you were arguing against the theorem, then I am sorry.

 

I think I have already explained relevant to statement given in bigger size. but please do not think that you are giving me unrest , I really do not feel matter anytime because of scientific conversations especially sometimes in mathematics there exist some details whiches seem unimportant ,but they are not unimportant.(if you have any doubts whethr fundamental theorem is correct or not, then I recommend that you check the related subjects repeatedly how required to best undertand) I also allege that in my opinion the details are the most important things in mathematics,I think we closely can/should use it as a modelling.or we should use its efficacy via studying cryptologic.but of course it will require to study hard and getting hard to take the best from our systematized results

[ajb]

if you have any doubts whethr fundamental theorem is correct or not, ...

We see to have some crossed wires. I have not expressed any doubt about the fundamental theorems of calculus.

Edited June 26, 2016 by ajb

[blue89]

I have not expressed any doubt about the fundamental theorems of calculus.

 

O.K. I accept this expresion that you have not had any problem with that theorem. I think I am expert of basic analysis. and just because of this,fundamental theorem is basic, I relied on again.

O.K. ajb

see you then.

Edited June 26, 2016 by blue89