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if most integrals aren't integrable to and can be evaluated numerically to any degree of accuracy why learn these esoteric techniques at all?
Math isn't just about finding solutions to equations - when it comes to proving theorems computers are rather useless in many cases. And even when computers can help us prove theorems - it's only when there're to many special cases for a person to examine, the four color theorem for example.matt grime said:Why bother learning any maths if a computer can solve it all numerically...?
Be careful here. For example, it is often claimed that [tex]e^{-x^2}[/tex] is not integrable. That of course is not true. The integral of this function is very well-known:ice109 said:if most integrals aren't integrable to and can be evaluated numerically to any degree of accuracy why learn these esoteric techniques at all?
D H said:Loop, while that is one way of defining integrability, it obviously is not the sense meant by the original poster. What the OP really meant was (my changes to the OP are in italics):
"if most antiderivatives cannot be expressed in terms of elementary functions but can be evaluated numerically to any degree of accuracy why learn these esoteric symbolic integration techniques at all?"
D H said:That one has to resort to numerical techniques to solve a numerical problem is not limited to the special functions. What are the exact values of [itex]\surd 2[/itex] and [itex]\sin 1[/itex]? We have to use numerical techniques to evaluate [itex]\surd 2[/itex] and [itex]\sin 1[/itex], even though both the square root and sine functions are elementary functions.
Gib Z said:Why learn how to differentiate when taking a very close finite difference will do?
ice109 said:that's not a very good rebuke because i don't need to resort to finite differences unless I'm taking discrete data which most people don't do.
D H said:Loop, while that is one way of defining integrability, it obviously is not the sense meant by the original poster. What the OP really meant was (my changes to the OP are in italics):
"if most antiderivatives cannot be expressed in terms of elementary functions but can be evaluated numerically to any degree of accuracy why learn these esoteric symbolic integration techniques at all?"
Gib Z said:I don't get what you mean by having "to resort to" them, my point was that taking close finite differences can give us the numerical value for the derivative at that point to any degree of accuracy, just as numerical integration techniques do for integrals. I thought this parallel might have made the answer a tiny bit easier to see, but it obviously doesnt, my bad :(
My point was, exact answers are always nice =] And when we can't get an exact answer, call it something new.
i'm going to get lynched for this one but that again is a completely pointless thing I am my humble opinon. anyway then symbolic integration is a tool for research engineers and physicists not practicing engineers. meaning of course i had in mind all the phenomenology that goes on in physics when i asked this question hence i restricted it to engineering. i guesss i should've been more specific about what kind.dx said:Integration is not always about finding the numerical value of an integral. Just pick up any book on pure mathematics. You'll most likely find integrals on every page, but not one of them will be evaluated to give a numerical answer. In fact, you can even just look at any book on physics or engineering. Most results will involve symbolic integration as intermediate steps in the derivations.
ice109 said:i'm going to get lynched for this one but that again is a completely pointless thing I am my humble opinon. anyway then symbolic integration is a tool for research engineers and physicists not practicing engineers. meaning of course i had in mind all the phenomenology that goes on in physics when i asked this question hence i restricted it to engineering. i guesss i should've been more specific about what kind.
ice109 said:i'm going to get lynched for this one but that again is a completely pointless thing I am my humble opinon. anyway then symbolic integration is a tool for research engineers and physicists not practicing engineers. meaning of course i had in mind all the phenomenology that goes on in physics when i asked this question hence i restricted it to engineering. i guesss i should've been more specific about what kind.
dx said:You clearly have no idea what engineering is. Post this in the engineering forum and see what they say.
ice109 said:For what it's worth I'm a pure math student so I'm not some lazy dunce trying to argue their way out of learning abstract concepts.
daudaudaudau said:Say you have to evaluate this integral 100,000 times for various values of "a" (this is a completely realistic scenario if you are simulating something)
[tex]
\int_0^a \frac{1}{x^{0.999}}dx
[/tex]
which do you think is faster: Waiting for some numerical algorithm to converge, or to do it by hand?
Or what about a function that oscillates like crazy? It takes a while for that one to converge too.
DavidWhitbeck said:Actually in your case doing it by hand is still slower because you still have to evaluate the antiderivative 100,000 times. Simply use an algorithm that's quadrature based on rational functions (as opposed to polynomials) in C code and a home pc will beat you to the punchline. Even if you allow yourself the use of a scientific calculator when you evaluate that function 100,000 times!
daudaudaudau said:Obviously I was going to evaluate the antiderivative on a computer. I'm not doing simulations with pen and paper.
Gib Z said:Pure mathematics must aim to get exact answers in the most elementary form possible, without the aid of anything other than pen, paper, and ones mind.
daudaudaudau said:Say you have to evaluate this integral 100,000 times for various values of "a" (this is a completely realistic scenario if you are simulating something)
[tex]
\int_0^a \frac{1}{x^{0.999}}dx
[/tex]
which do you think is faster: Waiting for some numerical algorithm to converge, or to do it by hand?
Or what about a function that oscillates like crazy? It takes a while for that one to converge too.
Gib Z said:Pure mathematics must aim to get exact answers in the most elementary form possible, without the aid of anything other than pen, paper, and ones mind.
Doesn't the four colour theorem fall into such a category?Eidos said:Sure I understand that numerically, computers can do stuff we can't. This question is more aimed at asking if computers have discovered anything new. As in we posed it a mathematical question which we did not know the answer too, it gave us an answer which was later verified to be correct.
Gib Z said:Pure mathematics must aim to get exact answers in the most elementary form possible, without the aid of anything other than pen, paper, and ones mind.
jimvoit said:The same question of "utility" has been directed to the teaching of Euclidean geometry in high school. I think the utiliterians prevailed on this one and computer assisted learing products such as The Geometers Sketchpad have prevailed to the exclusion of the hard cold logic necessary to follow a set of axioms to a correct conclusion.
I'm not an educator so maybe someone in the education community can amplify these comments.
ice109 said:yes that was a mistake on my part.
but calculus classes are for engineers, why do they need to learn these techniques? I'm not an engineering student i don't know but i would guess their integrals are simple? even if they are why waste their time when everyone these days has access to some way of evaluating them numerically.
te integral and derivative are very important, it's the rigamarole i don't see the necessity of.