Evaluating multi-dimensional integrals

[loom91]

Hi,

I was wondering how I can evaluate various types of multi-dimensional integrals directly. In this context, I'll also be very grateful if you could provide me with links to some free online resource for an application-based approach to multi-variate calculus. I don't want to learn proofs or rigorous technical definitions, just an applicative approach that will help me 'do' the problems in physics. Thank you.

Molu

 

[arildno]

What do you mean by "directly"?

 

[HallsofIvy]

Unfortunately there are no "algorithms" that will apply to all possible problems, even those in "problems in physics". Deciding how to solve some problems will require thinking which typically requires knowing "technical definitions". What you've written sounds an awful lot like "give me a formula so I won't have to think".

 

[loom91]

That's not what I meant. I meant that I will like to learn only as much as is required for solving physical problems. Much of calculus is pure mathematics which is not used in Physics (at least not until the next fancy theory comes along with its own bag of obscure mathematical techniques), I'm not interested in learning that part. Are there any good resources on the net?

 

[loom91]

By directly I mean some method (possibly laborious and time-consuming) that can compute any Multi-D integrals, without making use of tricks like Fubini's theorem.

 

[pivoxa15]

I use to think along those lines but than I learned that the only way is to do lots of maths problems yourself. Once you've done that, you will realize that tricks like Fubini's theorem is very handy and good. And physics becomes much easier as a result of your fluency in the mathematics.

 

[loom91]

I know Fubini's theorem is very useful. What I want to know is a method to compute the integral of an arbitrary p-form over an arbitrary subset of R^p.

 

[HallsofIvy]

I know Fubini's theorem is very useful. What I want to know is a method to compute the integral of an arbitrary p-form over an arbitrary subset of R^p.

There is no such "method". Physics, like mathematics, requires thinking!

That's not what I meant. I meant that I will like to learn only as much as is required for solving physical problems. Much of calculus is pure mathematics which is not used in Physics (at least not until the next fancy theory comes along with its own bag of obscure mathematical techniques), I'm not interested in learning that part.

What you are saying is that you want to be able to do physics problems without having to learn the physics.

 

[loom91]

There is no such "method". Physics, like mathematics, requires thinking!

It seems unlikely to me that there is no such method. An arbitrary 1-form can be integrated over an arbitrary measurable subset of R using the brute force method of coming from definition (taking the elimit of the summation). Why shouldn't there be a similar procedure for higher-dimensional integrals?

What you are saying is that you want to be able to do physics problems without having to learn the physics.

I fail to understand why you insist on taking such an approach even though I implied nothing of the sort. What I said is that I will like to learn Physics, and only that much mathematics as is required to learn it.

 

[loom91]

So, isn't there any method of evaluating the integral of an arbitrary p-form over a p-dimensional measurable subset of R^p?

 

[lance]

That's not what I meant. I meant that I will like to learn only as much as is required for solving physical problems. Much of calculus is pure mathematics which is not used in Physics (at least not until the next fancy theory comes along with its own bag of obscure mathematical techniques), I'm not interested in learning that part. Are there any good resources on the net?

I hate to break it to you, but calculus came into existence for the express purpose of use in physics. if you don't understand parts of calculus, it just means you haven't done enough physics. I would suggest you take a multivariable calc course, because it is the first calc course in our dismal education system that usually really links the calculus to the reason it was invented (solving physical problems).

 

[loom91]

I hate to break it to you, but calculus came into existence for the express purpose of use in physics. if you don't understand parts of calculus, it just means you haven't done enough physics. I would suggest you take a multivariable calc course, because it is the first calc course in our dismal education system that usually really links the calculus to the reason it was invented (solving physical problems).

You mean every thing in calculus has uses in physics?

 

[HallsofIvy]

Just about everything in mathematics has uses in physics: even abstract group theory is used in quantum physics. I would suggest that you stop looking for the easiest possible way to learn physics until you actually know a little physics.

 

[eljose]

I heard about "Montecarlo Integration2 to handle with those integrals over R^{N} with N usually a "big" number..these integrals happen in Quantum-Path integral formulation of Physics..the problem is that Montecarlo method is finite and can Handle the problem of integrating a function $$f(x_0 ,x_1 ,x_2 ,....,x_N)$$ when you take the limit N--->oo