Equality of integrals VS equality of integrands

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Discussion Overview

The discussion revolves around the implications of the equality of integrals versus the equality of integrands, specifically questioning whether the equality of two integrals over the interval from 0 to infinity implies that the functions being integrated are equal. The scope includes theoretical considerations, mathematical reasoning, and potential applications in calculus of variations.

Discussion Character

  • Debate/contested
  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • Some participants question whether $$\int_{t=0}^{\infty}f(t)dt=\int_{t=0}^{\infty}g(t)dt$$ implies $$f(t)=g(t)$$, suggesting that different functions can yield the same integral.
  • One participant provides examples of functions that have the same integral but are not equal, illustrating that the area under the curve can be the same for different functions.
  • Another participant discusses the injectivity of the mapping from integrable functions to their integrals, noting that a linear functional on an infinite-dimensional space is never injective.
  • Some participants propose that under certain conditions, such as continuity, one might be able to equate integrands if the integrals are equal over all intervals.
  • There is a suggestion that specific cases, such as those involving Gauss' Law, might allow for equating integrands under certain conditions.
  • Concerns are raised about the implications of setting integrals equal to zero and whether this leads to the conclusion that the functions must be zero everywhere.
  • Discussion includes the concept of functional derivatives in the context of optimization problems, with participants seeking clarification on the relationship between derivatives and integrals.
  • Some participants express uncertainty about the rigor of their statements and seek further resources or clarification on functional derivatives.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether the equality of integrals implies the equality of integrands. Multiple competing views remain, with some arguing for specific conditions under which equality might hold, while others provide counterexamples and express skepticism about the implications.

Contextual Notes

Limitations include the dependence on the continuity of functions and the nature of integrability, as well as the unresolved mathematical steps regarding the implications of equal integrals.

  • #31
AFAIK there are no problems up to here: $$\frac{d}{dx} \lim_{x \to \infty} \int_{0}^{x} \left[f(t) - g(t) \right] = 0$$ but yes I don't think you can generally interchange those limits to get ##\lim_{x \to \infty} \frac{d}{dx} \int_{0}^{x} \left[f(t) - g(t) \right] = 0##
 
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  • #32
Math_QED said:
so I'm not sure where your claim comes from.
People whom I know (in real life) learned by themselves, teachers only graded them.
 
  • #33
Adesh said:
People whom I know (in real life) learned by themselves, teachers only graded them.

My opinion is that most of mathematics is self-study. The professor is just there to motivate the concepts and give a first exposure, or to ask questions to.
 
  • #34
Math_QED said:
My opinion is that most of mathematics is self-study. The professor is just there to motivate the concepts and give a first exposure, or to ask questions to.
Then what does it mean when so many people say “Rudin is not good for self-study” ?
 
  • #35
Adesh said:
Then what does it mean when so many people say “Rudin is not good for self-study” ?

It means that it is a bad book to learn the material from, with which I agree. There are much better self-study books out there.
 
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  • #36
@Adesh I generally had very good professors in undergrad and learned a lot from their lectures. I can sometimes be a slow student though, so I generally had to re-work the parts I didn't follow afterwards, but this is all on me and certainly doesn't reflect negatively on my instructors.

I think Rudin is harder to self-study because it's terse and doesn't "hand-hold" the reader. This is much less of an obstacle when you have all the aspects of a class to help you along (other students, professors, teaching assistants, homeworks, etc.)

In the future, you might want to PM a user if you have a specific question for them, instead of posting on an unrelated thread.
 
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