How to solve this without infinitesimals

In summary, the conversation discusses using calculus without infinitesimals to solve a physics problem involving a uniform charge distribution. The main issue is whether the differential, dE, is a width or a rate of change. The general answer to solving this problem without infinitesimals is to set up an approximating Riemann Sum and take a limit. Some believe that physicists are being lazy and abusing notation, but it still gets the job done.
  • #1
jaydnul
558
15
I'm trying to find a way to use calculus without infinitesimals and I'm stuck on this physics problem.

It's a uniform charge distribution question. Basically a half circle with radius [itex]r[/itex] and you have to find the electric field at a point that is along its x-axis. The [itex]E_y[/itex] component will be 0 because of symmetry. So all you need is is [itex]E_x[/itex]. The equation ends up being:
[tex]E_x=∫dEcosθ[/tex]
The only way I know how to solve this is using infinitesimals and finding one in terms of the other, like this:
[tex]dE=k\frac{dQ}{r^2}[/tex]
[tex]dQ=\frac{Q}{πr}dy[/tex]
[tex]dy=rdθ[/tex]
Then substituting those in all the way up so I have my integral in terms of θ.

So how would I go about solving this without using infinitesimals? Thanks
 
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  • #2
This is always difficult to explain.

##E_x = \int cos(\theta(E)) dE##.

On the one hand, ##E_x## is the area under ##y = cos(\theta(E))##, a Riemann sum of rectangles having width ##dE##. This seems to imply that ##dE## is a width, an infinitesimal width.

On the other hand, we have a rate of change ##dE## and a related rate of change ##cos(\theta(E)) dE## which when integrated, gives ##E_x##. The rate of change of ##E_x## is ##cos(\theta(E)) dE##, it is related to ##dE##, the rate of change of ##E##.

Which is it? Is ##dE## a width or is it a rate? This is a little like asking what is a number. One person will say, the thing you count with, another will say, an equivalence class of sets by cardinality, another will say, a Church numeral, etc, etc. There is no answer to that question. In the same way, it doesn't matter what ##dE## is, what matters is how differentials are used.

Of course, there is also the view that ##dE## is a finite width and ##cos(\theta(E))## is the gradient of the tangent to ##E_x## at ##E##, so ##cos(\theta(E)) dE## is the ##E_x## offset of the tangent at distance ##dE## from ##E##.
 
  • #3
Jd0g33 said:
So how would I go about solving this without using infinitesimals? Thanks

I think the general answer to this question (as it pertains to setting up integrals) is "Set up an approximating Riemann Sum and take a limit." Most of the time it's just a matter of changing all of the [itex]d[/itex]s to [itex]\Delta[/itex]s. In my experience, the physicists aren't using infinitesimals so much as they are being a bit lazy with limits and abusing notation. It gets the job done, though. So more power to 'em.
 

1. Can you explain what infinitesimals are and why they are problematic?

Infinitesimals are mathematical objects that represent quantities that are infinitely small. They are problematic because they do not fit into the traditional framework of calculus, leading to inconsistencies and paradoxes.

2. What are some alternative methods for solving problems without using infinitesimals?

Some alternative methods include using limits, the epsilon-delta approach, and the method of exhaustion. These methods involve breaking down a problem into smaller, more manageable parts and using algebraic or geometric techniques to find a solution.

3. Is it possible to solve all problems without using infinitesimals?

Yes, it is possible to solve all problems without using infinitesimals. Infinitesimals are not necessary for solving mathematical problems, and there are alternative methods that can be used.

4. Are there any advantages to solving problems without infinitesimals?

Yes, there are several advantages to solving problems without infinitesimals. It allows for a more rigorous and consistent approach to calculus, and it also helps to avoid paradoxes and inconsistencies that can arise when using infinitesimals.

5. How can I improve my understanding of solving problems without using infinitesimals?

To improve your understanding, you can practice using alternative methods, attend workshops or seminars on the topic, and consult with other mathematicians or scientists who have experience in solving problems without infinitesimals. You can also read textbooks or articles that discuss these methods in detail.

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