Handling dx/dt as an ordinary fraction?

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In summary, the discussion revolves around the use of differential fractions in deriving formulas, particularly in physics and mathematics. While there has been controversy over treating differential fractions as ordinary fractions, in physics it is generally acceptable to do so. The focus should be on understanding the meaning of the formulas rather than the formulas themselves. Differential forms play a role in this discussion and are generally not proportional unless there is only one independent variable. An example is given to illustrate the potential issues that can arise when using differential fractions.
  • #1
jafisika
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I'm a South African undergraduate physics student.

In our textbook (Fundamentals of physics, John Wiley and sons, 2008) many formulas are derived by treating a differential fraction (like dq/dt) as an ordinary fraction -see example below. Can this differential fraction in all cases be treated as a normal fraction. If not in all cases, when can it be treated as such?
Example of deriving a formula from Fundamentals of physics (2008): Note: E is the emf of an ideal battery and i is the current through the battery.
dW=Edq=Eidt
From conservation of energy:
Eidt= (i^2)Rdt, which gives
E=iR and thus i=E/R
 
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  • #2
It doesn't look like it is being treated as an "ordinary fraction". You have in general f(x)dx=g(x)dx => f(x)=g(x).
 
  • #3
For a function y(x), the symbolisme dy/dx has a well defined meaning. But considering it as a quotient of infinitisimal values and treating it as an ordinary fraction was the subject of a long controversy.
In pure mathematics, the question is now overcome in the context of Non Standard Analysis, which provides a satisfactory justification.
In Physics, one can use it as an ordinary fraction without reservation : The models of physical phenomena involve more concret concepts.
About this controversial subject : A paper for general public "Une querelle des Anciens et des Modernes" (For French readers only. Sorry, presently there is no available translation)
http://www.scribd.com/JJacquelin/documents
 
  • #5
granpa said:
this is why i don't like the way calculus is taught.
the focus on slopes is confusing.
d in calculus is just the limit of Δ

http://en.wikipedia.org/w/index.php...s_for_calculus_of_finite_difference_operators
Do you think then that students should be taught non-standard analysis from the beginning? That would require either a long preliminary course on symbolic logic and model theory that or telling them "these are the formulas, don't ask where they came from"! I don't like either alternative.
 
  • #6
no. I think the focus should be on the meaning of the formulas not the formulas themselves.
the meaning of 'd' is Δ.
 
  • #7
HallsofIvy said:
Do you think then that students should be taught non-standard analysis from the beginning? That would require either a long preliminary course on symbolic logic and model theory that or telling them "these are the formulas, don't ask where they came from"! I don't like either alternative.
Students aren't taught a long preliminary course on mathematical foundations before they are first introduced to standard calculus. That would remain true if they were introduced to calculus the non-standard way, as in Keisler's book.



I too don't like the structure of the calculus curriculum, thinking differential forms should be made explicit (though presented in an introductory fashion) rather than lurking just behind the scenes in the introductory courses.

As to the opening poster's question, differential forms are generally not proportional. There generally does not exist any function f with the property that dy = f dx. Their algebra is a lot like that of vectors (in fact, they are fields vectors in a suitable abstract sense). But when there is only one (differentiably) independent variable, forms will be proportional, and it will just so happen the function we call dy/dx satisfies the equation
[tex]dy = \frac{dy}{dx} dx.[/tex]​
Of course, I imagine the notation for differential forms was chosen precisely so that suggestive equation would hold true.


Incidentally if f is continuous and y is differentiable, the equation
f dy = 0
implies that, if you choose any "point", at least one of the following is true:
  • dy is zero at that point. (what such a thing means is beyond the scope of this post)
  • f is zero at that point

If dx never vanishes (which is true if it is, in a suitable sense, an "independent variable" from which we are defining other things)
  • f(x)dx=g(x)dx
  • (f(x) - g(x))dx = 0
  • f(x) - g(x) is locally zero everywhere
  • f(x) - g(x) = 0
  • f(x) = g(x)


An example of what goes wrong is if we have:
y = x2 whenever x is positive
y = 0 whenever x is negative
f(x) = 0 whenever x is positive
f(x) = x2 whenever x is negative.​
In this case, we have f(x) dy = 0. However, neither dy=0 nor f(x)=0 is true. Of course, at particular points, one of them is true, and both of them are when x=0.
 
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Related to Handling dx/dt as an ordinary fraction?

1. How do you handle dx/dt as an ordinary fraction?

To handle dx/dt as an ordinary fraction, you need to use a technique called separation of variables. This involves isolating dx on one side of the equation and dt on the other side, and then integrating both sides to solve for x.

2. Can you explain why dx/dt is considered an ordinary fraction?

dx/dt is considered an ordinary fraction because it represents the change in x with respect to the change in t. This is similar to a regular fraction, where the numerator represents the change in the quantity and the denominator represents the change in the unit.

3. What are the key differences between handling dx/dt as an ordinary fraction and a regular fraction?

The key differences between handling dx/dt as an ordinary fraction and a regular fraction are the units and the concept of infinitesimals. In a regular fraction, the units of the numerator and denominator must match, whereas in dx/dt, the units of x and t may be different. Additionally, dx and dt represent infinitesimal changes, meaning that they are infinitely small and cannot be measured.

4. Are there any limitations to handling dx/dt as an ordinary fraction?

Yes, there are limitations to handling dx/dt as an ordinary fraction. This method is only applicable to certain types of equations, such as separable differential equations. It also assumes that the infinitesimal changes in x and t are continuous and not discrete.

5. How is handling dx/dt as an ordinary fraction useful in scientific research?

Handling dx/dt as an ordinary fraction is useful in scientific research because it allows us to analyze and understand the relationship between two variables that are continuously changing. This can help us make predictions, model complex systems, and solve real-world problems in fields such as physics, biology, and engineering.

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