Handling dx/dt as an ordinary fraction?

[jafisika]

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

 

[mathman]

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).

 

[JJacquelin]

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

 

[granpa]

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

 

[HallsofIvy]

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.

 

[granpa]

no. I think the focus should be on the meaning of the formulas not the formulas themselves.
the meaning of 'd' is Δ.

 

[Hurkyl]

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

$$dy = \frac{dy}{dx} dx.$$​

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.