Dx in an integral vs. differential forms

In summary, the conversation discusses the use of "dx" in integrals and derivatives. It is clarified that "dx" should never stand alone, but rather be accompanied by some other quantity such as "dy" or "df(x)". The full definitions of the derivative and integral are provided, which involve limits. The concept of infinitesimals is also briefly mentioned as a possible extension of calculus.
  • #1
Trying2Learn
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TL;DR Summary
what is the dx in an integral
Good Morning

To cut the chase, what is the dx in an integral?

I understand that d/dx is an "operator" on a function; and that one should never split, say, df, from dx in df/dx

That said, I have seen it in an integral, specifically for calculating work.

I do understand the idea of "differential forms." However, I am trying to understand if there is a difference between how dx is used in an integral (effectively splitting out the denominator in the "operator" of d/dx).

I do understand that in calculating "work" in physics, we integrate the force over dx. In that case, we realize that force
is a vector and dx is a co-vector.

So, in a nutshell, i can see "through a dark glass" (so to speak) that something is going on here.

However, with that background, I hope I can rephrase my initial question as

"What is the difference between the "dx" in, say the integral leading to work done, vs. the dx in the denominator of the operator (which, should NEVER, in theory, stand alone).

I am not sure I am asking this properly. Please forgive me for the poor wording -- that is half the problem I face.
 
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  • #2
My calc teacher said that the last infinitesimal in the infinite series was "treated as zero" and consequently discarded ##-## he didn't say that it was existentially nonexistent ##-## the Physics Forums aren't where to go on the net for discussions of Philosophy of Mathematics ##\dots##
 
  • #3
Trying2Learn said:
Summary:: what is the dx in an integral

and that one should never split, say, df, from dx in df/dx
That is not correct, we can split them because they are actually stand alone quantities. You should read about how the differential of a function is defined. In short it is defined as ##df=f'(x)dx##
In the Leibniz formalism the derivative is taken to be the quotient of two differentials: The differential of the function f, which is symbolized as ##df## and the differential of the identity function ##i(x)=x## which is ##di=dx##
So to answer directly your question, the dx in the integral and the dx in the derivative are the exact same thing:it is the differential of the identity function.
 
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  • #4
sysprog said:
My calc teacher said that the last infinitesimal in the infinite series was "treated as zero" and consequently discarded ##-## he didn't say that it was existentially nonexistent ##-## the Physics Forums aren't where to go on the net for discussions of Philosophy of Mathematics ##\dots##
OK, I deleted the word "philosophical."
 
  • #5
There is no big difference. In neither case does the 'dx' stand alone. In the differential case it has to keep company with the dy or df(x) in the numerator of the fraction, in order to have a meaning. In the integral case it has to keep company with the integral sign in order to have a meaning. That's because in both case a limit is implied. Here are the full statements, which hold for reasonably well-behaved functions f(x).

$$\frac{df(x)}{dx} \triangleq \lim_{dx\to 0} \left(\frac{f(x+dx)-f(x)}{dx}\right)$$

$$\int_a^b f(x)\,dx \triangleq \lim_{dx\to 0} \sum_{k=0}^{\lfloor \frac{b-a}{dx} \rfloor} f(a+k\,dx)\,dx$$

where ##\lfloor u \rfloor## denotes the greatest integer not exceeding ##u##.
 
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  • #6
andrewkirk said:
There is no big difference. In neither case does the 'dx' stand alone. In the differential case it has to keep company with the dy or df(x) in the numerator of the fraction, in order to have a meaning. In the integral case it has to keep company with the integral sign in order to have a meaning. That's because in both case a limit is implied. Here are the full statements, which hold for reasonably well-behaved functions f(x).

$$\frac{df(x)}{dx} \triangleq \lim_{dx\to 0} \left(\frac{f(x+dx)-f(x)}{dx}\right)$$

$$\int_a^b f(x)\,dx \triangleq \lim_{dx\to 0} \sum_{k=0}^{\lfloor \frac{b-a}{dx} \rfloor} f(a+k\,dx)\,dx$$

where ##\lfloor u \rfloor## denotes the greatest integer not exceeding ##u##.
Ah... thank you! That was what I was looking for!
 
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  • #7
Just as a fun fact, you can apparently actually extend calculus with 'infinitesimals', your dx would here then be an 'infinitesimal' if I understand correctly...(?) That is loosly defined a number smaller than any other number but not zero. This now can be used as a stand alone number (but not like any other number, special rules apply). Look e.g. here:
https://en.wikipedia.org/wiki/Nonstandard_calculus
https://www.math.wisc.edu/~keisler/calc.htmlI'm not a mathematician, so I would love to know if others have experience with this.
 
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Related to Dx in an integral vs. differential forms

1. What is the difference between integral and differential forms in Dx?

The main difference between integral and differential forms in Dx is the way in which the quantity is represented. In integral form, the quantity is represented as a single value, while in differential form, it is represented as a function of variables. This means that integral forms are used to find the total value of a quantity, while differential forms are used to analyze the rate of change of a quantity.

2. How are integral and differential forms related in Dx?

Integral and differential forms are closely related in Dx as they both involve the concept of integration. Integral forms use definite integrals to find the total value of a quantity, while differential forms use indefinite integrals to analyze the rate of change of a quantity. In some cases, integral forms can be converted into differential forms and vice versa using certain mathematical techniques.

3. What are the advantages of using integral forms in Dx?

One of the main advantages of using integral forms in Dx is that they provide a straightforward way to find the total value of a quantity. This is especially useful when dealing with physical quantities such as distance, area, or volume. Integral forms also allow for easier calculations and can be applied to a wide range of problems in various fields of science and engineering.

4. When should differential forms be used in Dx?

Differential forms are typically used in Dx when analyzing the rate of change of a quantity. This is particularly useful in situations where the quantity is constantly changing, such as in physics and engineering problems involving motion or fluid flow. Differential forms also allow for a more detailed understanding of the behavior of a quantity over time.

5. How can I determine which form to use in a Dx problem?

The choice between integral and differential forms in Dx depends on the specific problem and what information is needed. If the problem involves finding the total value of a quantity, integral forms would be more appropriate. If the problem involves analyzing the rate of change of a quantity, differential forms would be more suitable. It is important to carefully read and understand the problem to determine which form would be most effective in solving it.

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