What does the square of a differential mean?

In summary, the conversation discusses the calculation of potential energy of a spring and the unique properties of the integral used to calculate it. The integral's differential term is squared, which is not a common occurrence. The reasoning behind this and how to compute the integral are also discussed.
  • #1
Leo Liu
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Homework Statement
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Relevant Equations
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When I was following the calculations of finding the potential energy of a spring standing on a table under gravity, I encountered the integral shown below, where ##d\xi## is the compression of a tiny segment of the spring and ##k'## is the effective spring constant of that segment. The integral sums up the elastic potential energy of each segment along the spring. However, it is not a run of the mill integral in that its differential term is squared. From my experience, the square of a differential is 0, but in this case it obviously possesses some unique properties. Could someone explain why this makes sense? And how do you compute this integral?
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Origin: https://www.zhihu.com/question/324405110/answer/1860254582
In case you need it: https://www.deepl.com/translator
 
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  • #2
From your text
[tex]k'=A(dx)^{-1}[/tex] though I am not familiar at all with such a treatment of infinitesimal. Anyway
[tex]d\xi=B(dx)[/tex]
[tex]k'(d\xi^2)=C dx[/tex]
So
[tex]\int Dk'(d\xi^2)= \int E dx[/tex]
as we expect.
 
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  • #3
I don't think it's helpful to think of it as a square differential. Let's call u(x) the compression of a segment of length δx. Then
u(x) = λg(L0-x)δx/kL0
k' = kL0/δx
Evaluate 1/2 k'u(x)2, and THEN (only then) let δx → 0. That's when it becomes a differential.
 
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Related to What does the square of a differential mean?

1. What is the definition of a differential?

A differential is a mathematical concept that represents a small change in a variable or function. It is often denoted as "dx" and is used in calculus to calculate rates of change.

2. What does it mean to square a differential?

Squaring a differential means multiplying it by itself. In other words, it is the process of taking the differential and raising it to the power of 2.

3. How is the square of a differential used in mathematics?

The square of a differential is used in various mathematical applications, such as in finding the area under a curve or calculating the second derivative of a function. It is also used in physics to represent the change in position or velocity over time.

4. Can the square of a differential be negative?

Yes, the square of a differential can be negative. This can occur when the differential itself is a negative number, or when it is multiplied by a negative number. However, in some cases, the square of a differential may be restricted to positive values depending on the context in which it is being used.

5. What is the difference between the square of a differential and the differential of a square?

The square of a differential is the result of multiplying a differential by itself, while the differential of a square is the result of taking the derivative of a squared function. In other words, the square of a differential is a number, while the differential of a square is a function.

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