- #1
cubic
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Ok so what I want to know is, is this valid? If so what does it mean?
Infinitesimals as interval limits in integration
- Context: Graduate
- Thread starter cubic
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Discussion Overview
The discussion revolves around the validity and implications of using infinitesimals in integration, particularly in the context of calculating work done by a force over an infinitesimal distance. Participants explore the mathematical rigor behind this concept and the appropriate notation to use.
Discussion Character
- Exploratory, Technical explanation, Debate/contested
Main Points Raised
- One participant questions the validity of using infinitesimals and seeks clarification on its meaning.
- Another participant suggests that the notation may imply a function evaluated at zero multiplied by an infinitesimal distance.
- A participant explains the derivation of work done by a force and expresses a need to prove that the variability of the force does not affect the outcome over an infinitesimal distance.
- There is a suggestion that an integral should be defined with an upper limit of dx to represent the area of the first strip.
- One participant concludes that the area of the first strip can be calculated as the height (initial force value) multiplied by the width (dx).
- Several participants engage in a discussion about the appropriate notation, with some advocating for the use of Δx to represent a small non-zero width instead of dx.
- Another participant emphasizes that dx is specifically the width of each infinitesimal strip, while Δx refers to a finite width, which does not apply to the current context.
- There is a reiteration that dx should not be used to describe the width of a strip, with distinctions made between the widths of individual strips and finite sums of widths.
Areas of Agreement / Disagreement
Participants express differing views on the appropriate notation for infinitesimals, with some advocating for Δx while others maintain that dx is suitable for the context. The discussion remains unresolved regarding the best approach to rigorously prove the initial claims about work done.
Contextual Notes
There is a lack of consensus on the definitions and implications of using dx versus Δx, as well as on the mathematical rigor required to support the claims made about infinitesimals in integration.
- #2
mathman
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The notation is unusual. It might simply mean F(0)dx.
- #3
cubic
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The integral comes from finding the work done by a force over a distance dx. The force may or may not be variable so I needed to prove the variableness of F did not matter over an infinitesimal for the purposes of determining the amount of work done.
So dw = F⋅dx => F0⋅dx
So integrating over the distance dx would prove this.
I'm doing a proof that I'd like to make rigorous. This makes sense to me visually - we are simply looking at the area of the first strip of dx - which would be the initial value of F. However I cannot find a way to prove it mathematically.
So dw = F⋅dx => F0⋅dx
So integrating over the distance dx would prove this.
I'm doing a proof that I'd like to make rigorous. This makes sense to me visually - we are simply looking at the area of the first strip of dx - which would be the initial value of F. However I cannot find a way to prove it mathematically.
Last edited:
- #4
mathman
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You first need to define an integral with an upper limit dx.
- #5
cubic
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An upper limit of dx would be the area of the first strip of dx. I need to prove the above.
- #6
cubic
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Doh. The area of that first strip would be the height (F0) by the width (dx.) Ok problem solved.
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- #7
mathman
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Notation quibble: dx is used for the differential. If you are trying to describe a small non-zero width, use Δx.
- #8
cubic
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It has to be along a distance dx.
- #9
Mark44
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Like mathman said, use Δx. dx has its own meaning. Δx can represent some small distance along the x-axis.cubic said:
- #10
cubic
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dx is the width of each strip. dx would then be the width of the first strip. Δx is the width of a finite number of strips, which is not what I am looking for.
- #11
pwsnafu
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cubic said:
Thirding what the others said: the width of the first strip is ##\Delta x_0##, the width of the second strip is ##\Delta x_1## etc. The width of a finite number of strips is ##\Sigma \Delta x_n##.
dx is not used for the width of a strip.
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