How do the concepts of "boundary"and"infinitesimal" interact

In summary, space can always be more granular, so something with an edge would have an impact that is negligible in most cases.
  • #1
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How can something have a definitive edge if space can always be more granular?
 
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  • #2
It is all a question of scale. For example, if you are concerned with how a ball of radius r bounces off a sidewalk that is more or less flat, with perturbations up to +/- 1/r^10, those small perturbations will not likely impact the trajectory of the bouncing ball very much. If you are looking to get to that level of precision, you would probably also need to account for other physical influences as well. If you want to account for all the additional factors, you can work through the math to see what the maximum impact would be. In most cases, it is negligible.

Based on the scale of your problem and desired precision, you can assume away most of those questions of granularity. Clearly, if you are working with lasers in the optical frequencies, your definition of smooth will be very different from that of someone looking into bouncing a soccer ball on concrete.
 
  • #3
RUber said:
It is all a question of scale. For example, if you are concerned with how a ball of radius r bounces off a sidewalk that is more or less flat, with perturbations up to +/- 1/r^10, those small perturbations will not likely impact the trajectory of the bouncing ball very much. If you are looking to get to that level of precision, you would probably also need to account for other physical influences as well. If you want to account for all the additional factors, you can work through the math to see what the maximum impact would be. In most cases, it is negligible.

Based on the scale of your problem and desired precision, you can assume away most of those questions of granularity. Clearly, if you are working with lasers in the optical frequencies, your definition of smooth will be very different from that of someone looking into bouncing a soccer ball on concrete.
If you wanted to get down to the nth degree; it seems something must have a boundary but, on the other hand, it can't have a boundary because space can always be more granular.
 
  • #4
Why must one have a boundary that is perfectly sharp in order to have a boundary at all? Where, exactly, is the boundary between the trunk of a tree and its roots?
 
  • #5
Seems like calculus is as close as one can get to an answer.
 
  • #6
Pjpic said:
Seems like calculus is as close as one can get to an answer.
What is the question?
 

1. What is the definition of a boundary in mathematics?

A boundary is a mathematical concept that refers to the edge or limit of a set or region. It can also be thought of as the point of separation between two different sets or regions.

2. How does the concept of boundary relate to the concept of infinitesimal?

The concept of boundary and infinitesimal are closely related in mathematics. Infinitesimal quantities are considered to be infinitely small and are often used to describe the behavior of a function or set near its boundary.

3. Can an infinitesimal be a boundary?

No, an infinitesimal cannot be a boundary. Boundaries are defined as points or edges, while infinitesimals are infinitely small quantities. However, infinitesimals can play a role in describing the behavior of a function or set near its boundary.

4. How do boundaries and infinitesimals impact calculus?

Boundaries and infinitesimals are essential concepts in calculus. Infinitesimals are used to define derivatives and integrals, which are fundamental tools in calculus. Boundaries are also important in determining the behavior of a function or set at a specific point.

5. Can boundaries and infinitesimals be applied to real-world scenarios?

Yes, boundaries and infinitesimals have many real-world applications, especially in fields such as physics and engineering. For example, the concept of an infinitesimal is used in the study of motion, and boundaries are used in defining the limits of a system or object.

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