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

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The discussion centers on the interaction between the concepts of "boundary" and "infinitesimal," emphasizing that the perception of boundaries is highly dependent on the scale of the problem being analyzed. When considering physical phenomena, such as a ball bouncing on a surface, small perturbations may be negligible unless extreme precision is required. The need for a definitive boundary is questioned, as it seems contradictory to the idea of infinite granularity in space. Calculus is suggested as a tool to approach these complexities, providing a means to understand boundaries in a more nuanced way. Ultimately, the relationship between boundaries and infinitesimals is framed as a matter of perspective and context.
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How can something have a definitive edge if space can always be more granular?
 
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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.
 
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.
 
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?
 
Seems like calculus is as close as one can get to an answer.
 
Pjpic said:
Seems like calculus is as close as one can get to an answer.
What is the question?
 

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