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

[Pjpic]

How can something have a definitive edge if space can always be more granular?

 

[RUber]

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.

 

[Pjpic]

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.

 

[jbriggs444]

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?

 

[Pjpic]

Seems like calculus is as close as one can get to an answer.

 

[jbriggs444]

Seems like calculus is as close as one can get to an answer.

What is the question?