- #1

DrBanana

- 47

- 3

- TL;DR Summary
- Trying to learn about the right path to understanding 'differentials'

So I've been searching around for rigorous explanations for things like ##dx## in physics, I'm not looking to fully commit myself to reading the relevant literature at the moment but just want to know what I'll have to do in order to understand. Perhaps I'll make a separate thread about that.

In most introductory physics textbooks, the method for calculating the moment of inertia of a rigid body is like this: you consider it as a collection of really small particles, so the approximation for the moment of inertia is ##I = \sum m_i r_i ^2 ##

As the number of particles go to infinity, and consequently each one's mass goes to zero, ##I=\int r^2 dm##.

So, some questions:

1. This isn't limited to moment of inertia but other derivations in physics in general: I was only taught to thing of integrals as areas under curves, but what about the cases (like in moment of inertia) where we are adding up small things, and not directly computing the area under some curve? What is the rigorous way to do this?

2. In a single variable calculus textbook, the 'x' in ##\int f dx## just refers to the integral's variable. However in the above integral expression that can't be the case as the radial distance r doesn't directly depend on m. To reconcile this, I found this answer on stackexchange: https://physics.stackexchange.com/a/550227/259006

So my question for this point is, if I confronted the author of a single variable calculus textbook (e.g. Courant), what would he have to say? That is, is it possible to give a satisfactory meaning to ##\int r^2 dm## using what one learns in standard single variable calculus (no hyper reals, no differential forms etc), or does one have to learn advanced methods (by the way Courant

3. On wikipedia I find this expression for the moment of inertia:
Relevant section: https://en.wikipedia.org/wiki/Moment_of_inertia#Motion_in_a_fixed_plane

Even though I don't know what triple integrals are yet, I can get a feel if this and it's more satisfying than simply ##\int r^2 dm##. But, where does it come from? I don't know of any book that expresses it like this.

P.S. I now realise that there is a second part to Courant's book that might deal with stuff like this, but I would still like to know the answers to my questions.

In most introductory physics textbooks, the method for calculating the moment of inertia of a rigid body is like this: you consider it as a collection of really small particles, so the approximation for the moment of inertia is ##I = \sum m_i r_i ^2 ##

As the number of particles go to infinity, and consequently each one's mass goes to zero, ##I=\int r^2 dm##.

So, some questions:

1. This isn't limited to moment of inertia but other derivations in physics in general: I was only taught to thing of integrals as areas under curves, but what about the cases (like in moment of inertia) where we are adding up small things, and not directly computing the area under some curve? What is the rigorous way to do this?

2. In a single variable calculus textbook, the 'x' in ##\int f dx## just refers to the integral's variable. However in the above integral expression that can't be the case as the radial distance r doesn't directly depend on m. To reconcile this, I found this answer on stackexchange: https://physics.stackexchange.com/a/550227/259006

So my question for this point is, if I confronted the author of a single variable calculus textbook (e.g. Courant), what would he have to say? That is, is it possible to give a satisfactory meaning to ##\int r^2 dm## using what one learns in standard single variable calculus (no hyper reals, no differential forms etc), or does one have to learn advanced methods (by the way Courant

*does*actually calculate moment of inertia in his book but only in a special case where the mass is confined to a plane region bounded by a curve).3. On wikipedia I find this expression for the moment of inertia:

Even though I don't know what triple integrals are yet, I can get a feel if this and it's more satisfying than simply ##\int r^2 dm##. But, where does it come from? I don't know of any book that expresses it like this.

P.S. I now realise that there is a second part to Courant's book that might deal with stuff like this, but I would still like to know the answers to my questions.