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**1. Homework Statement**

I hate infinitesimals and differentials.

When I learned calculus, we used Liebniz notation df/dx only as a convenience for using the chain rule. In physics, apparently, people just play around with differentials and infinitesimals and expect to get the right answer. Physicists think that differentials are like regular numbers and you can just add them and multiply them and pretend they are meaningful outside of Liebniz notation. Even worse is the use of these "infinitesimal" quantities, which don't even have Liebniz notation to justify their use. Physicists simply made up these objects and go around pretending they are real mathematical objects that you can use in equations. THIS IS INSANE.

In thermodynamics and statistical mechanics, for example, many of the laws and relations are formulated in terms of differentials and infinitesimals. You can write down ridiculous things like

[tex]dU=\delta Q-\delta w [/tex]

and

[tex]\mathrm{d}U = \delta Q - \delta W+\sum_i \mu_i\,\mathrm{d}N_i[/tex]

and pretend they make sense.

In classical mechanics, you "define," the angular velocity vector as

[tex]\vec{\omega} = \frac{\delta \vec{\theta}}{\delta t} [/tex]

That is probably the second most absurd equation I have ever seen. Apparently, no one who writes classical mechanics or statistical mechanics books understands the concept of something being "well-defined." When you define a derivative, you DO NOT JUST SAY "its just the ratio of the infinitesimal change of f to the infinitesimal change in x," you define it as a limit of a quotient AND YOU PROVE IT EXISTS. I have never seen any proof that [tex]\vec{\omega} [/tex] exists and therefore I think all of classical mechanics that uses [itex] \omega[/tex] is downright wrong and should be reformulated with mathematical objects that actually exist.

BTW, I am taking differential geometry next year. Is that where I learn these things?

EDIT: I am being somewhat sarcastic, but this has been constantly bothering me for a long time

**2. Homework Equations**

**3. The Attempt at a Solution**

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