Symbolic manipulation inside integral

[RustyDoorknobs]

I'm an undergrad who has just completed the standard calculus sequence (1, 2, and multivariable). I've done well in the courses, however, things like the following, which is a derivation of kinetic energy, still confuse me:

Taken from here.

I want to understand the symbolic manipulation that often occurs when making meaningful integrations. I was taught that the ending 'dx' term simply signifies the variable to be integrated over. However, it is commonly used, for example, as a term to cancel things out. In general, I see a lot of symbolic manipulation with differential elements that I want to understand. Could you recommend something I could read to better understand this stuff?

Thank you.

 

[Matterwave]

These manipulations are basically the physicist's way to do it. A mathematician would perhaps chide you for treating the differential for something it's not technically meant to do.

In those expressions, think of the d's as deltas, so you are dealing with a finite change and then the derivatives give you averages. Then hopefully at the end of the calculation you can take a smooth limit and make the expression make sense. :)

 

[RustyDoorknobs]

These manipulations are basically the physicist's way to do it. A mathematician would perhaps chide you for treating the differential for something it's not technically meant to do.

In those expressions, think of the d's as deltas, so you are dealing with a finite change and then the derivatives give you averages. Then hopefully at the end of the calculation you can take a smooth limit and make the expression make sense. :)

Ok, I'm understanding more.

Could you explain how the dt's cancel out when dp/dt * v dt becomes dp * v? Are the differentials considered as numbers in these manipulations?

Thanks for your reply.

 

[Matterwave]

Ok, I'm understanding more.

Could you explain how the dt's cancel out when dp/dt * v dt becomes dp * v? Are the differentials considered as numbers in these manipulations?

Thanks for your reply.

Yes, they are considered as "small numbers" by physicists. Just think of the dt's as small, but non-zero numbers, and you'll get why the manipulations work out. Of course, at some point some rigor should be introduced into the argument, but for many physicists this is good enough.