Doing algebra on infenetesimal amounts

[Nikitin]

Hey. I always assumed that when calculating with infinitesimal amounts, like dV = infinitesimal change in volume, you can play around with them the same as you'd do with regular numbers.

For example, you if you multiply dV/dt with dr/dr, you have dV/dt=dV/dr * dr/dt. And so on.

What exactly are the restrictions? I'm having my first calculus exam at university level tomorrow, and I thought I could use some help.

Also: Do you guys have any tips on general strategies for solving hard/unexpected Calculus problems?

 

[Stephen Tashi]

Your question could be interpreted two different ways:

1) What are the guidelines for the informal manipulation of symbols like "dx", 'dV" when doing calculus?

or

2) Is there a system of mathematics that defines infintesimal numbers and the rules they obey?

The best advice for 1) is not to think of symbols such as "dx" as a kind of number. Think of them as pnemonics - things that help you remember facts about calculus. Physics texts often reason with "infinitesimals". I've never seen an orderly presentation of how to do this. Perhaps you learn only by studying many examples. The general idea seems to be that symbols for infinitesimals with the same "order" (exponent) have some meaningful result when they are added, multiplied or divided. Symbols of a different order don't necessaily. Usually "higher order" infinitesimals are "neglected" and the answer is given by the lower order infinitesimals.

The answer to 2) is yes (see "nonstandard analysis"), but the manipulations you see in your current studies aren't based on knowing that system and most people who use calculus never study nonstandard analysis.