What is the concept of dx in calculus?

[bcrowell]

Infinitessimal has a precise definition -- usually it's something like:

x is infinitessimal if the size of x is smaller than 1/n for every positive integer n​

The description you have doesn't really fit with how mathematicians use the term "infinitessimal". Among other things, in the real number system, there is only one infintiessimal, and it's called "zero".

Other number systems have infinitesimal objects that really aren't all that mysterious. Other algebraic structures capture the notion too -- such as the notions of "differential form" and "tangent vector".

In your definition, I think you want to include a condition that x is greater than zero. Otherwise -1 would be considered infinitesimal. The way the term is normally used, 0 is also not considered infinitesimal. Non-standard analysis http://en.wikipedia.org/wiki/Non-standard_analysis has infinitesimals that fit this definition. There is a very nice freshman calc book available online that does calculus using infinitesimals: http://www.math.wisc.edu/~keisler/calc.html

 

[Hurkyl]

In your definition, I think you want to include a condition that x is greater than zero.

"Size". In the case of elements an ordered field, that would mean absolute value. I suppose I should have been more explicit, though.

The way the term is normally used, 0 is also not considered infinitesimal.

Huh. I've only seen the opposite convention. e.g. Keisler's book:

Then the only real number that is infinitesimal is zero.​

It's good to know there are people who use the opposite convention.




At least, I only remember seeing the opposite convention. I can easily imagine having seen the other, but chalked it up to the tendency for people to (IMO) gratuitously exclude degenerate cases from definitions, and so didn't take it all that seriously.

 

[bcrowell]

Huh. I've only seen the opposite convention. e.g. Keisler's book:

Then the only real number that is infinitesimal is zero.​

It's good to know there are people who use the opposite convention.

Or maybe I was just wrong :-)

 

[vector22]

Whoopes I made a typo

I previously said 0 > dx < 1

but i meant

0 < dx < 1

I blame it on the cat tibels