What is the Definition and Purpose of Differentials in Mathematics?

[Pyroadept]

Homework Statement

Homework Equations

3. The Attempt at a Solution [/b

Hi everyone, this isn't a homework problem but rather just a question of definition. Is the differential (e.g. 'dx' for the 'differential of x') just when you differentiate without specifying by what you are differentiating by?
e.g. dx could stand for dx/dy, dx/dt etc.

Thanks for any help

 

[Mark44]

dx does not stand for dx/dy, dx/dt, etc. dx is an infinitesimally small, positive value of x that is different from zero.

 

[Hurkyl]

That requires explanation.

dx is not a "real number" -- i.e. it is not a member of the number system you've been learning since elementary school.

But it's not unfair to consider the differentials a number system in its own right. If x denotes a "generic" real number, then dx represents a "generic" differential.

The "infinitessimalness" comes not from there being any sort of ordering to compare a differential to a real number (or even to compare two differentials) -- it comes from the fact dx dx = 0. Also, dx dy = 0 if y is dependent on x. However, if x and y are independent, then dx dy is nonzero. And this multiplication is anticommutative:

dy dx = -dx dy​

However, if you are thinking in terms of functions -- e.g. you are considering x as the function that maps a point of the line to its coordinate -- then dx acquires a related interpretation. And then, the idea of "infinitessimal number" is supplied by the differential operators.