In Wikipedia, it is said that[tex]\mathrm dy=\frac{\mathrm

[dalcde]

In Wikipedia, it is said that
\mathrm dy=\frac{\mathrm dy}{\mathrm dx}\mathrm dx.
Can we divide both sides by \mathrm dx and say that the derivative is \mathrm dy divided by \mathrm dx?

 

[LCKurtz]

Yes. That is the usual convention. I think it is less confusing for a calculus student to introduce differentials using the prime notation like this: Given y = f(x) and y' = f'(x), if dx is a small nonzero change in x we define the symbol dy = f'(x) dx. dy can be thought of the change in the y value of the tangent line or the approximate change in y if dx is small. In any case, if you divide both sides by dx you have dy/dx = f'(x).

 

[Dr. Seafood]

It looks like ordinary division of numbers, but \mathrm dx and \mathrm dy are not ordinary numbers. However, we manipulate them symbolically in a way that appears like they are real numbers, for the sake of intuition. But we can do this without loss of precision! A good demonstration of it is using the substitution rule for integrals:

\int f'(g(x))g'(x) \mathrm dx

Substituting u = g(x), we get, by the chain rule for derivatives,

\int f'(u) \frac{\mathrm du}{\mathrm dx} {\mathrm dx} = f(u) + C = \int f'(u) \mathrm du

The last equality is given by the definition of antiderivative. Though we didn't actually "cancel out" the differentials like fractions, it does turn out that we can safely write \frac{\mathrm du}{\mathrm dx} {\mathrm dx} = \mathrm du, by the chain of equalities. Note that the justification for this, however, has nothing to do with fractions.

 

[dalcde]

I mean, if we define dy and dx like that, then is it totally correct to say that dy/dx is dy divided by dx?

 

[Dr. Seafood]

Don't you really have to define "divide" (namely, "division") in order to do that? But "divide" typically refers to an operation involving numbers -- and differentials aren't numbers.

 

[dalcde]

Well, we can define division for differentials, why not?

 

[Dr. Seafood]

Alright, then wouldn't we be interested in defining what a differential is first, so we can define operations on them?

I think in the elementary, traditional sense, "division" here doesn't really work. I've always thought \frac{\mathrm dy}{\mathrm dx} was cool-looking, but a bit notationally abusive.

 

[Rodrae]

In Wikipedia, it is said that
\mathrm dy=\frac{\mathrm dy}{\mathrm dx}\mathrm dx.
Can we divide both sides by \mathrm dx and say that the derivative is \mathrm dy divided by \mathrm dx?

You might do that.. Sometimes there are things that is written in a book, etc. that is really confusing to understand. Maybe a simple explanation might be helpful when you are just starting calculus rather than those with lots of formula. Like those that I red before about the law of derivatives that when i read it. It is so confusing so i just make my own shortcut formula rather do those.

 

[hunt_mat]

Hmmm, differentials are really part of of exterior algebra which is a very important tool for differential geometry. So I think you should look into exterior algebra for a firm understanding of a differential.

 

[dalcde]

Do you have any resources for studying exterior algebra? I'd prefer some online resources since I can't gain access to any bookstores or libraries that have some serious mathematics books (I'm still at junior high school).

 

[hunt_mat]

Any set of notes on elementary differential geometry will do but google is your friend...

 

[dalcde]

Thanks. Just needed to know what to search for (I tried to search for exterior algebra but every result turned out very difficult.