Question on differentials (afraid to ask)

[mnb96]

This is one of those questions I'd be afraid to ask, but here I go:

If I have a quantity $\Delta y= \Delta x+ (\Delta x)^2 + (\Delta x)^3 + (\Delta x)^4+\ldots$

and I let $\Delta x$ tend to 0, and denote it with $dx$, is it correct to state that $dy=dx$ ?

If that is correct, then what's the reasoning behind it?

 

[Hurkyl]

$$\lim_{\Delta x \to 0} \Delta y = 0$$

However,

$$\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = 1$$


Also, on the differential forms,

$$d(\Delta y) = (1 + 2 \Delta x + 3 (\Delta x)^2 + \cdots) d(\Delta x)$$

and so restricted to the specific point $\Delta x = 0$, there is an equality $d(\Delta y) = d(\Delta x)$. But at other points near $\Delta x = 0$, equality doesn't hold.

These equations might look a little nicer if you define new variables x and y via the relationships $y = y_0 + \Delta y$ and $x = x_0 + \Delta x$. (Where $x_0, y_0$ are constants) (e.g. $d\Delta y = dy$)

 

[Hurkyl]

If you are just learning calculus, it's probably better to just work with derivatives, differential approximation (and Taylor series) rather than trying to work with differential forms.

While differential forms are a good idea to learn, you probably don't have good reference materials or a good text to learn them from, so it might wind up being more confusing than useful.

 

[stevenb]

While differential forms are a good idea to learn, you probably don't have good reference materials or a good text to learn them from, so it might wind up being more confusing than useful.

That is probably good advice for now, but when the OP feels ready to study differential forms, I recommend the book "A Geometrical Approach to Differential Forms", by David Bachman.

 

[DrRocket]

While differential forms are a good idea to learn, you probably don't have good reference materials or a good text to learn them from, so it might wind up being more confusing than useful.

Differential forms are a very nice thing to learn. But it is quite impossible to do so until one understands calculus and derivatives.

"More confusing than useful" in this context is a masterful understatement.

 

[mnb96]

Thanks to you all.
I took a look to Bachman´s book, and at least in the beginning it looks very accessible.
I now pose to everyone two questions:

1) What is the main benefit that one could receive from mastering differential-forms?

2) Has anyone else seen the http://geocalc.clas.asu.edu/html/CA_to_GC.html" ? If so, how do its techinques relate to differential forms?

I have been previously exposed to this subject (Geom. Algebra) and I find it extremely intuitive.

 

[DrRocket]

Thanks to you all.
I took a look to Bachman´s book, and at least in the beginning it looks very accessible.
I now pose to everyone two questions:

1) What is the main benefit that one could receive from mastering differential-forms?

2) Has anyone else seen the http://geocalc.clas.asu.edu/html/CA_to_GC.html" ? If so, how do its techinques relate to differential forms?

I have been previously exposed to this subject (Geom. Algebra) and I find it extremely intuitive.

This is getting bizarre.

The level of this question has become completely opaque.

"Differentials" as encountered in calculus are basically an approximation to perturbations of a functin related to the ordinary derivative, and are pretty much extraneous to the heart of the subject. It is a very elementary topic.

On the other hand, differential forms are path invented by Elie Cartan to the study of modern differential geometry. They are not difficult, but the level is distinctly beyond that of introductory calculus. Similarly Cllifford algebras are usually a rather advanced topic.

In any case differential forms are sometimes thought of, in applied areas, as an alternative to tensor analysis. That is not really the case as differential forms and tensors complement one another, but perhaps it gets across one aspect of the flavor of differential forms. To appreciate them one needs to study the theory of smooth manifolds, and to do that one needs to be familiar with multivariable calculus, at the very leas to the level of understanding the inverse function theorem and the iimplicit function theorem.

If you are prepared for it (meaning have already a solid background in calculus and in classical vector analysis in three dimensions) the classic introductory book on differential forms Harley Flander's book Introduction to Differential Forms with Applications to the Physical Sciences. For a more systematic treatment one would normally look to a book on differential geometry, and there Sterenberg's book is quite good but assumes some background in multivariable calculus and point set topology. None of this stuff is accessible to anyone just learning calculus for the first time.

So again, it would be most helpful if the level of the question were to be clarified. Are we talking about a calculus student or an advanced graduate student ?