Manipulation of differentials.

[Yuqing]

Hey everyone!

This has always been a source of confusion for me that everyone seems to play around with dx and dy as if they were variables while in many sources it was stated that they are purely symbolic. For example, in the integral, dx is purely symbolic. If I'm not misunderstanding, dx is in fact part of the integral symbol. Yet all my professors manipulate these "symbols" endlessly when performing integration. Even worse is when the integral is set up for volumes and dx seems to play the part of the missing dimension (infinitesimal width, etc).

Now my rough understanding is this: although dy/dx is just a "symbol" for derivative, fraction like properties can be proven for it, justifying our manipulation of these symbols as fractions. Similarly for the integral, substitution by parts allows us to prove fraction like properties as if the dx were being multiplied into the integrand. Yet when I did further research I found a wikipedia page that states "does not always behave exactly like a fraction (e.g. dx/dy is not always equal to 1/(dy/dx))". Ironically, this was under the page for "Abuse of Notation". I'm not sure whether wikipedia is correct or not but my question is then:

What is the justification for these fraction like manipulations (I would love to see something like a proof, but if that is too long or complicated then a simple verbal comment would be just as good) and when are (if there indeed are) there exceptions?

 

[mathwonk]

read spivak's calculus on manifolds, or cartan's book on differential forms.

 

[Berko]

Or, Do Carmo's book on Differential Forms.

Bottom line...use them like fractions now and learn what they really are when you get to that level.

 

[AlephZero]

If you don't have time to read the recommended books:

Remember a derivative is the limit of an expression involving small quantities, as those quantities go to zero.

You can often justify "treating dy/dx like a fraction" by replacing it with the full written-out definition of the limit, ie.

[tex]dy/dx = \lim_{\Delta x \to 0} \Delta y / \Delta x[/itex]

You can use $\Delta y$ and $\Delta x$ like any other algebraic expressions, and take the limit at the end.

Integration is the inverse of differentiation, so operations with integrals can be treated the same way.

But as Berko said, the beauty of the notation is that (almost all of the time) you can forget about the details and just do what "looks right" algebraically.

 

[Stephen Tashi]

The differential notation is a good way to remember procedures. It isn't a completely safe way to reason about mathematics but it's reliable enough for most applications.

For example, the chain rule

$$\frac{df}{dx} = \frac{df}{dg} \frac {dg}{dx}$$

looks very convincing but the form $$f'(x) = f'(g(x)) g'(x)$$ reminds you that you must evaluate the derivative of $$f$$ at $$g(x)$$ , not at $$x$$.

 

[vector22]

I'll put my 2 cents in here. The notation dx actually means "the infitesimal of x" Math uses the symbols such as dy/dx when the exact derivitive can be found example

y = x^2

y' = d(x^2)/dx = dy/dx = 2x

The value of the derivitive of x^2 is dependant on x in this case
so if x = 2 then dy/dx = 4

Further, an infitesimal is a value that is small but mathematicians really don't need to know what the value is, and that is where the symbolism comes into play.
An infitesimal does not equal 0 otherwise dy/dx would be indeterminate and as we know it is not. The one important thing that can be said of the infitesimal is this

0 > dx < 1 and that is all we need to remember.

 

[HallsofIvy]

I'll put my 2 cents in here. The notation dx actually means "the infitesimal of x" Math uses the symbols such as dy/dx when the exact derivitive can be found example

y = x^2

y' = d(x^2)/dx = dy/dx = 2x

The value of the derivitive of x^2 is dependant on x in this case
so if x = 2 then dy/dx = 4

Further, an infitesimal is a value that is small but mathematicians really don't need to know

This is not true. An "infinitesmal" has a very precise definition and value in "non-standard" Calculus. In "standard" calculus, in which limits are used to define the derivative, "infinitesmal" is simply an "abuse of terminology".

what the value is, and that is where the symbolism comes into play.
An infitesimal does not equal 0 otherwise dy/dx would be indeterminate and as we know it is not. The one important thing that can be said of the infitesimal is this

0 > dx < 1 and that is all we need to remember.

 

[vector22]

OK let's get out the popcorn

hallsofivy said

This is not true. An "infinitesmal" has a very precise definition and value in "non-standard" Calculus. In "standard" calculus, in which limits are used to define the derivative, "infinitesmal" is simply an "abuse of terminology".

my question to hallsofivy is : what is this very precise definition of an infitesimal then??
I do know that there is a large crowd out there that cowers at even the though of an "infitesimal" and what it represents. It is the same crowd that does not believe that man walked on the moon - heh. Anyway, reread my post and what i say i stand behind 100%
Is this the right room for an argument?

 

[Stephen Tashi]

OK l Anyway, reread my post and what i say i stand behind 100%
Is this the right room for an argument?

You should show some references that support your side of that argument, otherwise you will only be making claims about what "mathematicians really don't need" or do need without any identifiable mathematicians standing beside you. On hallsofivy's side of the question, there are articles like http://en.wikipedia.org/wiki/Non-standard_analysis

 

[chiro]

This is not true. An "infinitesmal" has a very precise definition and value in "non-standard" Calculus. In "standard" calculus, in which limits are used to define the derivative, "infinitesmal" is simply an "abuse of terminology".

Can you be more specific about the definition of infinitesimals in non-standard calculus?

I agree with vector22 about the value of dx being non-zero because of the non-determinant form, but if you know something that contradicts this, I would love to know about it.

 

[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