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Definition of Derivative and its relationship with the concept of differential
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Shmuel (Seymour J.) Metz1
science forum Guru


Joined: 03 May 2005
Posts: 604

PostPosted: Tue Jul 11, 2006 3:33 pm    Post subject: Re: Definition of Derivative and its relationship with the concept of differential Reply with quote

In <1152615776.490635.195090@m79g2000cwm.googlegroups.com>, on
07/11/2006
at 04:02 AM, greenishguy@gmail.com said:

Quote:
On the other hand, the definition of the differential of a function
y=f(x), is dy=f'(x)dx. (2)

No; that isn't a definition, because you haven't defined dx.

Quote:
i) What's the relationship between (1) and (2)?

(1) is a definition, (2) is just a heuristic.

Quote:
Is the derivative the limit of an increment quotient or a quotient
of differentials?

The former; the latter is meaningless absent a definition. The closest
that you can come to the latter is via Nonstandard Analysis, where
there is a well defined concept of differential and if dx is a
differential then the standard parts of (f(x+dx)-f(x))/dx and f'(x)
are equal.

Quote:
ii) Is it reasonable to consider a differential of a differential,
e.g. d(dx)=d^2(x)?

Not unless you can provide a meaningful definition and prove useful
properties. There is a context where it is defined, but there ddf=0.

--
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Han de Bruijn
science forum Guru


Joined: 18 May 2005
Posts: 1285

PostPosted: Tue Jul 11, 2006 12:33 pm    Post subject: Re: Definition of Derivative and its relationship with the concept of differential Reply with quote

G.E. Ivey wrote:

Quote:
By the way, [ ... ]

http://plato.stanford.edu/entries/continuity/
http://en.wikipedia.org/wiki/Hyperreal_number

Han de Bruijn
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Han de Bruijn
science forum Guru


Joined: 18 May 2005
Posts: 1285

PostPosted: Tue Jul 11, 2006 12:27 pm    Post subject: Re: Definition of Derivative and its relationship with the concept of differential Reply with quote

G.E. Ivey wrote:

Quote:
By the way, it is possible, using some very deep concepts from logic,

Very deep, NO. Very much CRAP, yes.

Quote:
to show that it is possible to extend the real number system to include
both "infinitely large" and "infinitely small" numbers (that's called
"non-standard analysis"). In that case we CAN define "dx" and "dy"
separately and THEN define the differential as the quotient dy/dx.

http://en.wikipedia.org/wiki/Nonstandard_analysis

Quote:
However, dx and dy are not ordinary numbers, they are "infinitesmals".
The rules of arithmetic are quite different for infinitesmals and
infinite numbers than they are for ordinary numbers.

As I've said, the theory of infinitesimals in "standard" mathematics is
very much CRAP. It bears no relationship to the way infinitesimals are
actually employed in the exact sciences, such as, for example, Physics.

In physics, infinitesimals _are_ just small real numbers and the rules
of their arithmetic are not quite different from real number arithmetic,
of course. The _real_ problem is that infinitesimals cannot be embodied,
at all, in contemporary mathematics. These topics have been discussed at
length in the thread starting with:

http://groups.google.nl/group/sci.math/msg/b686cb8d04d44962?hl=en&

In probability theory, the paradoxes, emerging from the inability of
standard mathematics to deal with infinitesimals, are most apparent.

http://hdebruijn.soo.dto.tudelft.nl/QED/klassiek.htm (: bottom)

Han de Bruijn
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JEMebius
science forum Guru Wannabe


Joined: 24 Mar 2005
Posts: 209

PostPosted: Tue Jul 11, 2006 12:06 pm    Post subject: Re: Definition of Derivative and its relationship with the concept of differential- Reply with quote

The best thing you can do is IMO to prove, right from the modern
definition of differentiability, that a real-valued function F on the
reals which is twice differentiable at x can be written as

F(x+h) = F(x) + h.F'(x) + h^2.F''(x)/2 + o(h^2) (just 2nd-order Taylor
with remainder),

to proceed with digging out some sensible interpretations of dx and
d^2x, and finally to compare your findings with classical literature
like Euler's Introduction in Analysin Infinitorum = Introduction to
Analysis of the Infinite.

Happy studies: Johan E. Mebius


greenishguy@gmail.com wrote:

Quote:
Hi there!

We know that the definition of the derivative of a function nowadays is
made by using the concept of the limit though traditionally it wasn't
so. Thus, considering a function y=f(x), the limit when delta x tends
to zero of the incremental quotient delta y over delta x (delta y =
f(x+delta x) - f(x)) is called y' or f'(x), i.e. the derivative of
f(x). (1)

On the other hand, the definition of the differential of a function
y=f(x), is dy=f'(x)dx. (2)

Now, according to (2) we can algebraically isolate y'=f'(x)=dy/dx. So,
the derivative can be expressed as a quotient of differentials.

The questions are:

i) What's the relationship between (1) and (2)? Is the derivative the
limit of an increment quotient or a quotient of differentials? If they
both are the same, the limit of the quotient would be the quotients of
limits (being dy= limit when delta x tends to zero of delta y, and
dx=limit when delta x tends to zero of delta x), which is not true
since this property of limits is only correct if the denominator is
different from zero.

ii) Is it reasonable to consider a differential of a differential, e.g.
d(dx)=d^2(x)? What's it equal to? Can it be dismissed compared to dx?
What's the value of d(dy)=d^2(y)? Is it reasonable to talk abou the
n-th differential as we talk about the n-th derivative?

Thanks.

greenishguy.


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G.E. Ivey
science forum Guru


Joined: 29 Apr 2005
Posts: 308

PostPosted: Tue Jul 11, 2006 11:28 am    Post subject: Re: Definition of Derivative and its relationship with the concept of differential Reply with quote

Quote:
Hi there!

We know that the definition of the derivative of a
function nowadays is
made by using the concept of the limit though
traditionally it wasn't
so. Thus, considering a function y=f(x), the limit
when delta x tends
to zero of the incremental quotient delta y over
delta x (delta y =
f(x+delta x) - f(x)) is called y' or f'(x), i.e. the
derivative of
f(x). (1)

On the other hand, the definition of the differential
of a function
y=f(x), is dy=f'(x)·dx. (2)

Now, according to (2) we can algebraically isolate
y'=f'(x)=dy/dx. So,
the derivative can be expressed as a quotient of
differentials.

The questions are:

i) What's the relationship between (1) and (2)? Is
the derivative the
limit of an increment quotient or a quotient of
differentials?
The way you have defined them above, the derivative cannot be DEFINED as a quotient of differentials because you have to use the derivative to define the differential! AFTER you have defined them, then yes, it is true that y'= dy/dx (right side being a quotient of differentials).


Quote:
If they
both are the same, the limit of the quotient would be
the quotients of
limits (being dy= limit when delta x tends to zero of
delta y, and
dx=limit when delta x tends to zero of delta x),
which is not true
No. dx and dy, the differentials are NOT defined by limits. They are defined by dy= y'(x)dx just as you said.


Quote:
since this property of limits is only correct if the
denominator is
different from zero.

A differential is never 0.
ii) Is it reasonable to consider a differential of a
differential, e.g.
d(dx)=d^2(x)? What's it equal to? Can it be dismissed
compared to dx?
What's the value of d(dy)=d^2(y)? Is it reasonable to
talk abou the
n-th differential as we talk about the n-th
derivative?

Thanks.

greenishguy.

Yes, it is possible to talk about "the differential of a differential"- that sort of thing is done in differential geometry. However, d(dx)= 0 always. In general d(df)= 0 unless f has singularities. That being the case higher order differentials are necessarily 0 also.


By the way, it is possible, using some very deep concepts from logic, to show that it is possible to extend the real number system to include both "infinitely large" and "infinitely small" numbers (that's called "non-standard analysis"). In that case we CAN define "dx" and "dy" separately and THEN define the differential as the quotient dy/dx. However, dx and dy are not ordinary numbers, they are "infinitesmals". The rules of arithmetic are quite different for infinitesmals and infinite numbers than they are for ordinary numbers.
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greenishguy@gmail.com
science forum beginner


Joined: 11 Jul 2006
Posts: 1

PostPosted: Tue Jul 11, 2006 11:02 am    Post subject: Definition of Derivative and its relationship with the concept of differential Reply with quote

Hi there!

We know that the definition of the derivative of a function nowadays is
made by using the concept of the limit though traditionally it wasn't
so. Thus, considering a function y=f(x), the limit when delta x tends
to zero of the incremental quotient delta y over delta x (delta y f(x+delta x) - f(x)) is called y' or f'(x), i.e. the derivative of
f(x). (1)

On the other hand, the definition of the differential of a function
y=f(x), is dy=f'(x)dx. (2)

Now, according to (2) we can algebraically isolate y'=f'(x)=dy/dx. So,
the derivative can be expressed as a quotient of differentials.

The questions are:

i) What's the relationship between (1) and (2)? Is the derivative the
limit of an increment quotient or a quotient of differentials? If they
both are the same, the limit of the quotient would be the quotients of
limits (being dy= limit when delta x tends to zero of delta y, and
dx=limit when delta x tends to zero of delta x), which is not true
since this property of limits is only correct if the denominator is
different from zero.

ii) Is it reasonable to consider a differential of a differential, e.g.
d(dx)=d^2(x)? What's it equal to? Can it be dismissed compared to dx?
What's the value of d(dy)=d^2(y)? Is it reasonable to talk abou the
n-th differential as we talk about the n-th derivative?

Thanks.

greenishguy.
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