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January 7th, 2017, 03:24 PM   #11
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I don't think that division by a differential has any meaning. It's the similar to division by zero.
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January 7th, 2017, 03:42 PM   #12
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Quote:
Originally Posted by v8archie View Post
I don't think that division by a differential has any meaning. It's the similar to division by zero.
I think it's even worse.

The symbols $0 \div 0$ have at least enough meaning that we are forced to declare it undefined.

The symbols, $\partial x \div \partial y$ don't have any meaning.
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January 8th, 2017, 10:27 PM   #13
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Originally Posted by zylo View Post
Definition:
$\displaystyle \frac{\partial u}{\partial x}=Lim\frac{u(x+\Delta x,y,z)-u(x,y,z)}{\Delta x} \equiv \frac{\mathrm{d} u|_{y,z}}{\mathrm{d} x}$
$\displaystyle \partial u$ can be interpreted as the differential of u with all variables except one fixed.

Perhaps a review from elementary calculus will help:

SUMMARY:
$\displaystyle \Delta y = y'\Delta x + \epsilon$
dy = y'dx, any dx.


Definition of derivative y' at x:
y' = Lim $\displaystyle \frac{\Delta y}{\Delta x}$

Having defined y', increments $\displaystyle \Delta y= f(x+\Delta x)-f(x)$, and $\displaystyle \Delta x$ are related by:
$\displaystyle \Delta y = y'\Delta x + \epsilon$,
Where Lim $\displaystyle \epsilon / \Delta x$ = 0.

Definition of differentials dy and dx at x:
dx = $\displaystyle \Delta x$
dy = y'dx

Last edited by zylo; January 8th, 2017 at 11:06 PM.
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January 9th, 2017, 11:05 AM   #14
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See also:

Rigorous definition of "Differential"
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January 9th, 2017, 04:10 PM   #15
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Quote:
Originally Posted by zylo View Post
Definition of derivative y' at x:
y' = Lim $\displaystyle \frac{\Delta y}{\Delta x}$
That's not a definition. I notice that your usually meaningless catch-all "any dx" that represents the point at which you have run out of knowledge appears particularly early on.

All of the post is vague hand waving that, in particular, fails to define division by a differential.
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January 9th, 2017, 04:22 PM   #16
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Quote:
Originally Posted by zylo View Post
$\displaystyle \partial u$ can be interpreted as the differential of u with all variables except one fixed.

Perhaps a review from elementary calculus will help:

SUMMARY:
$\displaystyle \Delta y = y'\Delta x + \epsilon$
dy = y'dx, any dx.


Definition of derivative y' at x:
y' = Lim $\displaystyle \frac{\Delta y}{\Delta x}$

Having defined y', increments $\displaystyle \Delta y= f(x+\Delta x)-f(x)$, and $\displaystyle \Delta x$ are related by:
$\displaystyle \Delta y = y'\Delta x + \epsilon$,
Where Lim $\displaystyle \epsilon / \Delta x$ = 0.

Definition of differentials dy and dx at x:
dx = $\displaystyle \Delta x$
dy = y'dx
Is all this supposed to imply that division of differentials is a meaningful operation? If so well, you're wrong.
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January 9th, 2017, 04:57 PM   #17
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It's clear to me that differentials are no more than Leibnitz and Newton's infinitesimals which we can now access via hyperreals. Zylo is trying to look knowledgeable, but I'd encourage the reader to take note of other posters in the thread he referenced rather than Zylo.
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January 11th, 2017, 08:50 AM   #18
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If meaningless jargon means nothing to you, you can (should) look up differentials in standard textbooks such as Thomas (calculus) or Kaplan (advanced calculus) which are obtainable used for a few dollars.

Meaningless jargon stems from basic ideas which have been stripped of the content which makes them meaningful, and renamed. Jargon can become meaningful if you understand the basic ideas underlying it. It is often used to disguise a lack of understanding.
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January 11th, 2017, 08:57 AM   #19
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So you have stripped away the context that gives your post any meaning. I can guess why.
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January 11th, 2017, 10:54 AM   #20
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Differentials of a function are defined by equation of tangent line or plane at a point.

As such, they can be treated as ordinary variables, including cancellation.

When small, they approximate the function in the neighborhood of the point.
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