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#1 2015-01-02 06:16:46

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
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Inexact Differential

What is a function that does not meet the euler criterion of exactness?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#2 2015-01-02 12:43:03

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Inexact Differential

The Euler Criterion is for quadratic residues. There is an Euler test for DE's to determine whether they are exact.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#3 2015-01-02 14:01:01

Agnishom
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From: Riemann Sphere
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Re: Inexact Differential

I was talking about this test:

Last edited by Agnishom (2015-01-02 14:08:07)


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#4 2015-01-02 14:53:39

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Inexact Differential

That is not in the form I know.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#5 2015-01-02 15:09:48

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: Inexact Differential

Do you have an example?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#6 2015-01-02 15:13:18

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Inexact Differential

Hmmm yes, to m you listen!

p[x_, y_] := x*y
q[x_, y_] := (3 x^2)

D[p[x, y], y] - D[q[x, y], x]


p[x_, y_] := -(5 x^2 - 2 y^2 + 11)
q[x_, y_] := (Sin[y] + 4 x*y + 3)

D[p[x, y], y] - D[q[x, y], x]


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#7 2015-01-02 15:17:57

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: Inexact Differential

But that is not what the test is


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#8 2015-01-02 15:20:53

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Inexact Differential

http://www.chem.arizona.edu/~salzmanr/4 … idiff.html

Step #4, that is the way he does the Euler test and the way Wolfram does it and so do I.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#9 2015-01-02 15:24:08

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: Inexact Differential

Shouldn't you be doing D[D[p[x,y],x],y] and D[D[p[x,y],y],x]?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#10 2015-01-02 15:26:27

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Inexact Differential

Did you try that on those functions?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#11 2015-01-02 15:33:46

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: Inexact Differential

Yes, but they are equal!


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#12 2015-01-02 15:38:29

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Inexact Differential

Then something is wrong with his method or you have misunderstood it.

The first example is not exact while the second one is.

Shouldn't you be doing D[D[p[x,y],x],y] and D[D[p[x,y],y],x]?

Nope, that looks like an identity, try this and let m show you.

{D[D[p[x, y], x], y], D[D[p[x, y], y], x]}


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#13 2015-01-02 15:57:46

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: Inexact Differential

Not exactly.

I read his page and understood the idea.


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#14 2015-01-02 15:59:08

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Inexact Differential

Did you see why his way as you have described it, is useless.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#15 2015-01-02 16:03:58

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: Inexact Differential

No,


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#16 2015-01-02 16:06:17

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Inexact Differential

See post #12


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#17 2015-01-02 16:09:35

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: Inexact Differential

Yes, and if that identity does not hold then it is not a function. E.g, Distance, Work in a non-conservative field, etc

Yes?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#18 2015-01-02 16:14:19

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Inexact Differential

That does not hide the fact that it does not work on the above 2 expressions. I suggest doing it the way that page says and the way m does it.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#19 2015-01-02 16:39:42

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: Inexact Differential

It is not supposed to work with that function.


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#20 2015-01-02 16:53:17

anonimnystefy
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From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: Inexact Differential

Hi Agnishom

What's happening is you need to do this test on two functions. You take the derivative of one w.r.t. x and the other w.r.t. y and see if they're equal. If they are, you can conclude they are the y and x (in that order) derivatives of the same function. Why does it work? Because of the identity you posted! Every functions second order mixed partial derivative is unique, no matter which variable you derive with respect to first. Thus, given two functions p(x,y) and q(x,y), if there is a function f(x,y) such that d/dx f(x,y)=p(x,y) and d/dy f(x,y)=q(x,y), then we know that this equation holds:
d/dy d/dx f(x,y)=d/dx d/dy f(x,y) which is equivalent to:
d/dy p(x,y)=d/dx q(x,y)

As you can see, this is an obvious statement and is not of much use. But what the Euler's criterion states is the converse of the above i.e. "if two functions, p(x,y) and q(x,y), satisfy the last equation I wrote above, then there is a function f(x,y), such that d/dx f(x,y)=p(x,y) and d/dy f(x,y)=q(x,y).


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#21 2015-01-02 17:15:56

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: Inexact Differential

Hi;

Thanks for explaining

What happens when I have an equation like

df = xy dx + 3x^2 dy

And I want to integrate both sides?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#22 2015-01-02 19:32:59

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Inexact Differential

It is not supposed to work with that function.

I gave you a test for whether or not a function is exact.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#23 2015-01-20 04:07:10

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: Inexact Differential

I suppose I can use this concept to determine if a vector field is conservative?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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