Relationship between total derivative and integral

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samspotting
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Relationship between total derivative and integral

Postby samspotting » Fri Apr 17, 2009 4:53 pm UTC

For multivariable functions on R^n, the total derivative can be taken so long and the function is smooth, and the riemann integral is defined for functions from R^n to R that are Riemann integrable.

Is there a connection between these two? For R, there is the fundamental theorem of calculus between antiderivatives and integrals.

Is there such a thing as an indefinite riemann integral in R^n?

So far I am seeing multivariable calculus as a study of derivatives and integrals seperately, with little connection between the two.

hnooch
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Re: Relationship between total derivative and integral

Postby hnooch » Fri Apr 17, 2009 5:35 pm UTC

What you're looking for is Stoke's Theorem. It's an absolutely incredible generalization of the Fundamental Theorem of Calculus, but it's beyond the level of a usual introductory course in multivariable calculus, where you'll probably be learning things like iterated integrals and Lagrange multipliers.

You might see things like Green's Theorem and the Divergence Theorem, which are special cases of Stoke's Theorem.

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t0rajir0u
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Re: Relationship between total derivative and integral

Postby t0rajir0u » Fri Apr 17, 2009 6:29 pm UTC

General rule: whenever you think two mathematical objects aren't connected, you're wrong.

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Yakk
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Re: Relationship between total derivative and integral

Postby Yakk » Fri Apr 17, 2009 6:54 pm UTC

That would be an interesting challenge-response game.

You name two mathematical concepts that you don't think are related, and someone else tries to relate them. :)
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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samspotting
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Re: Relationship between total derivative and integral

Postby samspotting » Fri Apr 17, 2009 7:55 pm UTC

Woah ok that makes sense. I whipped out my undergrad calendar and its covered in differential geometry so I guess im going to look forward to that.

itaibn
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Re: Relationship between total derivative and integral

Postby itaibn » Fri Apr 17, 2009 10:20 pm UTC

Yakk wrote:That would be an interesting challenge-response game.

You name two mathematical concepts that you don't think are related, and someone else tries to relate them. :)

The halting problem and matrixes.

Edit: Added quote.
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megabnx
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Re: Relationship between total derivative and integral

Postby megabnx » Sat Apr 18, 2009 1:02 am UTC

itaibn wrote:
Yakk wrote:That would be an interesting challenge-response game.

You name two mathematical concepts that you don't think are related, and someone else tries to relate them. :)

The halting problem and matrixes.

Edit: Added quote.


Hey, that's an easy one.

Find a markov chain that represents the conditions of you're problem and determine if there is any steady state for the matrix.

Now Lets try a real challenge.

The Riemann hypothesis and the Goldbach conjecture. It can probably be done, but they are so specific that it should take some working.

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t0rajir0u
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Re: Relationship between total derivative and integral

Postby t0rajir0u » Sat Apr 18, 2009 2:56 am UTC

megabnx wrote:Find a markov chain that represents the conditions of you're problem and determine if there is any steady state for the matrix.

In a little more detail: if you weaken Turing machines to finite automata, you can associate with any finite automaton its adjacency matrix by interpreting it as a directed graph, and the eigenvalues of this matrix solve the "halting problem" for finite automata.

Obviously, this doesn't work for Turing machines because the total number of possible states of a given Turing machine is countably infinite, but for certain very restricted types of problems I believe it's possible to perform some kind of eigenvalue analysis on the operator that takes a state of the Turing machine to its next state interpreted as a linear operator over formal sums of states.

megabnx wrote:The Riemann hypothesis and the Goldbach conjecture.

From Wikipedia:

http://www.ams.org/era/1997-03-15/S1079-6762-97-00031-0/S1079-6762-97-00031-0.pdf wrote:In 1997, Deshouillers, Effinger, te Riele and Zinoviev showed[1] that the generalized Riemann hypothesis implies Goldbach's weak conjecture. This result combines a general statement valid for numbers greater than 1020 with an extensive computer search of the small cases.

That was a bad example considering how closely both are related to the distribution of the primes :P This is a very fun game, though!

Alright, I've got one that isn't too hard: graphs and algebraic number theory.

SteveG
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Re: Relationship between total derivative and integral

Postby SteveG » Sat Apr 18, 2009 3:15 pm UTC

t0rajir0u wrote:Alright, I've got one that isn't too hard: graphs and algebraic number theory.


The drawings of children: http://www.institut.math.jussieu.fr/~le ... isseFr.pdf

Thanks Grothendieck!

[imath]a[/imath] and [imath]b[/imath] relate to [imath]a \Rightarrow b[/imath] in a poset the same way that the sets [imath]A[/imath] and [imath]B[/imath] relate to the set [imath]\{ f:A \rightarrow B\}[/imath]. What is the relationship?

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t0rajir0u
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Re: Relationship between total derivative and integral

Postby t0rajir0u » Sat Apr 18, 2009 6:10 pm UTC

Interesting - that's much deeper than what I was thinking of. To every algebraic integer [imath]\alpha[/imath] of degree [imath]n[/imath] we can associate the matrix that gives its action on [imath]\{ 1, \alpha, ... \alpha^{n-1} \}[/imath], which behaves as the "weighted" adjacency matrix of a graph with vertices [imath]1, \alpha, ... \alpha^{n-1}[/imath]; then [imath]\alpha[/imath] is an eigenvalue of this matrix. In particular, when [imath]\alpha[/imath] is a root of unity we get the corresponding Cayley graph. There is a very natural way to add and multiply eigenvalues: the Cartesian product and tensor product respectively, which act on adjacency matrices by Kronecker sum and Kronecker product. It follows that the sum and product of algebraic integers is algebraic. :) (That is, using a combinatorial interpretation of the powers of two algebraic integers and their conjugates we can find a combinatorial interpretation of the powers of their sums and products.)

As for your question, are you just talking about the category axioms?

SteveG
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Re: Relationship between total derivative and integral

Postby SteveG » Thu Mar 04, 2010 1:16 pm UTC

Sorry to be such a necromancer, but the intended link was that they are both examples of exponential objects in their respective categories.

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BlackSails
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Re: Relationship between total derivative and integral

Postby BlackSails » Thu Mar 04, 2010 1:49 pm UTC

This is a fun idea, necromancy or not.

Legendre transforms and Mobius transforms.


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