My professor said that this integral = 0

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Discussion Overview

The discussion revolves around the evaluation of the integral \(\int_{\Omega } f^{5}(f_{x}+2f_{y})=0\) using the divergence theorem. Participants explore the implications of the theorem, the nature of the function \(f\), and the characteristics of the integration domain \(\Omega\).

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant mentions that the integral equals zero by applying the divergence theorem, but seeks clarification on the reasoning.
  • Another participant asks for the meaning of the integration area \(\Omega\) to better understand the problem.
  • It is clarified that \(\Omega\) is a closed domain in \(\mathbb{R}^2\), and the function \(f\) is continuous with continuous derivatives.
  • A participant questions whether the absence of poles or singularities affects the integral's value.
  • There is a suggestion that the integral of \(f^4(2f_x + f_y)\) might also be zero, leading to a discussion about the significance of odd and even powers of \(f\).
  • One participant speculates that the partial derivatives in the integrand could be manipulated to yield zero.
  • Another participant tests the case where \(f=x\) and finds that the integral does not necessarily equal zero, prompting further inquiry into the nature of the integral.
  • Clarification is sought regarding whether the integral is a standard double integral over the region \(\Omega\), with a participant noting that more notation is needed to determine its value.
  • A participant provides a specific example using \(f(z)=Re(z)=x\) to illustrate the integrand and its implications.

Areas of Agreement / Disagreement

Participants express differing views on whether the integral is indeed zero, with some supporting the idea based on the divergence theorem while others challenge it by providing counterexamples and questioning the assumptions made.

Contextual Notes

Participants note the importance of the integration domain and the properties of the function \(f\), but there are unresolved questions regarding the specific form of the integral and its evaluation.

Weilin Meng
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He said that the integral below equal to zero by an easy application of the divergence theorem.

\int_{\Omega }^{}f^{5}(f_{x}+2f_{y})=0

where f is some function

Can somebody explain how?
 
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Hello Weilin
What does the integration area omega mean here? Perhaps I can try to solve your question if I know that.
greetings Janm
 
In this case, omega is just some closed domain lying in R^2
 
Weilin Meng said:
In this case, omega is just some closed domain lying in R^2
So the equation has no poles = singularities?
 
ah yes, the function is continuous and the derivatives are continuous too. It's basically a very nice domain to work with. no strings attached.
 
Weilin Meng said:
ah yes, the function is continuous and the derivatives are continuous too. It's basically a very nice domain to work with. no strings attached.
Hello Weilin
my function theory was a lot of years ago and my book complex variables is somewhere lost for a long time. At the time I bought it together with a colleague student... The same thing as with comic books. Just the nice ones get lost... I still want to find a intuitive answer to your question. May I do that with a counter question.
Would the integral of f^4*(2f_x+f_y) also be zero?

So has it to do with odd or even powers of f? If so we could reduce it to the integral f*(f_x+2f_y)...

greetings Janm
 
Yes I think in that case it still would be zero. I think the main thing is the partial derivatives in the parenthesis. I'd imagine you would manipulate that somehow with the divergence theorem to make that parenthesis zero...That is my guess at least.
 
So what if you put f=x, is it then zero? Not as far as I can tell, if Omega is any domain in R^2.
 
Weilin Meng said:
In this case, omega is just some closed domain lying in R^2


Is it a standard double integral over the region omega?

\int_{\Omega}\int f^{5}\left(f_{x}+2f_{y}\right)dA

If so, then it is not necessarily 0. Please give more notation, path integral, line integral, double integral, Mdx + Ndy, F dot ds, F dot n ds, etc...
 
  • #10
daudaudaudau said:
So what if you put f=x, is it then zero? Not as far as I can tell, if Omega is any domain in R^2.

Hello daudaudaudau
if you put f(z)=f(x+iy)=Re(z)=x, then f_x=1 and f_y=0 so the integrand would be x^5(1+2*0)=x^5.
You get the integral of x^5 dA
f(z)= x^5 is a surface bended in one direction and can be made of flat paper.
greetings Janm
 

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