# My professor said that this integral = 0

• Weilin Meng
In summary, He said that the integral below equal to zero by an easy application of the divergence theorem.
Weilin Meng
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?

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...

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

## 1. What is an integral?

An integral is a mathematical concept that represents the area under a curve. It is used to find the total value of a quantity that is changing over an interval of time or space. It is denoted by the symbol ∫ and can be thought of as the opposite of a derivative.

## 2. How does an integral equal 0?

If your professor said that an integral equals 0, it means that the value of the integral is 0. This could be due to several reasons, such as the function being integrated is symmetrical around the x-axis, or the limits of integration cancel each other out resulting in a net value of 0.

## 3. Can an integral ever be negative?

Yes, an integral can be negative if the function being integrated is below the x-axis. This indicates that the total value of the quantity being measured is decreasing over the given interval.

## 4. What does it mean if my professor said that the integral is divergent?

If the integral is divergent, it means that it does not have a finite value. This could happen if the function being integrated has an infinite or undefined value at one or more points within the given interval.

## 5. How is an integral used in science?

An integral is used in many different areas of science, such as physics, engineering, and biology. It is used to calculate important quantities such as velocity, acceleration, and area under a curve, which are crucial for understanding and predicting various phenomena in the natural world.

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