Example of non-integrable partial derivative

In summary, we are looking for a function f:X\times Y\to\mathbb{R}, where X,Y\subset\mathbb{R}, such that the integral \int\limits_Y f(x,y) dy converges for all x\in X, the partial derivative \partial_x f(x,y) exists for all (x,y)\in X\times Y, and the integral \int\limits_Y \partial_x f(x,y) dy diverges at least for some x\in X. Two examples of such functions are f(x,y) = \frac{1}{y^2} x^{-y} for X,Y = [1,\infty), and f(x,y) = \frac{\
  • #1
jostpuur
2,116
19
Can you give an example of a function [itex]f:X\times Y\to\mathbb{R}[/itex], where [itex]X,Y\subset\mathbb{R}[/itex], such that the integral

[tex]
\int\limits_Y f(x,y) dy
[/tex]

converges for all [itex]x\in X[/itex], the partial derivative

[tex]
\partial_x f(x,y)
[/tex]

exists for all [itex](x,y)\in X\times Y[/itex], and the integral

[tex]
\int\limits_Y \partial_x f(x,y) dy
[/tex]

diverges at least for some [itex]x\in X[/itex]?
 
Physics news on Phys.org
  • #2
Have you tried working backwards? Rather than guess at what f has to be to give a [itex]\partial_x f[/itex] with the desired property, instead choose [itex]\partial_x f[/itex] first.
 
  • #3
If I choose [itex]\partial_x f[/itex] so that it cannot be integrated with respect to [itex]y[/itex], then I easily get a function [itex]f[/itex] which cannot be integrated with respect to [itex]y[/itex] either. It looks like a difficult task, either way you try it.
 
  • #4
I have a few infinite families of solutions. Do you want me to just post them or let you figure them out?

Hint: try [itex]X, Y = \left[ 1, \infty \right)[/itex]
 
  • #5
Feel free to post your example functions. I don't mind if you take, the right to the feel of discovery, away from me :smile:

I might try to obtain the feel of proving somebody wrong, when I check what's wrong with your example functions :wink:
 
  • #6
OK, the first one I thought of was

[tex]f(x,y) = \frac{1}{y^2} x^{-y}[/tex]

where X and Y are both the interval [1, infinity).

[tex]\int_1^\infty f(x,y) \; dy[/tex]

converges for all x in X, but

[tex]\int_1^\infty \partial_x f(x,y) \; dy = \int_1^\infty \frac{-1}{y} x^{-y-1} \; dy[/tex]

diverges for x = 1.
 
  • #7
I see. Very nice.
 
  • #8
And here is one where the sets X and Y are the entire real line:

[tex]f(x,y) = \frac{\sin{xy}}{y^2 + a^2}[/tex]

Then

[tex]\int_R \partial_x f(x,y) \; dy = \int_R \frac{y \cos{xy}}{y^2 + a^2} \; dy[/tex]

fails to converge for x = 0.
 

1. What is an example of a non-integrable partial derivative?

An example of a non-integrable partial derivative is the function f(x,y) = xy. This function has a partial derivative with respect to x of y, and a partial derivative with respect to y of x. However, these partial derivatives cannot be integrated with respect to either x or y, making it a non-integrable partial derivative.

2. Why is a non-integrable partial derivative important?

Non-integrable partial derivatives are important because they can help describe phenomena that cannot be fully understood with standard integrable functions. These derivatives can also be used to model complex systems in fields such as physics, engineering, and economics.

3. How can a non-integrable partial derivative be calculated?

A non-integrable partial derivative can be calculated using the limit definition of a derivative. This involves taking the limit as the change in the independent variable approaches zero, and calculating the ratio of the change in the dependent variable to the change in the independent variable.

4. What is the relationship between a non-integrable partial derivative and a total derivative?

A non-integrable partial derivative is a special case of a total derivative, which is the derivative of a multivariate function with respect to all of its variables. However, not all total derivatives are non-integrable partial derivatives, as some may be integrable with respect to certain variables.

5. Can a non-integrable partial derivative have practical applications?

Yes, non-integrable partial derivatives have many practical applications in various fields of science and engineering. They can be used to model complex systems and phenomena that cannot be fully understood with standard integrable functions. Examples include chaotic systems, turbulence, and diffusion in fluids.

Similar threads

Replies
1
Views
94
Replies
1
Views
812
Replies
6
Views
1K
  • Calculus
Replies
2
Views
2K
  • Calculus
Replies
9
Views
1K
Replies
5
Views
270
Replies
3
Views
1K
Replies
3
Views
1K
Replies
4
Views
2K
Replies
4
Views
1K
Back
Top