Talk:Multiple integral

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 This article is a rough translation of an article in Italian.
It may have been generated by a computer or by a translator with limited proficiency in English or Italian.
Feel free to enhance the translation. For reference, the original article may appear under "Italian" in the "languages" list.

While this article seems very technically-sound, I also found it almost incomprehensible as a non-mathematician. In particular, the language seems stilted and perhaps was not written by a native English speaker. In any case, since this article is of great potential interest to many people who may not be mathematically-inclined (that is, may find the information useful/necessary but be unable to decipher the swarm of symbols here), it would benefit from an overhaul. Usually I don't like to just stamp a CleanUp tag on and keep going, but I am in no way qualified enough to properly edit this category. Mineralogy 23:44, 15 January 2006 (UTC)

I think the author is a native speaker of Spanish and not of English. Perhaps the article double integral is in some ways clearer. Michael Hardy 01:12, 7 March 2006 (UTC)

Contents

[edit] Mathematical Error

In the [Formulas of Reduction -> Normal Domains on R^2] section, upon checking the math I found the answer (7/10) to be wrong. I solved the double integral both ways noted and got (13/20) for both values. The value has been corrected.

[edit] Issue in Change of Variable Section

"There exist three main "kinds" of changes of variable (one in R2, two in R3), however is possible to hand with this method so as to to operate the substitution that more thinks good."

The above sentence doesn't make any sense. Anybody care to venture what they are talking about? user:Feinstein


[edit] Some questionable amterial

Agree with the above. Also, the example:

Example (3-a):

The region is D = \{ x^2 + y^2 \le 9, \ x^2 + y^2 \ge 4, \ 0 \le z \le 5 \} (that is the "tube" whose base is the circular crown of the 2-d example and whose height is 5); if you apply the transformation you'll get this region: T = \{ 2 \le \rho \le 3, \ 0 \le \phi \le \pi, \ 0 \le z \le 5 \} (that is the parallelepiped whose base is the rectangle in 2-d example and whose height is 5).

Is questionable, as the "circular crown" of the 2-d example is not relevant, since in example 2-d, y > 0. This is falsely assumed in this example by setting the region to T = \{ 2 \le \rho \le 3, \ 0 \le \phi \le \pi, \ 0 \le z \le 5 \} instead of T = \{ 2 \le \rho \le 3, \ 0 \le \phi \le 2\pi, \ 0 \le z \le 5 \}

Am I not correct? Still learning...

[edit] Rough translation

The article seems to be a translation from the Italian version of that article. The article needs to be rewritten on some parts. For example, I guess when the author spoke of dominion, he/she actually meant domain (mathematics) (a false friend translation). --Abdull 13:51, 25 May 2006 (UTC)

I've made some changes to try to improve the clarity, but it needs further work, as I can't pick out the meaning throughout! –Ian 12:38, 15 June 2006 (UTC)

[edit] Question regarding Example 2-d

Example (2-d):

The domain is D = \{ x^2 + y^2 \le 9, \ x^2 + y^2 \ge 4, \ y \ge 0 \}, that is the circular crown in the semiplane of positive y (please see the picture in the example); you note that φ describes a plane angle while ρ varies from 2 to 3. Therefore the transformed domain will be the following rectangle:
T = \{ 2 \le \rho \le 3, \ 0 \le \phi \le \pi \}.
The Jacobian determinant of that transformation is the following:
\frac{\partial (x,y)}{\partial (\rho, \phi)} = \begin{vmatrix} \cos \phi & - \rho \sin \phi \\ \sin \phi & \rho \cos \phi \end{vmatrix} = \rho
which has been got by inserting the partial derivatives of x = ρ cos(φ), y = ρ sin(φ) in the first column respect to ρ and in the second respect to φ, so the dx dy differentials in this transformation becomes ρ dρ dφ.
Once transformed the function and evaluated the domain, it's possible to define the formula for the change of variables in polar coordinates:
\iint_D f(x,y) \ dx\, dy = \iint_T f(\rho \cos \phi, \rho \sin \phi) \rho \, d \rho\, d \phi.
Please note that φ is valid in the [0, 2π] interval while ρ, because it is a measure of a length, can only have positive values.

Can someone explain why the Jacobian determinant had to be used here, which then produced an additional φ to the equation? --Abdull 13:51, 25 May 2006 (UTC)

[edit] Spherical Coordinates

In the section on formulae of reduction, the article makes reference to the spherical coordinate system.

In R3 some domains have a spherical symmetry, so it's possible to determinate the coordinates of every point of the integration's region by two angles and one distance. It's possible to use therefore the passage in spherical coordinates; the function is transformed by this relation: f(x,y,z) \longrightarrow f(\rho \sin \theta \cos \phi, \rho \sin \theta \sin \phi, \rho \cos \theta)\,\!

This system, while correct, has the standard American placements of theta and phi reversed. This has the ability to cause confusion, especially when viewed alongside the Spherical Coordinate System article, which uses the American system of notation.

While I understand that the American placement is not universally accepted, it would make sense to standardize notation across the various English articles. So, should we change the article to use the American notation, or add a note stating the difference, and pointing to the Spherical Coordinate System article for further information?

Plumberwill 09:45, 10 April 2007 (UTC)

I agree on standardizing notation across articles in principle. MathWorld lists several conventions, which should be noted at spherical coordinate system. I'd prefer using a mathematical convention over a physics convention in articles like multiple integral, but articles about physics that talk about spherical coordinates can rightly reverse theta and phi (going by MathWorld, that's what they do more often). As long as context is given, i.e. define r, thetha and phi before referring to them, it should be clear what's going on in each case. Although I personally prefer r, writing ρ distinguishes it from the r in cylindrical coordinates. –Pomte 10:41, 10 April 2007 (UTC)
Sorry, I think Jacobian is sin phi, not sin theta. Integral on sin from 2pi to zero is zero.. 89.138.242.197 10:52, 24 July 2007 (UTC)

[edit] Translation

I tried to improve the translation. I hope that despite being italian, it seems now clearer.


[edit] poor article

The article does not have give a clear idea of what a multiple integral is. yes, multiple integrals may be used to find volumes, but this is not their only application. also the article does not provide a method for evaluating multiple integrals. —Preceding unsigned comment added by 212.159.75.167 (talk • contribs)

as my contributions are allways deleted i shall not bother to fix this. —Preceding unsigned comment added by 212.159.75.167 (talk • contribs)

[edit] Indefinite antiderivatives?

Hello. This page states that:

"Since it is impossible to calculate the antiderivative of a function of more than one variable, indefinite multiple integrals do not exist. Therefore all multiple integrals are definite integrals."

However, Wolfram Mathworld says that

"In an indefinite multiple integral, the order in which the integrals are carried out can be varied at will; for definite multiple integrals, care must be taken to correctly transform the limits if the order is changed."

Which seems to imply the contrary. I leave the hyperlink. [[1]]

Cheers. —Preceding unsigned comment added by 200.6.195.225 (talk) 03:45, 6 February 2008 (UTC)

[edit] Article Break-Up

This article is too long, but it still does not devote enough space to applications (area, plane laminas, avg. value, SA).

I propose we create new articles devoted specifically to applications of double and triple integrals and just leave brief explanations on this page. Any thoughts?

-Nmoo (talk) 06:24, 2 April 2008 (UTC)