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Double Integral Query

bugatti79

Member
Feb 1, 2012
71
Folks,

Self reading a book in which an equation is given as

[tex]I_{mn}\equiv\int_{\Delta} x^m y^n dx dy[/tex]

where we are integrating an expression of the form [tex]x^m y^n[/tex] over an arbirtrary triangle.

Is the above actually a double integral because of the dxdy term? Ie can this be written

[tex]I_{mn}\equiv\int_{\Delta} x^m y^n dx dy= \int \int_{D} x^m y^n dA[/tex] where D is the triangle?

Thanks
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,191
Technically, the $dA$ differential is a double integral, and $dx \, dy$ differentials signify an iterated integral. I think many authors don't make a huge distinction between the two. The double integral is the more general concept - a particular iterated integral is coordinate dependent, usually.
 

bugatti79

Member
Feb 1, 2012
71
Technically, the $dA$ differential is a double integral, and $dx \, dy$ differentials signify an iterated integral. I think many authors don't make a huge distinction between the two. The double integral is the more general concept - a particular iterated integral is coordinate dependent, usually.
Thanks for that.

I have found a nice link - Double integrals as iterated integrals - Math Insight

Cheers
 

bugatti79

Member
Feb 1, 2012
71
Folks,

Self reading a book in which an equation is given as

[tex]I_{mn}\equiv\int_{\Delta} x^m y^n dx dy[/tex]

where we are integrating an expression of the form [tex]x^m y^n[/tex] over an arbirtrary triangle.

Is the above actually a double integral because of the dxdy term? Ie can this be written

[tex]I_{mn}\equiv\int_{\Delta} x^m y^n dx dy= \int \int_{D} x^m y^n dA[/tex] where D is the triangle?

Thanks
In the book I am reading they evaluate the following integral to be

[tex] \int_{\Delta} x dx dy= A \hat x[/tex] where

[tex]\displaystyle \hat x= \frac{1}{3} \Sigma_{i=1}^3 x_i[/tex] and [tex]A=\int_{\Delta} dx dy=xy[/tex]

Where does [tex]\hat x[/tex] come from? I realise its to do with the 3 coordinates of the triangle...