Integrate on a triangular domain

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Discussion Overview

The discussion revolves around the integration of a scalar function over a triangular domain in three-dimensional space, specifically addressing the challenges of calculating the Jacobian in this context. Participants explore the implications of using simplex coordinates and the dimensionality of the integral involved.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant presents a method for integrating over a triangular domain using simplex coordinates and seeks clarification on the Jacobian calculation.
  • Another participant questions whether the term "triangular domain" refers to a two-dimensional surface, suggesting that a triple integral may not be appropriate due to insufficient independent variables.
  • A third participant asks for clarification on the function being integrated, indicating that the initial problem statement may have been misstated.
  • One participant clarifies that the integration involves a mesh of triangles tessellated on a sphere, emphasizing the three-dimensional aspect of the triangles.
  • Another participant recalls a resource that explains how to integrate a differential 3-form over a two-dimensional surface, indicating familiarity with the topic but not direct experience.
  • A later reply mentions checking a library resource for further information on the topic, indicating ongoing exploration of the issue.

Areas of Agreement / Disagreement

Participants express differing views on the dimensionality of the integration problem and the appropriateness of the integral setup. There is no consensus on the correct approach to calculating the Jacobian or the integration process itself.

Contextual Notes

Participants acknowledge potential misunderstandings in the problem statement and the need for clarity regarding the function and the dimensional context of the integration.

daudaudaudau
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Hi!


I have to integrate on a triangular domain

[tex] \int_T f(x,y,z)dxdydz[/tex]

so I use simplex coordinates, i.e.
[tex]x=(1-\alpha-\beta)x_1+\alpha x_2+\beta x_3[/tex]
[tex]y=(1-\alpha-\beta)y_1+\alpha y_2+\beta y_3[/tex]
[tex]z=(1-\alpha-\beta)z_1+\alpha z_2+\beta z_3[/tex]

where [tex](x_i,y_i,z_i)[/tex] are the vertices of the triangle and [tex]0\le\alpha\le1,\ 0\le\beta\le1-\alpha[/tex]

So what is the Jacobian? When I try to calculate it, the matrix is not square because I have two variables and three equations!
 
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daudaudaudau said:
I have to integrate on a triangular domain

[tex]\int_T f(x,y,z)dxdydz[/tex]

So what is the Jacobian? When I try to calculate it, the matrix is not square because I have two variables and three equations!

Hi daudaudaudau! :smile:

By "triangular domain", do you mean a two-dimensional surface?

If so, you can't have a triple integral, can you (not enough independent variables)? :rolleyes:

Which is why the matrix isn't square! :smile:
 
Can you tell us what "f" is? Or did you already figure your problem out?

Tiny Tim's comment is "right on" given the way you stated your problem, but, maybe you mistated it.
 
"f" is just some scalar function, I don't think it matters.

Here is what I am doing: I have tessellated a sphere into a mesh of triangles, and I have to integrate over these individual triangles. So obviously the triangles are plane but they also have both x, y, and z coordinates because they make up the 3D sphere.

I hope you understand me now.
 
Oh, yeah, I got you now. You have a differential 3-form f(x,y,z)dxdydz on R3 and you need to cut it down so that it can be integrated over a two-dimensional surface in R3. I remember Apostle's calculus book had an explanation of how to do that that I understood when I was a first year graduate student. I never actually had to do it. If anything occurs to me, I'll let you know. At least I understand what you are trying to do.
Deacon John
 
I can see my library has one(don't know if it's the right one, it just says "Apostol - Calculus"), so I will go and have a look at it. Thank you.
 

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