2-form oriented triangle, Differential Forms

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SUMMARY

The discussion centers on calculating the value of the 2-form dxdy + 3dxdz over the oriented triangle defined by the vertices (0,0,0), (1,2,3), and (1,4,0). Participants express difficulty in deriving the correct answer, which is noted to be a negative rational number, from the provided coordinates. The reference text is Edward's 'Advanced Calculus: A Differential Forms Approach', which lacks detailed examples for calculating 2-forms, leading to confusion among users.

PREREQUISITES
  • Understanding of differential forms, specifically 2-forms.
  • Familiarity with vector calculus and coordinate systems.
  • Knowledge of the geometric interpretation of integrals over triangles in three-dimensional space.
  • Proficiency in using the concepts from 'Advanced Calculus: A Differential Forms Approach' by Edward.
NEXT STEPS
  • Study the geometric interpretation of 2-forms in differential geometry.
  • Learn how to compute integrals of differential forms over simple geometric shapes.
  • Review examples of calculating 2-forms from 'Advanced Calculus: A Differential Forms Approach'.
  • Explore the relationship between differential forms and rational numbers in calculus.
USEFUL FOR

This discussion is beneficial for students and educators in advanced calculus, particularly those focusing on differential forms, as well as mathematicians seeking to deepen their understanding of geometric integration techniques.

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Homework Statement


Find the value of the 2-form dxdy+3dxdz on the oriented triangle with (0,0,0) (1,2,3) (1,4,0) in that order.

Homework Equations

The Attempt at a Solution


I have tried various subtraction of these coordinates and applying them to the formula but the answer is in the back of the book and I fail to see how one could get that answer.
The book does not give one example of how to calculate with 2-forms there is but a brief description and then 4 problems.

This is a problem from Edward's, 'Advanced Calculus: A Differential Forms Approach'

The answer is a negative rational number. I also fail to see how I derive a rational number from this.
 
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This is an unusual formulation, but have you read the first chapter of that book ?
 

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