Conceptual question on greens theorem/line integrals

Click For Summary

Homework Help Overview

The discussion revolves around Green's Theorem and its relationship to line integrals, exploring the assumptions and conditions under which Green's Theorem applies. Participants are examining the connections between Green's Theorem, divergence, and Stokes' Theorem.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster questions whether it is correct to assume a direct correlation between Green's Theorem and line integrals, citing the formula for line integrals. Other participants discuss the conditions under which Green's Theorem holds true, particularly focusing on the necessity of a closed contour.

Discussion Status

The discussion is active, with participants providing insights into the conditions that may lead to the failure of Green's Theorem. There is an exploration of potential pitfalls, such as the nature of the contour and the behavior of functions involved.

Contextual Notes

Participants are considering specific cases where Green's Theorem may not apply, including non-closed contours and discontinuous functions, indicating a need for careful consideration of the theorem's assumptions.

Mdhiggenz
Messages
324
Reaction score
1

Homework Statement


Hey guys,

I just wanted to know, if it would be an incorrect assumption to say that greens theorem is directly correlated to a line integral.

The reason I am assuming that is because the formula for a line integral in my calc text is

∫f(x,y)dx+g(x,y)=∫f(x,y)dx+∫g(x,y)dy
c c c

Which is simply the left side of greens theorem.

I also am having a hard time putting the similarities between all three theorems. Greens, divergence, and stokes together in terms of what they all have in common.

Thanks

Higgenz


Homework Equations





The Attempt at a Solution

 
Physics news on Phys.org
Greens theorem is strict, the relation is true with a simple CLOSED contour but not in general. Take the line integral of a contour that isn't closed or simple and greens theorem will fail.
 
How would we know if it fails?
 
Mdhiggenz said:
How would we know if it fails?

Small things like discontinuous functions would cause it to fail, if the contour isn't closed then it will most likely fail, if the contour is too complicated (it overlaps) then it will probably fail.
 

Similar threads

Calculate this Integral around the Circular Path using Green's Theorem
  • · Replies 3 ·
Replies
3
Views
2K
Verify Green's Theorem in the given problem
  • · Replies 10 ·
Replies
10
Views
2K
Is the Calculation of the Vector Line Integral Over a Square Correct?
  • · Replies 12 ·
Replies
12
Views
2K
Using Green's Theorem for a quadrilateral
  • · Replies 3 ·
Replies
3
Views
2K
Stokes' Theorem for a Circle in a Plane
  • · Replies 7 ·
Replies
7
Views
2K
Calculating the Line Integral of F over C: Stokes' Theorem and Symmetry
  • · Replies 6 ·
Replies
6
Views
3K
A line integral about a closed "unit triangle" around the origin
  • · Replies 12 ·
Replies
12
Views
4K
Difficult Vector Field Integral
  • · Replies 3 ·
Replies
3
Views
1K
Green's Theorem vs Fundamental
  • · Replies 2 ·
Replies
2
Views
2K
Circulation of a triangular region
  • · Replies 2 ·
Replies
2
Views
2K