Green's Theorem & Line Integrals

In summary, the conversation discusses a problem involving Green's theorem and line integrals, with the goal of proving that a function f is constant on a given set U. The conversation covers the steps needed to solve the problem, including using Green's theorem, the vector version of the theorem, and the squeeze theorem. The conversation ends with a request for help in solving the final part (iii) of the problem and a question about how the results from the previous steps will aid in proving the overall problem.
  • #1
kingwinner
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0

Homework Statement


http://www.geocities.com/asdfasdf23135/advcal26.JPG
Note:
path-connected means arcwise-connected
C^0 means continuous
⊿f=Laplacian=(f)xx+(f)yy
df/dn = grad f ． n

2. Relevant material
Green's theorem, line integrals, ...


The Attempt at a Solution


The only part I can finish is part (ii):
(ii) follows from the vector version of Green's theorem
eq0046M.gif
by letting F=grad f

How can I do parts (i) and (iii)?
And provided that I have proved (i), (ii), and (iii), how would these results aid me in proving the actual problem?

Any help or hints would be greatly appreciated!https://www.physicsforums.com/images/smilies/smile.gif
 
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  • #2
I seriously need help with parts (i) and (iii)...
 
  • #3
Well, I've solved part (i) as well:
2008-03-25_232120.jpg

Then use squeeze theorem...


And now I am stuck only on part (iii), please help!
 
  • #4
Provided that I have proved (i), (ii), and (iii), how would these results aid me in proving the actual problem? (i.e. to prove that f is constant on U)
 

Question 1: What is Green's Theorem?

Green's Theorem is a mathematical relationship that connects the double integral of a two-dimensional region to the line integral along its boundary. In simpler terms, it allows us to calculate the area of a region by evaluating a line integral around its boundary.

Question 2: What is the purpose of Green's Theorem?

The main purpose of Green's Theorem is to simplify the calculation of certain types of integrals, such as area, flux, and work integrals, by relating them to line integrals. It is also used to prove other important theorems in mathematics, such as the Divergence Theorem and Stokes' Theorem.

Question 3: How is Green's Theorem used in real-world applications?

Green's Theorem is used in a variety of fields, including physics, engineering, and economics. It can be applied in calculations involving fluid flow, electric and magnetic fields, and surface integrals. It is also used in optimization problems and in calculating the work done by a force along a curved path.

Question 4: What is a line integral?

A line integral is a type of integral that calculates the sum of a function along a curve. It is used to calculate quantities such as work, circulation, and flux. The value of a line integral can depend on the path chosen, unlike a regular integral which only depends on the limits of integration.

Question 5: How do line integrals relate to Green's Theorem?

Green's Theorem relates the line integral around a closed curve to the double integral over the region enclosed by the curve. This means that we can use Green's Theorem to calculate the value of a line integral by evaluating a double integral instead. This can often simplify the calculation and make it more efficient.

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