Verify Greens theorem half done

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Homework Help Overview

The problem involves verifying Green's theorem for a specific line integral over a triangular contour defined by the vertices (0,0), (1,1), and (2,0). The original poster seeks to demonstrate that both sides of the theorem yield the same result.

Discussion Character

  • Exploratory, Problem interpretation, Assumption checking

Approaches and Questions Raised

  • The original poster expresses uncertainty about how to parametrize the contour for the line integral and mentions being stuck at the initial steps. Some participants suggest breaking the contour into segments and propose specific parameterizations, while others question the reasoning behind choosing certain parameters.

Discussion Status

Participants are actively exploring different parameterizations for the line segments of the contour. There is a focus on understanding how to derive these parameters and the relationship between points on the line segments. No consensus has been reached, but guidance on examining the contour and considering multiple parameterizations has been provided.

Contextual Notes

The discussion highlights the need for clarity on parameterization methods and the original poster's struggle with the initial steps of the verification process. There may be assumptions about the familiarity with line equations and parameterization techniques that are being questioned.

Jaqsan
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Homework Statement



Verify Greens theorem for the line integral ∫c xydx + x^2 dy where C is the triangle with vertices (0,0) (1,1) (2,0). This means show both sides of the theorem are the same.


Homework Equations


∫c <P,Q> dr = ∫∫dQ/dx -dP/dy dA
∫c xydx + x^2dy

The Attempt at a Solution



Ok, I know the how to verify it with Greens Theorem. My answer comes out to be 1, I just can't figure out the steps to parametrize it or whatever I need to do to solve it without Greens Theorem.

I'm honestly stuck at the first step
∫<P,Q> dr = ∫<xy, x^2> dr
 
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You have to break the contour up into three pieces. For example, the first leg might go from (0,0) to (1,1). The parameterization you could use would be x=t, y=t, where t runs from 0 to 1.
 
How do I come up with the parameters x=t and y=t. There's no equation to get it from.
 
You examine the piece of the contour under consideration. Remember, t is just a parameter. You are trying to find a relation using t which gives all of the (x,y) coordinates on a straight line segment starting with the point (0,0) and ending at the point (1,1). [Hint: you get to use your imagination. There may be more than one parameterization.]
 
For the given contour, the points (0,0) and (1,1) are connected by a line segment, right? So what would the equation of that line be?
 

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