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Evaluate the line integral

  • Thread starter Calpalned
  • Start date
  • #1
297
6

Homework Statement


Use Green's Theorem to evaluate the line integral along the given positively oriented curve. ##\int_C y^4 dx + 2xy^3 dy ##, C is the ellipse ##x^2 + 2y^2 = 2##.

Homework Equations


Change of variables: ##\int \int_S f(x(u,v),y(u,v)) |{\frac {\partial(x,y)}{\partial (u,v)}}| du dv ##

The Attempt at a Solution


How do I change the ellipse to a circle? Is there a way to determine u and v?
 

Answers and Replies

  • #2
Zondrina
Homework Helper
2,065
136
Please refer back to post #3 in your prior thread:

https://www.physicsforums.com/threads/using-greens-theorem.810989/

You could have posted there.

Anyway, after parametrizing the ellipse, you should know what the limits on the integral are for ##r## and ##\theta## without very much thought.

While a change of variables ##x = au## and ##y = bv## would map the ellipse to a circle of radius ##\sqrt{2}##, this is unnecessary and a bit of extra work since you still need to calculate the Jacobian.

I suggest just thinking about it for a second.
 

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