• Support PF! Buy your school textbooks, materials and every day products Here!

Greens theorem

  • #1
189
0

Homework Statement



##\mathscr{C}## is an ellipse ##\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1##
and ##\vec{F}(x,y) = <xy^2, yx^2>##

write ##\displaystyle \int_\mathscr{C} \vec{F} \cdot d\vec{s}## as a double integral using greens theorem and evaluate

Homework Equations



##\displaystyle \int_\mathscr{C} (Pdx+Qdy) = \iint_\mathscr{C} \Bigg(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\Bigg)dA##

The Attempt at a Solution



seems to be I need to use ##\nabla \times \vec{F} = \Bigg(0,0, \frac{\partial F_{2}}{\partial x} - \frac{\partial F_{1}}{\partial y}\Bigg) = (2xy-2yx)=0##


not sure about the double integral though, figured maybe this

##\displaystyle \int_{-a}^{a} \int_{-\sqrt{1-\frac{x^2}{a^2}-b^2}}^{\sqrt{1-\frac{x^2}{a^2}-b^2}}0dydx=0##
 
Last edited:

Answers and Replies

  • #2
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
16,416
6,239
Your problem is two-dimensional so the double integral should be over the interior of the ellipse.

If you necessarily want to see it as a 3D problem and apply Stoke's theorem, then the surface can be any surface with the ellipse as its boundary.
 
  • #3
189
0
Your problem is two-dimensional so the double integral should be over the interior of the ellipse.

If you necessarily want to see it as a 3D problem and apply Stoke's theorem, then the surface can be any surface with the ellipse as its boundary.
this was close to an example in my book, so where is it incorrect?

the book showed taking curl F then integrating over the boundary of the region in the given problem.

I don't know Stoke's theorem yet so I'd rather not try to apply that to this
 
  • #4
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
16,416
6,239
You need to rethink your limits for the y-integral, but they do not really matter since the integrand is zero.

Stoke's theorem (in the simple classical form) relates the integral of the curl of a vector field over a surface with the line integral of the same field along the border of that surface.
 
  • #5
189
0
You need to rethink your limits for the y-integral, but they do not really matter since the integrand is zero.

Stoke's theorem (in the simple classical form) relates the integral of the curl of a vector field over a surface with the line integral of the same field along the border of that surface.
ah okay thanks, ##-b \le y \le b## in similar fashion to how x was treated?
 
  • #6
vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
14,536
1,149
No, you need to check your algebra when you solved for ##y##.
 
  • #7
189
0
##y=\pm \frac{b\sqrt{a^2-x^2}}{a}##


##\displaystyle\int_{-a}^{a}\int_{-\frac{b\sqrt{a^2-x^2}}{a}}^{\frac{b\sqrt{a^2-x^2}}{a}} 0 dydx##
 
  • #8
HallsofIvy
Science Advisor
Homework Helper
41,770
911
What is the integral of 0 over any region?
 
  • #9
189
0
What is the integral of 0 over any region?
It would be zero
 
  • #10
Zero.
 
  • #11
HallsofIvy
Science Advisor
Homework Helper
41,770
911
Yes, so there was no reason to worry about the limits on the integral to begin with. That was what Orodruin meant when he said "they do not really matter since the integrand is zero".
 

Related Threads for: Greens theorem

  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
8
Views
1K
  • Last Post
Replies
2
Views
870
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
4
Views
511
  • Last Post
Replies
4
Views
2K
Top