Solve the following integral using Stokes Theorem.











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I am asked to evaluate the following integral:
$$intint text{curl} vec{F} cdot dvec{S}$$
where $F = (y,-x,zx^3y^2)$ and $S$ is the surface given by $x^2 + y^2 + 3z^2 = 1$ with $z leq 0$.



I did not have any problem with any other exercises of this kind. But this one is hard.



Any hint would be appreciated.






surface-integrals stokes-theorem







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  • What exactly are you having difficulties with? Have you found the boundary of the surface?
    – Maxim
    Nov 17 at 19:24










  • I am having difficulties with all the exercie. I dont know how to parametrize that boundary. Do you have any idea?
    – TheNicouU
    Nov 18 at 0:16










  • The surface is the lower half of the ellipsoid, lying below the plane $z = 0$. The boundary lies in that plane. Substitute $z = 0$ into the equation of the ellipsoid.
    – Maxim
    Nov 18 at 1:04












  • So I just parametrize $x^2 + y^2 = 1$ and then calculate $F$ over that parametrization?
    – TheNicouU
    Nov 18 at 1:16






  • 1




    Correct, then you integrate $mathbf F cdot dmathbf s$.
    – Maxim
    Nov 18 at 1:53















up vote
0
down vote

favorite
1












I am asked to evaluate the following integral:
$$intint text{curl} vec{F} cdot dvec{S}$$
where $F = (y,-x,zx^3y^2)$ and $S$ is the surface given by $x^2 + y^2 + 3z^2 = 1$ with $z leq 0$.



I did not have any problem with any other exercises of this kind. But this one is hard.



Any hint would be appreciated.






surface-integrals stokes-theorem







share|cite|improve this question






















  • What exactly are you having difficulties with? Have you found the boundary of the surface?
    – Maxim
    Nov 17 at 19:24










  • I am having difficulties with all the exercie. I dont know how to parametrize that boundary. Do you have any idea?
    – TheNicouU
    Nov 18 at 0:16










  • The surface is the lower half of the ellipsoid, lying below the plane $z = 0$. The boundary lies in that plane. Substitute $z = 0$ into the equation of the ellipsoid.
    – Maxim
    Nov 18 at 1:04












  • So I just parametrize $x^2 + y^2 = 1$ and then calculate $F$ over that parametrization?
    – TheNicouU
    Nov 18 at 1:16






  • 1




    Correct, then you integrate $mathbf F cdot dmathbf s$.
    – Maxim
    Nov 18 at 1:53













up vote
0
down vote

favorite
1









up vote
0
down vote

favorite
1






1





I am asked to evaluate the following integral:
$$intint text{curl} vec{F} cdot dvec{S}$$
where $F = (y,-x,zx^3y^2)$ and $S$ is the surface given by $x^2 + y^2 + 3z^2 = 1$ with $z leq 0$.



I did not have any problem with any other exercises of this kind. But this one is hard.



Any hint would be appreciated.






surface-integrals stokes-theorem







share|cite|improve this question













I am asked to evaluate the following integral:
$$intint text{curl} vec{F} cdot dvec{S}$$
where $F = (y,-x,zx^3y^2)$ and $S$ is the surface given by $x^2 + y^2 + 3z^2 = 1$ with $z leq 0$.



I did not have any problem with any other exercises of this kind. But this one is hard.



Any hint would be appreciated.






surface-integrals stokes-theorem




surface-integrals stokes-theorem






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 16 at 19:10









TheNicouU

171111




171111












  • What exactly are you having difficulties with? Have you found the boundary of the surface?
    – Maxim
    Nov 17 at 19:24










  • I am having difficulties with all the exercie. I dont know how to parametrize that boundary. Do you have any idea?
    – TheNicouU
    Nov 18 at 0:16










  • The surface is the lower half of the ellipsoid, lying below the plane $z = 0$. The boundary lies in that plane. Substitute $z = 0$ into the equation of the ellipsoid.
    – Maxim
    Nov 18 at 1:04












  • So I just parametrize $x^2 + y^2 = 1$ and then calculate $F$ over that parametrization?
    – TheNicouU
    Nov 18 at 1:16






  • 1




    Correct, then you integrate $mathbf F cdot dmathbf s$.
    – Maxim
    Nov 18 at 1:53


















  • What exactly are you having difficulties with? Have you found the boundary of the surface?
    – Maxim
    Nov 17 at 19:24










  • I am having difficulties with all the exercie. I dont know how to parametrize that boundary. Do you have any idea?
    – TheNicouU
    Nov 18 at 0:16










  • The surface is the lower half of the ellipsoid, lying below the plane $z = 0$. The boundary lies in that plane. Substitute $z = 0$ into the equation of the ellipsoid.
    – Maxim
    Nov 18 at 1:04












  • So I just parametrize $x^2 + y^2 = 1$ and then calculate $F$ over that parametrization?
    – TheNicouU
    Nov 18 at 1:16






  • 1




    Correct, then you integrate $mathbf F cdot dmathbf s$.
    – Maxim
    Nov 18 at 1:53
















What exactly are you having difficulties with? Have you found the boundary of the surface?
– Maxim
Nov 17 at 19:24




What exactly are you having difficulties with? Have you found the boundary of the surface?
– Maxim
Nov 17 at 19:24












I am having difficulties with all the exercie. I dont know how to parametrize that boundary. Do you have any idea?
– TheNicouU
Nov 18 at 0:16




I am having difficulties with all the exercie. I dont know how to parametrize that boundary. Do you have any idea?
– TheNicouU
Nov 18 at 0:16












The surface is the lower half of the ellipsoid, lying below the plane $z = 0$. The boundary lies in that plane. Substitute $z = 0$ into the equation of the ellipsoid.
– Maxim
Nov 18 at 1:04






The surface is the lower half of the ellipsoid, lying below the plane $z = 0$. The boundary lies in that plane. Substitute $z = 0$ into the equation of the ellipsoid.
– Maxim
Nov 18 at 1:04














So I just parametrize $x^2 + y^2 = 1$ and then calculate $F$ over that parametrization?
– TheNicouU
Nov 18 at 1:16




So I just parametrize $x^2 + y^2 = 1$ and then calculate $F$ over that parametrization?
– TheNicouU
Nov 18 at 1:16




1




1




Correct, then you integrate $mathbf F cdot dmathbf s$.
– Maxim
Nov 18 at 1:53




Correct, then you integrate $mathbf F cdot dmathbf s$.
– Maxim
Nov 18 at 1:53















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