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Math Help - Using Stokes theorem

  1. #1
    MHF Contributor arbolis's Avatar
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    Using Stokes theorem

    Let F(x,y,z)=x \vec i + y \vec j + yz \vec k.
    Calculate the flux of the curl of F through the surface of the semi-sphere x^2+y^2+z^2=4, z\leq 0, oriented positively. Use Stokes theorem. Then use another way that does not involve Stokes' theorem to solve the problem.

    My attempt : \iint_ S curl F \hat n dS= \int _{\partial S} F dS.
    curl F = z \vec i.

    n=\frac{-x}{\sqrt{4-x^2-y^2}}.
    Hence \iint_ S curl F \hat n dS=\iint _S \frac{-xz}{\sqrt{4-x^2-y^2}}dA= \iint _S -xdA=-\int _0^{2\pi} \int_ 0^2 r^2 \cos (\theta) drd\theta=0. How is that possible? The field is not conservative and the surface is closed, hence it is impossible that the net flux passing through the semi-sphere give 0.
    I want to tackle it by calculating \int _{\partial S} F dS but with no success. I parametrize \partial S : r(t)=(2\cos t , 2\sin t, 0). Thus r'(t)=(-2\sin t, 2 \cos t , 0).
    Now I don't know what to do, is it a line integral of a scalar field or vector field? Seems more like a vector field to me, let's try.
    \int _0^{2\pi} -4 \cos (t) \sin (t)+4 \cos (t) \sin (t)+0 dt =0. What's wrong?!
    Using the divergence theorem I got -\frac{64 \pi ^3}{3}. However I do not know where to check if I didn't made a mistake of sign, since they said the surface to be oriented positively. The divergence is worth 2+y so it can be positive or negative.

    I'm all confused. What's the difference between Stokes and Gauss' theorems? They can both calculate the flux through a surface, hence they're equivalent?!
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  2. #2
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    Quote Originally Posted by arbolis View Post
    Let F(x,y,z)=x \vec i + y \vec j + yz \vec k.
    Calculate the flux of the curl of F through the surface of the semi-sphere x^2+y^2+z^2=4, z\leq 0, oriented positively. Use Stokes theorem. Then use another way that does not involve Stokes' theorem to solve the problem.

    My attempt : \iint_ S curl F \hat n dS= \int _{\partial S} F dS.
    curl F = z \vec i.

    n=\frac{-x}{\sqrt{4-x^2-y^2}}.
    Hence \iint_ S curl F \hat n dS=\iint _S \frac{-xz}{\sqrt{4-x^2-y^2}}dA= \iint _S -xdA=-\int _0^{2\pi} \int_ 0^2 r^2 \cos (\theta) drd\theta=0. How is that possible? The field is not conservative and the surface is closed, hence it is impossible that the net flux passing through the semi-sphere give 0.
    I want to tackle it by calculating \int _{\partial S} F dS but with no success. I parametrize \partial S : r(t)=(2\cos t , 2\sin t, 0). Thus r'(t)=(-2\sin t, 2 \cos t , 0).
    Now I don't know what to do, is it a line integral of a scalar field or vector field? Seems more like a vector field to me, let's try.
    \int _0^{2\pi} -4 \cos (t) \sin (t)+4 \cos (t) \sin (t)+0 dt =0. What's wrong?!
    Using the divergence theorem I got -\frac{64 \pi ^3}{3}. However I do not know where to check if I didn't made a mistake of sign, since they said the surface to be oriented positively. The divergence is worth 2+y so it can be positive or negative.

    I'm all confused. What's the difference between Stokes and Gauss' theorems? They can both calculate the flux through a surface, hence they're equivalent?!
    your \bold{n} is very wrong! it's not even a vector! the correct one \bold{n}=\frac{-xi - yj - zk}{2} and the boundary of your surface, i.e. the circle C: \ x^2+y^2=4 is considered counter-clockwise.

    the answer would eventually be zero. using Stoke's theorem we have \int_C F \cdot d \bold{r} = \int_C xdx + ydy + yz dz = \int_C xdx + ydy = \int_C d \left( \frac{x^2+y^2}{2} \right) = 0.

    (you may also use Green's theorem in the last step to show that \int_C xdx + ydy = 0.)
    Last edited by NonCommAlg; August 1st 2009 at 07:12 PM.
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  3. #3
    MHF Contributor Calculus26's Avatar
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    Stokes Theorem is used to calculate a line integral in this case the line integral along the circle x^2 + y^2 = 4.

    Recall the line integral is the surface integral of curl F*n ds --

    If you will the circulation along the path.

    Gauss's Thm or the Divergence theorem as it sometimes known calculates
    the Flux of the vector field through the surface . If you Calculate the flux

    you use int(F*nds) you should get the same result as the divergence thm.

    But note this is not int( curlF*nds) which is the circulation

    The flux and circulation are entirely different quantities so there is no reason to expect them to be the same.

    The fact that the line integral is 0 does not imply the field is conservative-- If the field is conservative the line integral is 0 but the converse is not necessarily true.

    For Eg consider F = x^2 i + xy j along x= 2cos(t) y = 2sin(t)

    F is not conservative but the line integral is 0.

    Again the line integral is not computing the flux through the surface.

    To answer another question there is no such thing as the line integral of a scalar field --it only makes sense to talk about the line integral of a vector field.
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  4. #4
    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by NonCommAlg View Post
    your \bold{n} is very wrong! it's not even a vector! the correct one \bold{n}=\frac{-xi - yj - zk}{2}
    Oops! I have it as a vector with 3 components in my draft, but when I do the dot product of \bold n with curl F, it remains what I wrote. (Precisely my \frac{-x}{\sqrt{4-x^2-y^2}} was the \vec i component)
    By the way how did you get such an \bold n? Mine depends on x and y.

    and the boundary of your surface, i.e. the circle C: \ x^2+y^2=1 is considered counter-clockwise.
    you mean x^2+y^2=2? Thanks for the orientation.

    the answer would eventually be zero. using Stoke's theorem we have \int_C F \cdot d \bold{r} = \int_C xdx + ydy + yz dz = \int_C xdx + ydy = \int_C d \left( \frac{x^2+y^2}{2} \right) = 0.

    (you may also use Green's theorem in the last step to show that \int_C xdx + ydy = 0.)
    Thank you very much!
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  5. #5
    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by Calculus26 View Post
    Stokes Theorem is used to calculate a line integral in this case the line integral along the circle x^2 + y^2 = 4.

    Recall the line integral is the surface integral of curl F*n ds --

    If you will the circulation along the path.

    Gauss's Thm or the Divergence theorem as it sometimes known calculates
    the Flux of the vector field through the surface . If you Calculate the flux

    you use int(F*nds) you should get the same result as the divergence thm.

    But note this is not int( curlF*nds) which is the circulation

    The flux and circulation are entirely different quantities so there is no reason to expect them to be the same.

    The fact that the line integral is 0 does not imply the field is conservative-- If the field is conservative the line integral is 0 but the converse is not necessarily true.

    For Eg consider F = x^2 i + xy j along x= 2cos(t) y = 2sin(t)

    F is not conservative but the line integral is 0.

    Again the line integral is not computing the flux through the surface.

    To answer another question there is no such thing as the line integral of a scalar field --it only makes sense to talk about the line integral of a vector field.
    Wow, thank you very much for the clarifications.
    So when they say "the flux of the curl", they mean the circulation and not the flux through the surface?
    So I can't use Stokes and the divergence theorem to calculate the flux passing through a surface? (Only the divergence theorem works for this). This makes sense, otherwise I could write \iiint _{E} div F dV= \iint _S curl F \hat n dS. Which would be a connection between Stokes and Gauss' theorems.
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  6. #6
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    Quote Originally Posted by arbolis View Post

    you mean x^2+y^2=2? Thanks for the orientation.
    i meant x^2+y^2=4. it's fixed now.

    \bold{n}=\pm \frac{\nabla f}{||\nabla f||}, where f=x^2+y^2+z^2-4. we choose - in here because the direction of C is counter-clockwise (positive) and thus, by the right-hand rule, \bold{n} must be upward.
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  7. #7
    MHF Contributor Calculus26's Avatar
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    Wow, thank you very much for the clarifications.
    So when they say "the flux of the curl", they mean the circulation and not the flux through the surface?
    So I can't use Stokes and the divergence theorem to calculate the flux passing through a surface? (Only the divergence theorem works for this). This makes sense, otherwise I could write . Which would be a connection between Stokes and Gauss' theorems.
    You've got it Stokes thm gives the value of a line integral and
    Gauss's Theorem gives the flux ---there is no connection between the theorems of Gauss and Stokes
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  8. #8
    MHF Contributor arbolis's Avatar
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    By the way, how would you solve the problem without using Stokes' theorem? It is asked in the question.
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  9. #9
    MHF Contributor Calculus26's Avatar
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    For this eg as was suggested by NonCommAlg you could have used Green's Thm

    In General if you don't have a curve in the plane you'd have to find a parameterization and use int (F*r ' (t)dt)
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