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Math Help - Integral field vector

  1. #1
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    Integral field vector

    Explain how to relate the integral curvilinear of field vector F(x,y) = (x-y)i + (x+y)j with the area the region limited by a regular curve, simple and closed ? Apply the result and calculate the area of the ellipse:
    \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
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  2. #2
    MHF Contributor Calculus26's Avatar
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    Consider Green's Theorem

    For F = (x-y) i +(x +y) j

    d(x+y)/dx = 1 d(x-y)/dy = -1

    The line integral then is 2times the double integral over R , or twice the area of the region enclosed by your ellipse

    It is easier to compute this with the line integral than with a double integral.

    so the area enclosed is 1/2 the line integral

    parameterize C with x = acos(t) y = bsin(t)

    and now you are good to go.

    Remember you should get pi*ab
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  3. #3
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    Quote Originally Posted by Calculus26 View Post
    Consider Green's Theorem

    For F = (x-y) i +(x +y) j

    d(x+y)/dx = 1 d(x-y)/dy = -1

    The line integral then is 2times the double integral over R , or twice the area of the region enclosed by your ellipse

    It is easier to compute this with the line integral than with a double integral.

    so the area enclosed is 1/2 the line integral

    parameterize C with x = acos(t) y = bsin(t)

    and now you are good to go.

    Remember you should get pi*ab

    I find 2 \pi ab because my integral \int_0^{2 \pi} the correct is \int_0^{ \pi} ?
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  4. #4
    MHF Contributor Calculus26's Avatar
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    No-- remember the area is 1/2 the line integral--so 0 t0 2pi is correct

    if you use 0 to pi you don't have a closed curve
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  5. #5
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    Quote Originally Posted by Calculus26 View Post
    No-- remember the area is 1/2 the line integral--so 0 t0 2pi is correct

    if you use 0 to pi you don't have a closed curve

    Ok thank you the answer is 2 \pi ab
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  6. #6
    MHF Contributor Calculus26's Avatar
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    No the answer is pi *ab The line integral is 2pi*ab and the AREA is 1/2 this
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  7. #7
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    Quote Originally Posted by Calculus26 View Post
    No the answer is pi *ab The line integral is 2pi*ab and the AREA is 1/2 this
    why the AREA is \frac{1}{2} ?
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  8. #8
    MHF Contributor Calculus26's Avatar
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    See my first post on this -- I explained it there
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  9. #9
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    Quote Originally Posted by Calculus26 View Post
    See my first post on this -- I explained it there
    Ok te green theorem is 2 AREA 1/2. If green theorem is 1 AREA 1, if 3 AREA 1/3 ??
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  10. #10
    MHF Contributor Calculus26's Avatar
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    That would make sense wouldn't it ?

    If F = f i +g j

    And if dg/dx- df/dy is a constant k then Green's theorem gives k *Area
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  11. #11
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    Quote Originally Posted by Calculus26 View Post
    That would make sense wouldn't it ?

    If F = f i +g j

    And if dg/dx- df/dy is a constant k then Green's theorem gives k *Area
    Yes thank you
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