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Math Help - Integral curvilinear

  1. #1
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    Integral curvilinear

    Prove that integral curvilinear:

    \int_C yzdx + xzdy + yx^2dz

    is dependent on the path

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  2. #2
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    Quote Originally Posted by Apprentice123 View Post
    Prove that integral curvilinear:



    \int_C yzdx + xzdy + yx^2dz


    is dependent on the path

    You can choose two different paths to show the dependence of path.
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Apprentice123 View Post
    Prove that integral curvilinear:

    \int_C yzdx + xzdy + yx^2dz

    is dependent on the path

    The line integral for conservative vector fields are independent of path. Note that we can describe the vector field here as \bold{F} = \left< yz, xz, yx^2 \right>. show that this vector field is conservative, that is, find a function f(x,y,z) such that \nabla f = \bold{F}. that will show independents of path.

    it is not enough to pick two random paths and show the line integral is the same.
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  4. #4
    MHF Contributor Calculus26's Avatar
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    Remember if Curl F = 0 if and only if the line integral is independent of path

    There are several equivalent statemnts

    1Curl F = 0 (Fis irrotational
    2.F= gradf
    3.The line Integral is independent of path
    4.F is conservative
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  5. #5
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    rotF = (x^2-y)i - (2xy -y)j +(z-z)k is different of 0, not is conservative, then depends the path
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Calculus26 View Post
    Remember if Curl F = 0 if and only if the line integral is independent of path

    There are several equivalent statemnts

    1Curl F = 0 (Fis irrotational
    2.F= gradf
    3.The line Integral is independent of path
    4.F is conservative
    ah yes, i always forget that curlF = 0 condition. it is perhaps easier than finding the potential function here.
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  7. #7
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Apprentice123 View Post
    rotF = (x^2-y)i - (2xy -y)j +(z-z)k is different of 0, not is conservative, then depends the path
    well, that is not what the curl is, but the correct curl is non-zero still. so this is not independent of path
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  8. #8
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    What is curl ?
    Is Rotational ?

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  9. #9
    MHF Contributor Calculus26's Avatar
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    You may have learned it as rotation

    Del x F
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Apprentice123 View Post
    What is curl ?
    Is Rotational ?

    i suppose it is the same for you. curl is defined as follows:

    let \bold{F} = P \bold{i} + Q \bold{j} + R \bold{k}

    then \text{curl}\bold{F} = \nabla \times \bold{F} = \left(\frac {\partial R}{\partial y} - \frac {\partial Q}{\partial z} \right) \bold{i} + \left(\frac {\partial P}{\partial z} - \frac {\partial R}{\partial x} \right) \bold{j} + \left(\frac {\partial Q}{\partial x} - \frac {\partial P}{\partial y} \right) \bold{k}
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  11. #11
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    I learned how rotational

    Thanks, yours are helping me a lot
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