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Math Help - Conservative Vector Fields

  1. #1
    Junior Member krtica's Avatar
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    Conservative Vector Fields

    q1:
    If C is the curve given by r(t)=<1+3sin(t), 1+5sin^2(t), 1+5sin^3(t)>, 0≤t≤π/2 and F is the radial vector field F(x, y, z)=<x, y, z>, compute the work done by F on a particle moving along C.

    Work= int (F dot dr)

    If F is the potential function(?), do I integrate F with respect to each variable, then substitute the values of x, y, and z in r(t)? Would this then just be dotted into 1 since d/dt sin(t) is cos(t), which is 0 at π/2? Would my answer be something like (4^2/2)+(6^2/2)+(6^2/2)?

    q2:
    For each of the following vector fields F , decide whether it is conservative or not. Type in a potential function f (that is, ). If it is not conservative, type N. A.

    B.

    C.

    D.

    E.


    A would be -7 +(-7), which is conservative? All of my answers are incorrect aside from B. I do not understand how D is incorrect though.
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  2. #2
    MHF Contributor
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    Quote Originally Posted by krtica View Post
    q1:
    If C is the curve given by r(t)=<1+3sin(t), 1+5sin^2(t), 1+5sin^3(t)>, 0≤t≤π/2 and F is the radial vector field F(x, y, z)=<x, y, z>, compute the work done by F on a particle moving along C.

    Work= int (F dot dr)

    If F is the potential function(?), do I integrate F with respect to each variable, then substitute the values of x, y, and z in r(t)? Would this then just be dotted into 1 since d/dt sin(t) is cos(t), which is 0 at π/2? Would my answer be something like (4^2/2)+(6^2/2)+(6^2/2)?

    q2:
    For each of the following vector fields F , decide whether it is conservative or not. Type in a potential function f (that is, ). If it is not conservative, type N. A.

    B.

    C.

    D.

    E.


    A would be -7 +(-7), which is conservative? All of my answers are incorrect aside from B. I do not understand how D is incorrect though.

    If \frac{\partial M}{\partial y}=\frac{\partial N}{\partial x} or \frac{\partial P}{\partial y}=\frac{\partial N}{\partial z}, \frac{\partial P}{\partial x}=\frac{\partial M}{\partial z}, and \frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}, then the field is conservative.

    F(x,y,z)=Mi+Nj+Pk
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  3. #3
    Senior Member Tinyboss's Avatar
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    To find a potential function, you need to do "partial integration" with respect to each variable--i.e., considering the other variable(s) to be constant(s), and where your "constant of integration" is an arbitrary function in the other variable(s). If you can choose these arbitrary functions such that everything is equal, then that's your potential function. If you can't, then the field is not conservative.
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