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Math Help - Conservative vector field

  1. #1
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    Exclamation Conservative vector field

    Hey, id love some help in doing this questions. Cheers
    Attached Thumbnails Attached Thumbnails Conservative vector field-mathq7.png  
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    Re: Conservative vector field

    Quote Originally Posted by Brennox View Post
    Hey, id love some help in doing this questions. Cheers
    1. Integrate \frac{\partial \phi}{\partial x} = 4y~cos(4xy) over x. This will be a function f(x, y, z) plus some function g(y, z).
    For example, let \frac{\partial \phi}{\partial x} = 3x + 4xy \implies \phi = \frac{3}{2}x^2 + 2x^2y + g(y, z)
    The function g is a constant as far as the integration is concerned, so we have g(y, z) as an arbitrary "constant."

    Problems 2 and 3 use the same idea. See if you can get those for now.

    -Dan
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    Re: Conservative vector field

    Quote Originally Posted by Brennox View Post
    Hey, id love some help in doing this questions. Cheers
    Here is a sure fire way to reconstruct the primitive of a conservative field.

    Let I(x,y,z)=4y\cos(4xy),~J(x,y,z)=4x\cos(4xy),~\&~K(x  ,y,z)=3z^2 then

    \nu  = \int_0^x {I(t,0,0)dt}  + \int_0^y {J(x,t,0)dt}  + \int_0^z {K(x,y,t)dt}
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    Re: Conservative vector field

    so i got sin(4xy) once integrated with respect to x. How do i get g(y,z)?

    Quote Originally Posted by topsquark View Post
    1. Integrate \frac{\partial \phi}{\partial x} = 4y~cos(4xy) over x. This will be a function f(x, y, z) plus some function g(y, z).
    For example, let \frac{\partial \phi}{\partial x} = 3x + 4xy \implies \phi = \frac{3}{2}x^2 + 2x^2y + g(y, z)
    The function g is a constant as far as the integration is concerned, so we have g(y, z) as an arbitrary "constant."

    Problems 2 and 3 use the same idea. See if you can get those for now.

    -Dan
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    Re: Conservative vector field

    Quote Originally Posted by Plato View Post
    Here is a sure fire way to reconstruct the primitive of a conservative field.

    Let I(x,y,z)=4y\cos(4xy),~J(x,y,z)=4x\cos(4xy),~\&~K(x  ,y,z)=3z^2 then

    \nu  = \int_0^x {I(t,0,0)dt}  + \int_0^y {J(x,t,0)dt}  + \int_0^z {K(x,y,t)dt}
    so i got sin(4xy) for 1st, sin(4xy) for 2nd, z^3 for 3rd and 2sin(4xy) + z^3 for a final answer. But i got it wrong, where am i wrong?
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    Re: Conservative vector field

    Quote Originally Posted by Brennox View Post
    so i got sin(4xy) for 1st, sin(4xy) for 2nd, z^3 for 3rd and 2sin(4xy) + z^3 for a final answer. But i got it wrong, where am i wrong?
    The fist one is 0, so the final answer is \sin(4xy)+z^3~.
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