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Thread: Conservative Vector Field Function

  1. #1
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    Conservative Vector Field Function

    Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F= ∇f.

    F(x, y, z) = 2xy i + (x^2 + 2yz) j + y^2 k

    I found the vector field to be conservative but how do I find the function?

    The book's answer is x^2 y + y^2 z + K
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  2. #2
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    Re: Conservative Vector Field Function

    Quote Originally Posted by USNAVY View Post
    Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F= ∇f.
    F(x, y, z) = 2xy i + (x^2 + 2yz) j + y^2 k
    I found the vector field to be conservative but how do I find the function?
    The book's answer is x^2 y + y^2 z + K
    I like the notation: $F(x,y,z)=I(x,y,z)\bf{i}+J(x,y,z)\bf{j}+K(x,y,z) \bf {k}$ it makes it to keep up with which is which.
    In the above, $I(x,y,z)=2xy,~J(x,y,z)=x^2+2yz,~\&~K(x,y,z)=y^2$
    For conservative note: $\begin{array}{l}K_y=J_z=2y\\K_x=I_z=0\\J_x=I_y=2x \end{array}$

    This will always give you the primitive:
    $\displaystyle{\Theta (x,y,z) = \int_0^x {I(t,0,0)dt} + \int_0^y {J(x,t,0)dt} + \int_0^z {K(x,y,t)dt}}$

    See if you get what the text has. I did.
    Thanks from USNAVY
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    Re: Conservative Vector Field Function

    Can you explain the "conservative note" in your reply?
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    Re: Conservative Vector Field Function

    Quote Originally Posted by USNAVY View Post
    Can you explain the "conservative note" in your reply?
    You figure it out.
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