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Math Help - explanation of how to find vector field lines

  1. #1
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    explanation of how to find vector field lines

    Hello,
    I have a function f=(x^2+y^2)/z , F=grad(f)=2x/z i + 2y/z j + (x^2+y^2)/z^2 k.
    I am supposed to find the field lines of F but I can't really understand how to do this from reading the book.
    I did it in 2D and kind of got it but in 3D I am lost.
    Any help here is really appreciated.
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  2. #2
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    Re: explanation of how to find vector field lines

    First, if there exist such an "f" then we must have the mixed derivatives the same. That works for "xy" but
    \frac{\partial f}{\partial x\partial z}= \frac{\partial}{\partial x}\frac{x^2+ y^2}{z^2}= \frac{2x}{z^2}
    but \frac{\partial^2 f}{\partial z\partial y}= \frac{\partial}{\partial z}\frac{2x}{}= -\frac{2x}{z^2}.
    That is, there is a incorrect sign so there is NO SUCH f.

    I am going to assume that you meant \grad(f)= 2x/z i+ 2y/z j- \frac{x^2+y^2}{z^2} k

    The concept is the same as in two dimensions, there is just one more equation. If grad f= 2x/z i+ 2y/z j+ (x^2+ y^2)/x^2 k then we must have \frac{\partial f}{\partial x}= 2x/z, \partial f/\partial y}= 2y/z, \partial f}{\partial z}=\frac{x^2+ y^2}{z^2}.

    Starting from \partial f}{\partial x}= \frac{2x}{z} and integrating, f(x,y,z)= \frac{x^2}{z}+ g(y, z). The "constant of integration" can be any function of y and z. Differentiating that with respect to y, \frac{\partial f}{\partial y}= 0+ \frac{\partial g}{\partial y}= 2y/z so that g(x, y)= \frac{y^2}{z}+ h(z). Since g(y, z) is a function of y and z, that "constant of integration" can be a function of z but not x or y.

    Now, we have that f(x,y,z)= \frac{x^2}{z}+ g(y,z)= \frac{x^2}{z}+ \frac{y^2}{z}+ h(z)= \frac{x^2+ y^2}{z}+ h(z). Differentiate that with respect to z: \frac{-x^2+ y^2}{z^2}+ h'(z)= -\frac{x^2+ z^2}{z^2}. That just says that h'(z)= 0 so that h is a constant: f(x, y, z)= \frac{x^2+y^2}{z}+ C.
    Thanks from lytwynk
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  3. #3
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    Re: explanation of how to find vector field lines

    A yes... Got it. The question also asks though to describe the equipotential surfaces as well as find the field lines of F. I understand the equipotential surfaces ie. level surfaces of f=C therefore here it would be z=C(x^2+y^2) however I don't get how to find the field lines... Can you help with this?
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