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.

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

but .

That is, there is a incorrect sign so there is NO SUCH f.

I am going to assume that you meant

The concept is the same as in two dimensions, there is just one more equation. If then we must have , , .

Starting from and integrating, . The "constant of integration" can be any function of y and z. Differentiating that with respect to y, so that . 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 . Differentiate that with respect to z: . That just says that h'(z)= 0 so that h is a constant: .

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?