Understanding Parameterization in Divergence-Free Vector Fields

[JD571]

Hi, I am having a lot of trouble on the parameterization part of this problem:

Suppose there is a cylindrical container of liquid with circular flow velocity given in cylindrical coordinates by v(r,$$\theta$$,z) = <-f(r,z)sin$$\theta$$, f(r,z)cos$$\theta$$,0> for some function f(r,z) which we will determine. The vector field is divergence free.

Find a curve, x(t), that represents the path followed by a particle of liquid at radius r. So, x’(t) = v(x(t)) and x(0) = (r,0,z). Use this to find$$\theta$$(t), v(t) and a(t) = v’(t). Then write a in therms of r,$$\theta$$, z.

This is a long problem and I understand how to do the rest of it but this section is really confusing to me and I can't think of anything that would work. Thanks for any help.

 

[0rthodontist]

I'm going to assume that <-f(r,z)sin$$\theta$$, f(r,z)cos$$\theta$$,0> is in rectangular coordinates because the problem is much easier that way.

To solve this problem you have to find the integral curve of a particle at radius r. Look at the velocity field, v.

--What if v were <f(r,z)cos $$\theta$$, f(r,z)sin $$\theta$$,0> instead? Could you picture what that would look like?
--What is the relationship between <cos $$\theta$$, sin $$\theta$$> and <-sin $$\theta$$, cos $$\theta$$>? (try the dot product)
--From those two you should be able to get an idea of what v looks like, and then a good guess at what the integral curves are.

 

[JD571]

Thanks for the help but I have never heard of an integral curve before, I looked around online and I couldn't get a solid definition of what one was in terms that I understood. This is for a multivariable calc class where we just finished up the divergence theorem. I don't know if you can think of another way to do it or try to explain it to me but thanks for the effort.

 

[0rthodontist]

If you haven't covered integral curves yet, which would surprise me, you will soon so you might as well learn what they are.