Parameterization

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,[tex]\theta[/tex],z) = <-f(r,z)sin[tex]\theta[/tex], f(r,z)cos[tex]\theta[/tex],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[tex]\theta[/tex](t), v(t) and a(t) = v’(t). Then write a in therms of r,[tex]\theta[/tex], 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[tex]\theta[/tex], f(r,z)cos[tex]\theta[/tex],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 [tex]\theta[/tex], f(r,z)sin [tex]\theta[/tex],0> instead? Could you picture what that would look like?
--What is the relationship between <cos [tex]\theta[/tex], sin [tex]\theta[/tex]> and <-sin [tex]\theta[/tex], cos [tex]\theta[/tex]>? (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.