Parameterization problem (1 Viewer)

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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.
 
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0rthodontist

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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.
 
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

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

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