Can tubes be minimal surfaces?

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SUMMARY

The discussion centers on the proof that tubes cannot be minimal surfaces. A tube is defined as a surface generated by circles of constant radius in the normal plane of a space curve, represented by the parametric equations x = r cos(t), y = r sin(t), and z = t. To establish that tubes are not minimal surfaces, one must demonstrate that their mean curvature is non-zero. The relevant equation for the tube's surface is F(u, v) = γ(u) + R(cos(u)N(v) + sin(u)B(v)).

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Homework Statement



Prove that there are no tubes that are minimal surfaces

Homework Equations


F(u, v) = γ(u) + R(cosuN(v) + sinuB(v))


The Attempt at a Solution



A tube is defined to be the surface formed by drawing circles with constant radius in the normal plane in a space curve.

I know that a minimal surface is a surface with a mean curvature of zero. So to prove the tubes aren't minimal surfaces, I need to show that the mean curvature is non-zero. I just don't know what the first step to take here is. Any tips/suggestions?
 
Last edited:
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Assume the curve is given by the parametric equation x= f(t), y= g(t), z= h(t). Can you write parametric equations for a point on the tube?
 
Edited the first post for a relevant equation.

Wouldn't x=rcost, y=rsint and z=t?
 

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