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Proof: Everywhere Tangent to Curve?

  • Thread starter bakra904
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  • #1
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Proof: Everywhere Tangent to Curve??

If the function v depends on x and y, v(x,y) and we know there exists some function psi(x,y) such that
vx = partial w.r.t (y) of psi
vy= -(partial w.r.t (x) of psi)

show that the curves psi(x,y) = constant, are everywhere tangent to v.
 
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Answers and Replies

  • #2
Usually you are supposed to show effort to get there, but I think this is a case where either you get it or you don't.

[tex]\nabla \psi[/tex] is normal to surfaces of constant [tex]\psi[/tex] and [tex]v\cdot \nabla \psi = 0[/tex]. Fill in the rest.
 
  • #3
6
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Usually you are supposed to show effort to get there, but I think this is a case where either you get it or you don't.

[tex]\nabla \psi[/tex] is normal to surfaces of constant [tex]\psi[/tex] and [tex]v\cdot \nabla \psi = 0[/tex]. Fill in the rest.
Thanks a bunch! I'm a new poster and did not know about the effort rule...I had worked on it but did not post what I had worked on.

I was trying to use the fact that if v = [tex]\nabla \times[/tex] [tex]\psi[/tex],

then that would imply that [tex]\psi[/tex] is a stream function, which in cartesian co-ordinates would reduce to:

Vx = [tex]\frac{\partial\psi}{\partial y}[/tex] and Vy = - [tex]\frac{\partial\psi}{\partial x}[/tex]

which is basically what the problem had to begin with. Then, since I know that [tex]\psi[/tex] (x,y) is a stream function, doesn't it have to be tangent to v by virtue of the fact that its a streamline?
 
  • #4
Are you trying to curl a scalar field??
 
  • #5
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oh right...i overlooked that part. thanks!
 
  • #6
6
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so basically [tex] v. \nabla\psi = 0[/tex] which proves that [tex] v [/tex] and [tex]\nabla\psi[/tex] are perpendicular (since their dot product is 0) and so [tex]\psi[/tex] must be tangent to [tex] v [/tex]
 

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