Normal vector fields

  • #1
WannabeNewton
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Hi guys, I was wondering if anyone could post or point me to a proof of the statement that given a hypersurface [itex]\Sigma [/itex], specified by setting a function [itex]f(x) = const.[/itex], the vector field [itex]\xi ^{\mu } = \triangledown ^{\mu }f = g^{\mu \nu }\triangledown _{\nu }f [/itex] will be normal to [itex]\Sigma [/itex] in the sense that [itex]\xi ^{\mu }[/itex] will be orthogonal to all [itex]u\in T_{p}(\Sigma )[/itex] for some [itex]p\in \Sigma [/itex]. I tried to visualize it for trivial manifolds but I really couldn't. If there isn't really a proof of any kind could someone at least make the statement more intuitive. Thanks in advance.
 

Answers and Replies

  • #2
HallsofIvy
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The simplest way of looking at it is to use the "directional derivative". If f(X) is a real valued function (X is a point in some space) and [itex]\vec{v}[/itex]
is a vector in that space, then the "derivative of f in the diretion [itex]\vec{v}[/itex]", the rate at which f is increasing in that direction, is given by [itex]D_\vec{v} f= \nabla f\cdot\vec{v}[/itex]. In particular, if [itex]\vec{v}[/itex]
is tangent to the surface f(X)= constant then f does not change: [itex]D_\vec{v} f= \nabla f\cdot \vec{v}= 0[/itex] which says directly that [itex]\nabla f[/itex] and [itex]\vec{v}[/itex]
are perpendicular. Since [itex]\vec{v}[/itex]
could be any vector in the tangent plane, [itex]\nabla f[/itex] is perpendicular to the tangent plane, i.e. normal to the surface.
 
  • #3
WannabeNewton
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Oh, the fact that I saw it in component form was confusing me. If I could just restate what you said, for my own clarification: since [itex]\Sigma [/itex] is specified by [itex]f(x) = const.[/itex] then on [itex]\Sigma [/itex], [itex]\triangledown _{\nu }f = 0 [/itex] for all [itex]\nu [/itex] so [itex]\xi ^{\mu } = g^{\mu \nu }\triangledown _{\nu }f = 0[/itex] identically as well on the hypersurface. Then, for all [itex]\mathbf{v}\in T_{p}(\Sigma )[/itex], [itex]\xi ^{\mu }v_{\mu } = 0[/itex] and since [itex]\mathbf{v}[/itex] lies on the tangent space to [itex]\Sigma [/itex] at p, [itex]\boldsymbol{\xi}[/itex] must be orthogonal to it so it is then normal to [itex]\Sigma [/itex]. Is this basically the gist of what you said?
Thank you so very much by the way. Cleared my head up; I think my problem was that I kept thinking of the function everywhere outside the hypersurface and not just what happens on it.
 
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