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What does it mean that the gradient is perpendicular/paralell to a vector?

  1. Oct 20, 2011 #1
    For a solenoidal velocity field [ tex ] \nabla \cdot \mathbf{u} [ /tex ] which means that [ tex ] \nabla [/tex ] is perpendicular to [ tex ] \mathbf{u} [ /tex ].

    Similarly, for an irrotational velocity field [ tex ] \nabla \times \mathbf{u} [ /tex ] which means that [ tex ] \nabla [/tex ] is parallel to [ tex ] \mathbf{u} [ /tex ].

    So what exactly does it mean physically to have a gradient (of nothing) parallel/perpendicular to a vector?

    PS - whats up with latex not working?
     
    Last edited: Oct 20, 2011
  2. jcsd
  3. Oct 20, 2011 #2
    Don't put spaces in square brackets for tags. I fixed this issue in the quote above.

    What you've posted doesn't make much sense. There seems to be many problems with it. For a solenoidal velocity field [itex] \mathbf{u} [/itex], [itex] \nabla \cdot \mathbf{u} [/itex] is the divergence of u.

    [itex] \nabla [/itex] is known as http://mathworld.wolfram.com/Del.html" [Broken], and not "a gradient". It may be thought of like a vector of differential operators.
     
    Last edited by a moderator: May 5, 2017
  4. Oct 20, 2011 #3
    Re: What does it mean that del is perpendicular/paralell to a vector?

    Sorry, let me correct and ask again:


    For a solenoidal velocity field [tex] \nabla \cdot \mathbf{u} = 0[/tex] which means that [tex] \nabla [/tex] is perpendicular to [tex] \mathbf{u} [/tex].

    Similarly, for an irrotational velocity field [tex] \nabla \times \mathbf{u} = \mathbf{0}[/tex] which means that [tex] \nabla [/tex] is parallel to [tex] \mathbf{u} [/tex].

    So what exactly does it mean physically to have [tex] \nabla[/tex] parallel/perpendicular to the velocity vector?
     
  5. Oct 20, 2011 #4
    Del may be thought of as a vector of operators. Claiming that del is perpendicular to a vector in ℝ3 makes no sense, like assigning a real number value to the plus sign.
     
  6. Oct 20, 2011 #5
    Your answer is exactly what I was thinking but, as you can see from http://i.imgur.com/VmbKS.jpg"in my hydrodynamics lecture, it is not what my professor claims. Hence the confusion.
     
    Last edited by a moderator: Apr 26, 2017
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