Partial differential: partial scalar partial vector

  1. Hi,

    I have a problem to find the meaning of a special partial differential: partial scalar partial vector.

    i.e. dF/dn where F is a scalar and n is a i.e. normal vector. This is a partial diff.

    n could be a vector consisting of partial differentials, (dT/dx,dT/dy)

    I have looked in literature but found nothing.

    Can someone help me?

    Thank you very much
    /Andreas
     
  2. jcsd
  3. tiny-tim

    tiny-tim 26,041
    Science Advisor
    Homework Helper

    Welcome to PF!

    Hi Andreas! Welcome to PF! :smile:

    (have a curly d: ∂ :wink:)
    No such thing … you can't have d(scalar)/d(vector) or ∂(scalar)/∂(vector). :wink:

    But (for example, when calculating flux) you can have ∂F/∂n or d(F.n^)/dn, where n^ is the unit vector in the normal direction, and n is the distance in that direction. :smile:
     
  4. LCKurtz

    LCKurtz 8,391
    Homework Helper
    Gold Member

    Perhaps you are thinking about directional derivatives. If F(x,y,z) is a scalar function (perhaps the temperature at (x,y,z)), and V is a vector, then the rate of change of F in the direction of V is:

    [tex]\frac {\partial F}{\partial \hat v} = D_{\hat v}(F) = \nabla F \cdot \hat V[/tex]

    where [tex] \hat V[/tex] is a unit vector in the direction of V.
     
  5. Hi tiny-tim and LCKurtz,

    This is exactly the explaination I am looking for! I had a hard time to understand the meaning of such partial derivative. And it with [tex]\partial[/tex] it should be :smile:

    It is the rate of F in the direction of some unit vector n that is normal to an arbitrary surface. I have no problem to find the growth rate of F in x and y but when it came to a other direction depending on other things, it became a problem. But now I understand how to do it!

    I actually did not think of the thing that n must be a unit vector. I will make it a unit vector!

    Thank you very much for your help on this problem, I have been struggling to find the answer for a long time!
     
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