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Help me understand this:

  1. Feb 26, 2012 #1
    My teacher often writes for a scalar function F:

    dF = [itex]\nabla[/itex]F [itex]\bullet[/itex] dr

    Why is it you are allowed to do this. Shouldn't you use the pythagorean theorem?
     
  2. jcsd
  3. Feb 26, 2012 #2

    I like Serena

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    Hi aaaa202! :smile:

    It's an identity that follows from the definition of the derivative of a function of multiple coordinates.
    Consider that the total derivative is:
    $$dF = {\partial F \over \partial x}dx + {\partial F \over \partial y}dy + {\partial F \over \partial z}dz$$
    And the right hand side is:
    $$\nabla F \cdot d\mathbf{r} = \begin{bmatrix}{\partial F \over \partial x}\\{\partial F \over \partial y}\\{\partial F \over \partial x}\end{bmatrix} \cdot \begin{bmatrix}dx\\dy\\dz\end{bmatrix} = {\partial F \over \partial x}dx + {\partial F \over \partial y}dy + {\partial F \over \partial z}dz$$

    I guess you could say that he pythagorean theorem is not involved because any application of it cancels left and right in the identity.
     
  4. Feb 27, 2012 #3
    hmm yes I kinda got this already, but I'm just unsure how to interpret the term:

    dF = dF/dx * dx + dF/dy * dy + dF/dz *dz

    What does this infinitesimal bit of represent? Surely its not a vector, since the result is a scalar. Surely it can't be an infinitesimal part of its length either, since dF =√(dFx^2 + dFy^2 + dFz^2)
    So what does it mean this dF?
     
  5. Feb 27, 2012 #4

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    You can interpret infinitesimals as small delta's.

    dF is the change in F(x,y,z) if you change the coordinates by (dx,dy,dz) to F(x+dx,y+dy,z+dz).
    You can write this as: dF=F(x+dx,y+dy,z+dz)-F(x,y,z).
    This is the (scalar) difference in F between 2 points in space.
    As such dF =√(dFx^2 + dFy^2 + dFz^2) does not apply, since it is not a vector.
     
  6. Feb 27, 2012 #5
    Okay yes, but I just don't find the idea of adding up the changes of the function in respectfully the x-, y- and x- direction very interesting.
    So far I've seen it used, but rather length has been used (or maybe I'm wrong) - for instance in continuity considerations you require that f(x,y) is well defined inside the cirfumference of an infinitesimal circle.
     
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