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Δ in derivative and partial derivative notation

  1. Jul 2, 2012 #1
    1. The problem statement, all variables and given/known data
    What does it mean when lowercase Delta (δ) is used in partial derivative and derivative notation? Does it make any difference? Or is it just a personal preference?


    2. Relevant equations
    -


    3. The attempt at a solution
    Google
     
  2. jcsd
  3. Jul 2, 2012 #2

    HallsofIvy

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    If you means something like [itex]\delta x[/itex] or [itex]\delta f[/itex], it means "a slight change in" x or f. It does NOT occur in "partial derivative and derivative notation" but you may see it in the definition of the derivative.
     
  4. Jul 2, 2012 #3

    Mark44

    Staff: Mentor

    The Greek character δ (lower-case Delta) is not used in either kind of derivative. In ordinary derivatives, d is used, as in $$\frac{dy}{dx}$$
    Here y is a function of a single variable. IOW, y = f(x).
    It's probably not too far wrong to think of this as the quotient of two differentials: dy and dx.

    For partial derivatives, a different character is used. As far as I know, it's not part of any alphabet.

    If z is a function of two variables, say x and y, then we can talk about two partial derivatives:
    $$\frac{\partial z}{\partial x}$$
    and $$\frac{\partial z}{\partial y}$$
     
  5. Jul 2, 2012 #4
    Usually it denotes a "virtual" derivative, one where time is held constant.

    If you have a position vector [itex]\textbf{r}[/itex] which is a function of several variables, [itex]\left \{ q_1,q_2,q_3,...,q_n \right \}[/itex] and time [itex]t[/itex], the the total differential displacement is given by:

    [tex]d\textbf{r}=\frac{\partial \textbf{r}}{\partial t}dt+\sum_{i=1}^n \frac{\partial \textbf{r}}{\partial q_i}dq_i[/tex]

    This is just the chain rule. The virtual displacement, however, is given by:

    [tex]\delta \textbf{r}=\sum_{i=1}^n \frac{\partial \textbf{r}}{\partial q_i}\delta q_i[/tex]

    Note that it holds time constant. Virtual displacement is very useful in areas that use Calculus of Variations, such as Lagrangian mechanics.
     
  6. Jul 2, 2012 #5
    It looks more like variations to me.

     
  7. Jul 2, 2012 #6

    HallsofIvy

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    But elfmotat's suggestion is good too.
     
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