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Can saomebody explain me whats differential mean?

  1. Jul 23, 2007 #1


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    Can saomebody explain me whats differential mean??

    can saomebody explain me whats differential mean??
  2. jcsd
  3. Jul 23, 2007 #2
  4. Jul 24, 2007 #3


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    ı know that.. thanks
  5. Jul 24, 2007 #4


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    It depends upon what context you are using the word differential in. Could you describe what you're studying?
  6. Jul 25, 2007 #5


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    Well, while "differentials" are used in "differential equations", you need to go back to calculus to find a definition. I'm moving this thread to "Calculus and Analysis".
  7. Jul 26, 2007 #6
    let Δx = dx be an increment given to x.
    Δy = f (Χ +ΔΧ) - f(Χ)
    is called the increment in y=f (x). if f(x) is continuous and has a continuous first derivative in an interval, then
    Δy = f ' (Χ) ΔΧ + ε ΔΧ = f'(x) dx + ε dx

    where ε goes to 0 as ΔΧ goes to 0.
    the expression
    dy = f' (x) dx.
    where dy is called the differential of y
    Last edited by a moderator: Jul 26, 2007
  8. Aug 11, 2007 #7


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    A way of seeing a differential that makes sense to me is this:

    The differential measures the change of the function along the

    tangent line, plane (depending on wether you are in R,R^2, etc.)

    A function is differentiable , if the change in its values can be

    approximated locally by a linear function, to any degree of precision

    ( in a delta-epsilon sense). The linear function that does this approximation

    is itself the derivative.

    This is one way of seeing the definition:

    || f(x+h)-f(x) -L(h)||=0
    Limh_>0 __________

    An example:

    For f(x)=x^2, we have:

    df=f'(x)dx , so df=2xdx.

    This means that the change in the value of f(x)=x^2 in

    a 'hood ( 'hood = neighborhood) of a point can be approximated

    by the change in the values in 2x.

    Take a small 'hood of, say, 10 on the real line, take

    (9.9,10.1). The change of f(x)=x^2 from the value 10 is:

    i) |10.1^2 -10^2| =0.201

    ii) |9.9^2-10^2| =0.199

    Now, consider the approximation to the change of x^2 ,using the derivative:

    i') |2(10.1)-2(10)|= 0.2

    ii') |2(9.9)-2(10)| = 0.2

    The error is pretty small, right?. There are, of course, analogies to this
    in higher dimensions, with approximations along tangent planes, etc.
    Unfortunately things get much hairier outside of R^n, where you have
    sometimes just local Euclidean, like in manifolds, without the standard
    tangent planes.

    Hope that helped
  9. Aug 11, 2007 #8


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    a curve is tyhe graph of a non linear function. its family of tangent lines are translates of the graphs of a family of linear approximations to this functions, and this family of linear functions is called the differential of the function. at each point, the tangent line is the graph of the differential at that point. It has equation f'(a)(x-a), at a. (A linear approximation to f(x)-f(a).)
  10. Aug 12, 2007 #9


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    Nicely written!. Sorry I could not find a clearer way of writing it. Do
    you think my post was overall correct (tho, I admit, not too clear
    nor to the point.).

    I had a follow-up question, please:

    Does the Jacobian J(f) map a point to its differential?. If so, what
    is the relevance of the differential being 0 at a point?.

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