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Derivatives and Linear transformations

  1. May 12, 2015 #1
    Let G be a non-empty open connected set in Rn, f be a differentiable function from G into R, and A be a linear transformation from Rn to R. If f '(a)=A for all a in G, find f and prove your answer.

    I thought of f as being the same as the linear transformation, i.e. f(x)=A(x). Is this true?
     
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  3. May 12, 2015 #2

    Svein

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    No, but observe: [itex] f'(x) = (\frac{\partial f}{\partial x_{1}}, \frac{\partial f}{\partial x_{2}}, \dotso , \frac{\partial f}{\partial x_{n}})[/itex] and since A is linear, [itex] A=(a_{1}, a_{2}, \dotso , a_{n})[/itex]. Therefore [itex] \frac{\partial f}{\partial x_{1}}=a_{1}[/itex] etc. SInce all ak are constants, ...
     
  4. May 13, 2015 #3

    mathwonk

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    is a function uniquely determined by its derivative?
     
  5. May 14, 2015 #4

    Svein

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    Of course not - g(x) and g(x)+C have the same derivatives. I am a mathematician - I leave the details as an exercise for the student.
     
  6. May 15, 2015 #5

    HallsofIvy

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    In general, with f a function from, say, Rn to Rm, we can define the derivative of f, at point p in Rn as "the linear transformation, from Rn to Rm that best approximates f in some neighborhood of p".

    To make that "best approximates f" more precise, note that we can write any function, f, as [tex]f(x)= A(x- p)+ D(x)[/tex] where A is a linear transformation and D(x) has the property that the limit, as x approaches p, of D(x)/||x- p|| is 0. Then A is the "derivative of f at p".
     
  7. May 15, 2015 #6

    WWGD

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    That's where it gets confusing: some call it the differential.
     
  8. May 16, 2015 #7
    Really? I've never heard someone call it the differential. As far as I know, the linear transformation that best approximates a function at a given point is always called the derivative (http://en.wikipedia.org/wiki/Fréchet_derivative)

    To answer the OP, it is almost true, you're just missing the constant. The answer is [itex]f(x) = Ax + c[/itex]. The linear transformation that best approximates this [itex]f[/itex] is clearly [itex]A[/itex], in other words [itex]f'(a) = A[/itex] for every element in [itex]G[/itex]. And since [itex]G[/itex] is connected, any other function with derivative equal to [itex]A[/itex] in [itex]G[/itex], must differ only by a constant.
     
  9. May 16, 2015 #8

    WWGD

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    I think that A as a function is usually called, in my experience, the differential. Once you plug in numbers , it is called the derivative in this layout, so that, e.g., for ##f(x)=x^2##, ##2x## is the differential, but the derivative at a fixed ##x_0## is ##2x_0##.
     
  10. May 16, 2015 #9

    HallsofIvy

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    No. If f(x)= x2, the derivative is [itex]df/dx= 2x[/itex]. The "differential" is [itex]df= 2xdx[/itex].
    And, as you say, the "derivative at fixed x0" is 2x0.
     
  11. May 17, 2015 #10

    WWGD

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    It would help if you quoted actual definitions: the differential is the best linear map approximating the local change of the function near the point. The derivative is the rate of change (modulo higher dimensions).
     
  12. May 23, 2015 #11

    Mark44

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    I agree with Halls here, and would add only "differential of f."

    If f is a function of two variables, say z = f(x, y), then the differential of f (also called the total differential of f) is ##df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy##.
     
  13. May 23, 2015 #12

    WWGD

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    Yes, I corrected myself in my post after that one.
     
  14. Jun 1, 2015 #13

    mathwonk

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    what used to be called the Frechet derivative some 50 years ago, e.g. in Dieudonne's Foundations of modern analysis, is now usually called the differential.
     
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