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Homework Help: Derivative matrices

  1. Mar 12, 2009 #1
    1. The problem statement, all variables and given/known data
    Suppose that the mapping F:Rn[tex]\rightarrow[/tex]Rm is continuously differentiable and that there is a fixed mxn matrix A so that
    DF(x)=A for every x in Rn.

    Prove that then F is a mapping such that F(x)=Ax+c for some c[tex]\in[/tex]Rm

    2. Relevant equations

    DF(x)ij= [tex]\partial[/tex]Fi(x)/[tex]\partial[/tex]xj (the ijth entry of the derivative matrix)

    3. The attempt at a solution
    I tried to solve this with the first-order approximation F(x)=F(xo)+F´(xo)(x-xo) +[tex]\epsilon[/tex](x-xo)||x-xo||

    Where [tex]\epsilon[/tex](x-xo)||x-xo|| approaches naught.
    But it didn't amount to anything sensible and I'm not sure what to do... What should I try next?
  2. jcsd
  3. Mar 12, 2009 #2


    User Avatar
    Science Advisor
    Homework Helper

    Actually, when you write F´(xo)(x-xo) like in a Taylor series, you mean
    DF(xo) . (x-xo)
    which is a multiplication of a matrix by a vector. You know that DF(xo) = A... if your expansion is otherwise correct that should enable you to actually identify c explicitly.
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