Differentiation on Euclidean Space (Calculus on Manifolds)

  1. 1. The problem statement, all variables and given/known data

    This is from Spivak's Calculus on Manifolds, problem 2-12(a).

    Prove that if f:Rn [tex]\times[/tex] Rm [tex]\rightarrow[/tex] Rp is bilinear, then

    lim(h, k) --> 0 [tex]\frac{|f(h, k)|}{|(h, k)|}[/tex] = 0

    2. Relevant equations

    The definition of bilinear function in this case: If for x, x1, x2 [tex]\in[/tex] Rn, y, y1, y2 [tex]\in[/tex] Rm, and a [tex]\in[/tex] R, we have

    f(ax, y) = af(x, y),
    f(x1 + x2, y) = f(x1, y) + f(x2, y),
    f(x, y1 + y2) = f(x, y1) + f(x, y2)

    3. The attempt at a solution

    Because f(x, y) is bilinear, I think |f(h, k)| goes to 0 as (h, k) goes to 0. But I am still trying to find a way to deal with bilinearity and how (h, k) comes from R^n x R^m (so (h, k) is in R^(n+m), right?). I do realize I need to think about this more on my own, but I was wondering if someone could lead me to the right direction.

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