(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

Prove that if f:R^{n}[tex]\times[/tex] R^{m}[tex]\rightarrow[/tex] R^{p}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, x_{1}, x_{2}[tex]\in[/tex] R^{n}, y, y_{1}, y_{2}[tex]\in[/tex] R^{m}, and a [tex]\in[/tex] R, we have

f(ax, y) = af(x, y),

f(x_{1}+ x_{2}, y) = f(x_{1}, y) + f(x_{2}, y),

f(x, y_{1}+ y_{2}) = f(x, y_{1}) + f(x, y_{2})

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.

Thanks

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# Differentiation on Euclidean Space (Calculus on Manifolds)

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