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Differentiation on Euclidean Space (Calculus on Manifolds) 
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#1
Mar3110, 12:19 AM

P: 186

1. The problem statement, all variables and given/known data
This is from Spivak's Calculus on Manifolds, problem 212(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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