# Differentiation on Euclidean Space (Calculus on Manifolds)

by PieceOfPi
Tags: calculus, differentiation, euclidean, manifolds, space
 P: 178 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 $$\times$$ Rm $$\rightarrow$$ Rp is bilinear, then lim(h, k) --> 0 $$\frac{|f(h, k)|}{|(h, k)|}$$ = 0 2. Relevant equations The definition of bilinear function in this case: If for x, x1, x2 $$\in$$ Rn, y, y1, y2 $$\in$$ Rm, and a $$\in$$ 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. Thanks

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