- #1

- 186

- 0

## Homework Statement

This is from Spivak's

*Calculus 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

## Homework 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})

## 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