Register to reply

Differentiation on Euclidean Space (Calculus on Manifolds)

Share this thread:
PieceOfPi
#1
Mar31-10, 12:19 AM
P: 186
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.

Thanks
Phys.Org News Partner Science news on Phys.org
An interesting glimpse into how future state-of-the-art electronics might work
Tissue regeneration using anti-inflammatory nanomolecules
C2D2 fighting corrosion

Register to reply

Related Discussions
Spivak Calculus on Manifolds Calculus & Beyond Homework 0
Differentiation on Smooth Manifolds without Metric Differential Geometry 21
Metric space and subsets of Euclidean space Calculus & Beyond Homework 18
Taking a course in calculus on manifolds. Differential Geometry 9
Euclidean and Non Euclidean Space? Differential Geometry 1