Differentiation on Euclidean Space (Calculus on Manifolds)

In summary, differentiation on Euclidean space is a mathematical concept that involves calculating the rate of change of a function with respect to its input variables. It extends traditional calculus to functions with multiple variables and is important in various fields such as science, engineering, and optimization. It can be applied to non-linear functions and is used in real-world applications to analyze and model systems, as well as optimize processes for efficiency.
  • #1
PieceOfPi
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Homework Statement



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

Homework 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)

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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  • #2
for your help!
Thank you for your post. To prove this statement, we can use the definition of limit. Let (h, k) be a sequence in R^n x R^m that converges to 0. Then for any \epsilon > 0, there exists an N \in \mathbb{N} such that for all n > N, |(h_n, k_n)| < \epsilon.

Now, we can use the bilinearity of f to write:

|f(h_n,k_n)| = |f(h_n,0) + f(0,k_n)|

= |f(h_n,0)| + |f(0,k_n)|

= |h_n||f(1,0)| + |k_n||f(0,1)|

= |h_n||f(1,0)| + |k_n||f(1,0)|

= (|h_n| + |k_n|)|f(1,0)|

= |(h_n, k_n)||f(1,0)|

Therefore, we have:

\frac{|f(h_n,k_n)|}{|(h_n, k_n)|} = \frac{|(h_n, k_n)||f(1,0)|}{|(h_n, k_n)|}

= |f(1,0)|

Since we know that (h_n, k_n) converges to 0, we can say that |(h_n, k_n)| also converges to 0. Therefore, as n approaches infinity, the expression \frac{|f(h_n,k_n)|}{|(h_n, k_n)|} approaches |f(1,0)|. But we can choose \epsilon to be as small as we want, so we can make |f(1,0)| as small as we want. This means that the limit of \frac{|f(h_n,k_n)|}{|(h_n, k_n)|} as (h, k) approaches 0 is 0, as required.

I hope this helps guide you in the right direction. Keep in mind that the definition of limit and the properties of bilinear functions are key in solving this problem. Good luck!
 

1. What is differentiation on Euclidean space?

Differentiation on Euclidean space is a mathematical concept in calculus that involves finding the rate of change of a function with respect to its input variables. It is essentially the process of calculating the slope or gradient of a curve at a specific point.

2. How is differentiation on Euclidean space different from traditional calculus?

Traditional calculus typically deals with functions on a single variable, while differentiation on Euclidean space extends this concept to functions with multiple variables. This allows for a more comprehensive understanding of the behavior of complex systems.

3. What is the importance of differentiation on Euclidean space?

Differentiation on Euclidean space is important in many fields of science and engineering, as it provides a way to analyze and model the behavior of complex systems. It is also a fundamental tool in optimization, as it allows for the determination of the minimum or maximum values of a function.

4. Can differentiation on Euclidean space be applied to non-linear functions?

Yes, differentiation on Euclidean space can be applied to non-linear functions. It is not limited to linear functions, and can be used to analyze any type of function, regardless of its complexity.

5. How is differentiation on Euclidean space used in real-world applications?

Differentiation on Euclidean space is used in a variety of real-world applications, such as physics, economics, and engineering. It is used to model and predict the behavior of systems, as well as to optimize processes and systems for maximum efficiency.

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