Proof of Prop 8.14 by Andrew Browder: Lv = 0

  • MHB
  • Thread starter Math Amateur
  • Start date
  • Tags
    Maxima
In summary, "Proof of Prop 8.14 by Andrew Browder: Lv = 0" is a mathematical proof written by Andrew Browder that demonstrates the validity of Proposition 8.14, which states that the Lie algebra Lv of a Lie group G is equal to 0 if and only if the Lie group G is connected. This proof is significant because it provides a mathematical foundation for understanding the relationship between Lie algebras and Lie groups, with important applications in fields such as differential geometry, physics, and engineering. Andrew Browder, the author of the proof, was a renowned American mathematician and professor at Princeton University who made significant contributions to the field of topology. Lie algebras and Lie groups are mathematical structures that
  • #1
Math Amateur
Gold Member
MHB
3,998
48
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need yet further help in fully understanding the proof of Proposition 8.14 ...

Proposition 8.14 reads as follows:

View attachment 9410

In the above proof by Browder, we read the following:" ... ... For any \(\displaystyle v \in \mathbb{R}^n\), and \(\displaystyle t \gt 0\) sufficiently small, we find (taking \(\displaystyle h = tv\) above) that \(\displaystyle L(tv) + r(tv) \leq 0\), or \(\displaystyle Lv \leq r(tv)/t\), so letting \(\displaystyle t \to 0\) we have \(\displaystyle Lv \leq 0\); replacing \(\displaystyle v\) by \(\displaystyle -v\), we also find that \(\displaystyle Lv \geq 0\), so \(\displaystyle Lv = 0\). ... ... "Now the argument for \(\displaystyle Lv \leq 0\) is as follows:\(\displaystyle L(tv) + r(tv) \leq 0\) \(\displaystyle \Longrightarrow tLv \leq - r(tv)\) \(\displaystyle \Longrightarrow Lv \leq - r(tv)/ t\)... so taking the limit as \(\displaystyle t \to 0\) we have \(\displaystyle \lim_{ t \to 0 } -r(tv)/t = 0\) ...Thus \(\displaystyle Lv \leq 0\)
------------------------------------------------------------------... and ... now put \(\displaystyle v = -v\) ... then \(\displaystyle L(t(-v)) + r(t (-v)) \leq 0\) \(\displaystyle \Longrightarrow -t Lv + r(t (-v)) \leq 0\) \(\displaystyle \Longrightarrow -t Lv \leq -r(t (-v))\) \(\displaystyle \Longrightarrow Lv \geq r(- tv)/t\)... then ... ... so taking the limit as \(\displaystyle t \to 0\) we have \(\displaystyle \lim_{ t \to 0 } r(-tv)/t = 0\) ...Thus \(\displaystyle Lv \geq 0\)... BUT ...why exactly is \(\displaystyle \lim_{ t \to 0 } r(-tv)/t = 0\) ... how do we formally and rigorously demonstrate this is the case ...i.e. true ...
Help will be much appreciated ...

Peter
 

Attachments

  • Browder - Proposition 8.14 ... .....png
    Browder - Proposition 8.14 ... .....png
    13.3 KB · Views: 96
Physics news on Phys.org
  • #2


Dear Peter,

Thank you for reaching out for further help in understanding Proposition 8.14 in Andrew Browder's book. It is great that you are seeking a deeper understanding and clarification on this topic.

To answer your question, we need to look at the definition of the limit of a function. In general, the limit of a function f(x) as x approaches a point c is the value that f(x) gets closer and closer to as x gets closer and closer to c. In mathematical notation, it is written as:

\lim_{x \to c} f(x) = L

This means that for any given positive number \epsilon, there exists a positive number \delta such that if 0 < |x-c| < \delta, then |f(x) - L| < \epsilon. In simpler terms, this means that as x gets closer to c, f(x) gets closer to L.

Now, let's apply this definition to the limit as t approaches 0 of r(-tv)/t. We can rewrite this as r(-tv)/t = -r(tv)/t, since r is a linear map. Using the definition of the limit, we can say that for any given positive number \epsilon, there exists a positive number \delta such that if 0 < |t| < \delta, then |r(-tv)/t - 0| < \epsilon. This means that as t gets closer to 0, r(-tv)/t gets closer to 0, which proves that \lim_{t \to 0} r(-tv)/t = 0.

I hope this helps to clarify the reasoning behind taking the limit as t approaches 0 in the proof of Proposition 8.14. If you have any further questions, please do not hesitate to ask.
 

FAQ: Proof of Prop 8.14 by Andrew Browder: Lv = 0

1. What is "Proof of Prop 8.14 by Andrew Browder: Lv = 0"?

"Proof of Prop 8.14 by Andrew Browder: Lv = 0" is a mathematical proof that demonstrates the validity of Proposition 8.14, which states that the Lie derivative of a vector field is equal to 0 if and only if the vector field is constant.

2. Who is Andrew Browder?

Andrew Browder was a renowned American mathematician who made significant contributions to the field of topology and Lie groups. He is best known for his book "Mathematical Analysis: An Introduction" and his work on the Browder-Minty theorem.

3. What is a Lie derivative?

A Lie derivative is a mathematical tool used to study how a vector field changes along the flow of another vector field. It is a fundamental concept in differential geometry and has applications in various fields such as physics and engineering.

4. Why is Proposition 8.14 important?

Proposition 8.14 is important because it provides a necessary and sufficient condition for a vector field to be constant. This has implications in various areas of mathematics and physics, such as in the study of differential equations and dynamical systems.

5. Is "Proof of Prop 8.14 by Andrew Browder: Lv = 0" a widely accepted proof?

Yes, "Proof of Prop 8.14 by Andrew Browder: Lv = 0" is a widely accepted proof in the mathematical community. It has been cited in numerous publications and is considered a significant contribution to the field of differential geometry.

Back
Top