MHB Recifiable Paths in C .... Conway, Section 1, Ch. 4 ....

  • Thread starter Thread starter Math Amateur
  • Start date Start date
  • Tags Tags
    Section
Math Amateur
Gold Member
MHB
Messages
3,920
Reaction score
48
I am reading John B. Conway's book, "Functions of a Complex Variable I" (Second Edition) ...

I am currently focussed on Chapter IV: Complex Integration ... Section 1: Riemann-Stieljes Integral ... ...

I need help in fully understanding some notes by Conway on rectifiable paths in $$\mathbb{C}$$ on page 63 ... ...

The notes on rectifiable paths on page 63 read as follows:https://www.physicsforums.com/attachments/7446
My question regarding the above text from Conway is as follows:Why exactly does $$\phi$$ need to be non-decreasing in order for $$\gamma \circle \phi$$ to be a path with the same trace as $$\gamma$$ ... ... ?Help will be much appreciated ... ...

Peter
 
Physics news on Phys.org
He's not saying that $\varphi$ must be nondecreasing, but that if $\varphi$ is nondecreasing, then $\gamma\circ \varphi$ has the same trace as $\gamma$. It's a sufficient condition he's giving, not a necessary condition.
 

Thread 'About the definition of topological manifold using closed sets'

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in ##\mathbb R^n##. It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on ##\mathbb R^n## ? I'm positive for this. Perhaps the definition of smooth manifold would be problematic, though.
View full post »

Similar threads

MHB Why Does Theorem 1.9 Use f(t) Instead of f(γ(t))?
  • · Replies 2 ·
Replies
2
Views
2K
MHB Complex Valued Functions BV: John B. Conway Prop 1.3 Explained
  • · Replies 3 ·
Replies
3
Views
2K
MHB The Riemann Integral .... Conway, Proposition 3.1.4 ....
  • · Replies 2 ·
Replies
2
Views
2K
MHB Understanding Cantor's Theorem and Diameter
  • · Replies 3 ·
Replies
3
Views
2K
MHB Understand Proposition 1.3 in Conway's Functions of Complex Variables I
  • · Replies 1 ·
Replies
1
Views
2K
MHB How does Conway's Theorem 2.29 Derive the Simplified Expression for f(z)?
  • · Replies 2 ·
Replies
2
Views
2K
MHB Law of Cosines for C .... Remmert Section 1.3, Ch. 0 ....
  • · Replies 1 ·
Replies
1
Views
1K
MHB Complex and Real Differentiability .... Remmert, Section 2, Ch. 1 .... ....
  • · Replies 1 ·
Replies
1
Views
1K
I Complex & Real Differentiability ... Remmert, Section 2, Ch 1
  • · Replies 4 ·
Replies
4
Views
3K
MHB Proof of Lemma 2.1, Part (vi) in Palka's Ch.4: Explaining Inequality
  • · Replies 2 ·
Replies
2
Views
1K