Difference between revisions of "User talk:Harold"

Revision as of 22:15, 14 February 2017 (view source) Alec (Talk | contribs) (→‎Some copy-and-paste-help: new section) ← Older edit		Revision as of 19:29, 15 February 2017 (view source) Harold (Talk | contribs) m Newer edit →
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		+	== A collection of thoughts on Morse theory ==
		+
		+	I'm currently trying to figure out why in Morse homology, the degree of the attaching map of a certain <m>n</m>-cell is somehow equivalent to the number of gradient flow lines.
		+	The setup is as following. Let <m>(M, g)</m> be a closed (i.e., compact and connected) smooth Riemannian manifold (without boundary), and suppose <m>f: M \to \mathbb{R}</m> is a smooth map satisfying the following properties:
		+	# for each <m>x \in \mathrm{Crit}(f) := { p \in M : df_p = 0 }</m>, the Hessian <m>\mathrm{Hess}(f): T_pM \times T_pM \to \mathbb{R}</m> is non-degenerate. '''TODO Define the Hessian.'''
		+	# <m>f|_{\mathrm{Crit}(f)}</m>: \mathrm{Crit(f)} \to \mathbb{R}</m> is injective.
		+
	== Caveat with xymatrix ==		== Caveat with xymatrix ==

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Revision as of 19:29, 15 February 2017

A collection of thoughts on Morse theory

I'm currently trying to figure out why in Morse homology, the degree of the attaching map of a certain [ilmath]n[/ilmath]-cell is somehow equivalent to the number of gradient flow lines. The setup is as following. Let [ilmath](M, g)[/ilmath] be a closed (i.e., compact and connected) smooth Riemannian manifold (without boundary), and suppose [ilmath]f: M \to \mathbb{R}[/ilmath] is a smooth map satisfying the following properties:

1. for each [ilmath]x \in \mathrm{Crit}(f) := { p \in M : df_p = 0 }[/ilmath], the Hessian [ilmath]\mathrm{Hess}(f): T_pM \times T_pM \to \mathbb{R}[/ilmath] is non-degenerate. TODO Define the Hessian.
2. [ilmath]f|_{\mathrm{Crit}(f)}[/ilmath]: \mathrm{Crit(f)} \to \mathbb{R}</m> is injective.

Caveat with xymatrix

Hey, try this page:

• /Xymatrix caveat

See how you can scroll right? Alec (talk) 22:08, 14 February 2017 (UTC)

Some copy-and-paste-help

It's good to render diagrams in tables, if only because they look a bit sparse with the white background (unless they're huge), try these:

[math]\text{YOUR MATH HERE}[/math]
Comment

To float to the right:

Now you can write here and reference the diagram on the right

• Lists and everything
  • Baby

Hope it helps Alec (talk) 22:15, 14 February 2017 (UTC)