Title: Theory / Important / Math / Analysis / Zorich{ / 12 Surfaces=section 8.7 on surfaces in R^n
Tags: examplesSwitchExample1 fromMath hasSeenConjDef mathToSee

Question: The standard model for a \(k\)-dimensional surface
Answer: \(\mathbb R^k\)
Page: 163
Chapter: 12
Section: 1
Variables: \(k,n\in\mathbb K\)
Kind: Def

Name: A surface of dimension \(k\)
Definition: A set \(S\subseteq \mathbb R^n\) such that each point of \(S\) has a neighborhood in \(S\) homeomorphic to \(\mathbb R^k\).
Page: 163
Chapter: 12
Section: 1
Variables: \(k,n\in\mathbb K\)
Kind: Def
Name2: A \(k\)-dimensional surface
Denoted by: \(S\)
Name3: A \(k\)-dimensional manifold

Name: A chart of \(S\)
Definition: An homeomorphic mapping \(\mathbb R^k\to U\subseteq S\)
Page: 163
Chapter: 12
Section: 1
Variables: \(k,n\in\mathbb K\),
\(S\subseteq\mathbb R^n\) a \(k\)-dimensional surface
Index: 2
Kind: Def
Name2: A local chart of \(S\)
Denoted by: \(\phi\)

Name: The parameter domain of \(\phi\)
Definition: \(D\)
Page: 163
Chapter: 12
Section: 1
Variables: \(k,n\in\mathbb K\),
\(S\subseteq\mathbb R^n\) a \(k\)-dimensional surface
\(\phi:D\to U\subseteq S\) a chart
Index: 2
Kind: Def

Name: The range of \(\phi\)
Definition: \(U\)
Page: 163
Chapter: 12
Section: 1
Variables: \(k,n\in\mathbb K\),
\(S\subseteq\mathbb R^n\) a \(k\)-dimensional surface
\(\phi:D\to U\subseteq S\) a chart
Index: 2
Kind: Def
Name2: The domain of action of the chart on the surface \(S\)

Name: The curvilinear coordinates in \(U\) induced by \(\phi\)
Definition: \((t^1,\dots, t^k)\) denotes \(\phi(t^1,\dots, t^k)\)
Page: 470, 163
Chapter: 8, 12
Section: 4.6, 1
Variables: \(k,n\in\mathbb K\),
\(S\subseteq\mathbb R^n\) a \(k\)-dimensional surface
\(\phi:\mathbb R^k\to U\subseteq S\) a chart
Index: 2
Kind: Def
Conjdef: The coordinate system for \(U\) where

Definition: The open cube of dimension \(k\)
Page: 164
Chapter: 12
Section: 1
Variables: \(k\in\mathbb K\),
Notation: \(I^k\)

Definition: The open ball of dimension \(k\)
Page: 164
Chapter: 12
Section: 1
Variables: \(k\in\mathbb K\),
Notation: \(B^k\)

Name: An elementary surface
Definition: A surface which can be defined by a single chart.
Page: 164
Chapter: 12
Section: 1
Example: For \(f:I^k\to\mathbb R\) a continuous function, the graph of \(f\) in \(\mathbb R^{k+1}\)

Name: An atlas of \(S\)
Definition: A countable set of local charts \(\phi_i:I_i^k\to U_i\subseteq S\) such that \(S=\bigcup U_i\)
Page: 164
Chapter: 12
Section: 1
Variables: \(k,n\in\mathbb K\),
\(S\subseteq\mathbb R^n\) a \(k\)-dimensional surface
Index: 3
Kind: Def

Name: The set of atlases
Page: 164
Section: 1
Chapter: 12
Index: 3
Variables: \(k,n\in\mathbb K\),
\(S\subseteq\mathbb R^n\) a \(k\)-dimensional surface
Kind: Def
Under: union

Smaller0: \(S\) is a smooth \(k\)-dimensional surface
Greater0: \(S\) is a \(k\)-dimensional surface
Page: 164
Chapter: 12
Section: 1
Variables: \(k,n\in\mathbb K\),
\(S\subseteq\mathbb R^n\)
Counterexample: Alexander horned sphere

Name: Alexander Horned Sphere
Definition: starting with a standard torus:
Remove a radial slice of the torus.Connect a standard punctured torus to each side of the cut, interlinked with the torus on the other side.Repeat steps 1–2 on the two tori just added ad infinitum

Definition: \(\{(t_1,\dots,t_k,0,\dots,0)\mid 0\le t_1,\dots,t_k\le 1\}\)
Page: 164
Chapter: 12
Section: 1
Variables: \(0\le k\le n\)
Context: A subset of \(\mathbb R^n\)
Notation: \(I^k\)
Index: 3
Kind: Def

Definition: \(U(x_0)\cap S\)
Page: 164
Chapter: 12
Section: 1
Variables: \(0\le k\le n\)
\(S\subseteq\mathbb R^n\)
\(x_0\in S\)
\(U(x_0)\) a neighborghood of \(x_0\)
Notation: \(U_S(x_0)\)
Index: 3
Kind: Def