1036: Standard Basis for Tangent Vectors Space at Point on C∞ Manifold with Boundary w.r.t. Chart

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definition of standard basis for tangent vectors space at point on $C^{\mathrm{\infty }}$ manifold with boundary w.r.t. chart

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Topics

About: $C^{\mathrm{\infty }}$ manifold

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of tangent vectors space at point on $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of Euclidean $C^{\mathrm{\infty }}$ manifold.
• The reader knows a definition of closed upper half Euclidean $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of open submanifold with boundary of $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of differential of $C^{\mathrm{\infty }}$ map between $C^{\mathrm{\infty }}$ manifolds with boundary at point.

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Target Context

• The reader will have a definition of standard basis for tangent vectors space at point on $C^{\mathrm{\infty }}$ manifold with boundary with respect to chart.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$M$: $\in \{\text{ the }d\text{ -dimensional }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$(U\subseteq M,\varphi )$: $\in \{\text{ the charts }\}$
$m$: $\in U$
$T_{m}M$: $=\text{ the tangent vectors space at }m$
$\varphi (U)$: $=\text{ the open submanifold with boundary of }{\mathbb{R}}^d\text{ or }{\mathbb{H}}^d$
$\{{\partial }_j{|}_{\varphi (m)}|j\in \{1,...,d\}\}$: $=\text{ the standard basis for }{T}_{\varphi (m)}\varphi (U)$
$\ast \{\partial /\partial x^j{|}_m:=d{\varphi }^{-1}{|}_{\varphi (m)}({\partial }_j{|}_{\varphi (m)})|j\in \{1,...,d\}\}$, $\in \{\text{ the bases for }{T}_{m}M\}$
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Conditions:
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2: Note

$\{{\partial }_j{|}_{\varphi (m)}|j\in \{1,...,d\}\}$ is indeed a basis for $T_{\varphi (m)}\varphi (U)$, by the proposition that for any open subset of ${\mathbb{R}}^d$ or ${\mathbb{H}}^d$, there is the standard basis for the tangent vectors space at each point on the open subset, where ${\partial }_j{|}_{\varphi (m)}$ is the partial derivative at $\varphi (m)$ by the $j$-th component.

By the proposition that for any diffeomorphism from any $C^{\mathrm{\infty }}$ manifold with boundary onto any neighborhood of any point image on any $C^{\mathrm{\infty }}$ manifold with boundary, the differential of the diffeomorphism or any its codomain extension at the point is a 'vectors spaces - linear morphisms' isomorphism, $\{\partial /\partial x^j{|}_m:=d{\varphi }^{-1}{|}_{\varphi (m)}({\partial }_j{|}_{\varphi (m)})|j\in \{1,...,d\}\}$ is a basis for $T_{m}M$, because ${\varphi }^{-1}:\varphi (U)\to U$ is a diffeomorphism.

While $\partial /\partial x^j{|}_m$ deceptively looks like the partial derivative, it is not so because $\partial /\partial x^j{|}_m\in T_{m}M$ and it operates on a function, $f:M\to \mathbb{R}$, not on any function from ${\mathbb{R}}^d$ or its open subset.

According to the definition of differential, $\partial /\partial x^j{|}_m(f)={\partial }_j{|}_{\varphi (m)}(f\circ {\varphi }^{-1})$ where $f\circ {\varphi }^{-1}:\varphi (U)\to \mathbb{R}$.

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References

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