1044: For $C^\infty$ Map Between $C^\infty$ Manifolds with Boundary and Corresponding Charts, Components Function of Pullback of $(0, q)$-Tensors w.r.t. Standard Bases Is This|T.B.P.

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description/proof of that for

C^{\infty}

map between

C^{\infty}

manifolds with boundary and corresponding charts, components function of pullback of

(0, q)

-tensors w.r.t. standard bases is this

Topics

About: $C^{\infty}$ manifold

Starting Context

The reader knows a definition of pullback of $(0, q)$ -tensors at point by $C^{\infty}$ map between $C^{\infty}$ manifolds with boundary.
The reader admits the proposition that for any field and any $k$ finite-dimensional vectors spaces over the field, the tensors space with respect to the field and the vectors spaces and the field has the basis that consists of the tensor products of the elements of the dual bases of any bases of the vectors spaces.
The reader admits the proposition that for any $C^{\infty}$ map between any $C^{\infty}$ manifolds with boundary and any corresponding charts, the components function of the differential of the map with respect to the standard bases is this.

Target Context

The reader will have a description and a proof of the proposition that for any $C^{\infty}$ map between any $C^{\infty}$ manifolds with boundary and any corresponding charts, the components function of the pullback of the $(0, q)$ -tensors with respect to the standard bases is this.

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.

Main Body

1: Structured Description

Here is the rules of Structured Description.

Entities:

M_{1}

\in {the d_{1} -dimensional C^{\infty} manifolds with boundary}

M_{2}

\in {the d_{2} -dimensional C^{\infty} manifolds with boundary}

f

: M_{1} \to M_{2}

\in {the C^{\infty} maps}

(U_{1} \subseteq M_{1}, ϕ_{1})

\in {the charts for M_{1}}

(U_{2} \subseteq M_{2}, ϕ_{2})

\in {the charts for M_{2}}

, such that

f (U_{1}) \subseteq U_{2}

m

\in U_{1}

f_{m}^{*}

: T_{q}^{0} (T_{f (m)} M_{2}) \to T_{q}^{0} (T_{m} M_{1})

= the pullback

{d x^{j_{1}} \otimes . . . \otimes d x^{j_{q}} | j_{l} \in {1, . . ., d_{1}}}

= the standard basis for T_{q}^{0} (T_{m} M_{1})

{d y^{j_{1}} \otimes . . . \otimes d y^{j_{q}} | j_{l} \in {1, . . ., d_{2}}}

= the standard basis for T_{q}^{0} (T_{f (m)} M_{2})

//

Statements:

\forall t = t_{j_{1}, . . ., j_{q}} d y^{j_{1}} \otimes . . . \otimes d y^{j_{q}} \in T_{q}^{0} (T_{f (m)} M_{2}) (f_{m}^{*} (t) = u_{j_{1}, . . ., j_{q}} d x^{j_{1}} \otimes . . . \otimes d x^{j_{q}} where u_{j_{1}, . . ., j_{q}} = t_{l_{1}, . . ., l_{q}} \partial {\hat{f}}^{l_{1}} / \partial x^{j_{1}} . . . \partial {\hat{f}}^{l_{q}} / \partial x^{j_{q}} where \hat{f} = ϕ_{2} \circ f \circ {ϕ_{1}}^{- 1})

{d x^{j_{1}} \otimes . . . \otimes d x^{j_{q}}}

and

{d y^{j_{1}} \otimes . . . \otimes d y^{j_{q}}}

are indeed some bases, by the proposition that for any field and any

k

finite-dimensional vectors spaces over the field, the tensors space with respect to the field and the vectors spaces and the field has the basis that consists of the tensor products of the elements of the dual bases of any bases of the vectors spaces:

{d x^{j} | j \in {1, . . ., d_{1}}}

and

{d y^{j} | j \in {1, . . ., d_{2}}}

are the dual bases of

{\partial / \partial x^{j} | j \in {1, . . ., d_{1}}}

and

{\partial / \partial y^{j} | j \in {1, . . ., d_{2}}}

.

In other words,

\hat{f}

is the components function of

f

with respect to

(U_{1} \subseteq M_{1}, ϕ_{1})

and

(U_{2} \subseteq M_{2}, ϕ_{2})

2: Proof

Whole Strategy: Step 1: compute

f_{m}^{*} (t) ((\partial / \partial x^{l_{1}}, . . ., \partial / \partial x^{l_{q}}))

t (d f_{m} (\partial / \partial x^{l_{1}}), . . ., d f_{m} (\partial / \partial x^{l_{q}}))

; Step 2: compare the result of Step 1 with

f_{m}^{*} (t) ((\partial / \partial x^{l_{1}}, . . ., \partial / \partial x^{l_{q}})) = u_{j_{1}, . . ., j_{l}} d x^{j_{1}} \otimes . . . \otimes d x^{j_{q}} ((\partial / \partial x^{l_{1}}, . . ., \partial / \partial x^{l_{q}}))

.

Step 1:

We know that

d f_{m} (\partial / \partial x^{l_{n}}) = \partial {\hat{f}}^{j} / \partial x^{l_{n}} \partial / \partial y^{j}

, by the proposition that for any

C^{\infty}

map between any

C^{\infty}

manifolds with boundary and any corresponding charts, the components function of the differential of the map with respect to the standard bases is this:

\partial / \partial x^{l_{n}}

has only the

l_{n}

-th component,

1

f_{m}^{*} (t) ((\partial / \partial x^{l_{1}}, . . ., \partial / \partial x^{l_{q}})) = t (d f_{m} (\partial / \partial x^{l_{1}}), . . ., d f_{m} (\partial / \partial x^{l_{q}})) = t_{j_{1}, . . ., j_{l}} d y^{j_{1}} \otimes . . . \otimes d y^{j_{q}} ((\partial {\hat{f}}^{m_{1}} / \partial x^{l_{1}} \partial / \partial y^{m_{1}}, . . ., \partial {\hat{f}}^{m_{q}} / \partial x^{l_{q}} \partial / \partial y^{m_{q}})) = t_{j_{1}, . . ., j_{q}} d y^{j_{1}} (\partial {\hat{f}}^{m_{1}} / \partial x^{l_{1}} \partial / \partial y^{m_{1}}) . . . \otimes d y^{j_{q}} (\partial {\hat{f}}^{m_{q}} / \partial x^{l_{q}} \partial / \partial y^{m_{q}}) = t_{j_{1}, . . ., j_{q}} δ_{m_{1}}^{j_{1}} \partial {\hat{f}}^{m_{1}} / \partial x^{l_{1}} . . . δ_{m_{q}}^{j_{q}} \partial {\hat{f}}^{m_{q}} / \partial x^{l_{q}} = t_{j_{1}, . . ., j_{q}} \partial {\hat{f}}^{j_{1}} / \partial x^{l_{1}} . . . \partial {\hat{f}}^{j_{q}} / \partial x^{l_{q}}

.

Step 2:

On the other hand,

f_{m}^{*} (t) ((\partial / \partial x^{l_{1}}, . . ., \partial / \partial x^{l_{q}})) = u_{j_{1}, . . ., j_{q}} d x^{j_{1}} \otimes . . . \otimes d x^{j_{q}} ((\partial / \partial x^{l_{1}}, . . ., \partial / \partial x^{l_{q}})) = u_{j_{1}, . . ., j_{q}} d x^{j_{1}} (\partial / \partial x^{l_{1}}) . . . d x^{j_{q}} (\partial / \partial x^{l_{q}}) = u_{j_{1}, . . ., j_{q}} δ_{l_{1}}^{j_{1}} . . . δ_{l_{q}}^{j_{q}} = u_{l_{1}, . . ., l_{q}}

.

So,

u_{l_{1}, . . ., l_{q}} = t_{j_{1}, . . ., j_{q}} \partial {\hat{f}}^{j_{1}} / \partial x^{l_{1}} . . . \partial {\hat{f}}^{j_{q}} / \partial x^{l_{q}}

, which is nothing but

u_{j_{1}, . . ., j_{q}} = t_{l_{1}, . . ., l_{q}} \partial {\hat{f}}^{l_{1}} / \partial x^{j_{1}} . . . \partial {\hat{f}}^{l_{q}} / \partial x^{j_{q}}

References

The Bias Planet
Definitions and Propositions
School Mathematics from Higher Viewpoints
To Disentangle Confusing Terms or Discourses
Information Tables
Exploiting LibreOffice or OpenOffice
Exposing the Java Programming Language
Exposing C++
Exposing Python
Exposing C#
Exposing Gradle
Exposing Git
UNO Dispatch Commands

T.B.P.

2025-03-23

1044: For $C^{\infty}$ Map Between $C^{\infty}$ Manifolds with Boundary and Corresponding Charts, Components Function of Pullback of $(0, q)$ -Tensors w.r.t. Standard Bases Is This

Topics

The table of contents of this article

Starting Context

Target Context

Orientation

Main Body

1: Structured Description

2: Proof

References

2025-03-23

1044: For C∞ Map Between C∞ Manifolds with Boundary and Corresponding Charts, Components Function of Pullback of (0,q)-Tensors w.r.t. Standard Bases Is This

Topics

The table of contents of this article

Starting Context

Target Context

Orientation

Main Body

1: Structured Description

2: Proof

References

1044: For $C^{\infty}$ Map Between $C^{\infty}$ Manifolds with Boundary and Corresponding Charts, Components Function of Pullback of $(0, q)$ -Tensors w.r.t. Standard Bases Is This