453: For Map C^\infty at Point, Coordinates Function with Any Charts Is C^\infty at Point Image

<The previous article in this series | The table of contents of this series | The next article in this series>

A description/proof of that for map $C^{\mathrm{\infty }}$ at point, coordinates function with any charts is $C^{\mathrm{\infty }}$ at point image

---

Topics

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

---

The table of contents of this article

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

---

Starting Context

• The reader knows a definition of map $C^{\mathrm{\infty }}$ at point.

---

Target Context

• The reader will have a description and a proof of the proposition that for any map $C^{\mathrm{\infty }}$ at point between any $C^{\mathrm{\infty }}$ manifolds, the coordinates function with any charts is $C^{\mathrm{\infty }}$ at the point image.

---

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: Description

For any $C^{\mathrm{\infty }}$ manifolds, $M_1,M_2$, any point, $p\in M_1$, any map $C^{\mathrm{\infty }}$ at $p$, $f:M_1\to M_2$, and any charts, $(U_1\subseteq M_1,{\varphi }_1)$ and $(U_2\subseteq M_2,{\varphi }_2)$, such that $f(U_1)\subseteq U_2$, the coordinates function with the charts, $\tilde{f}={\varphi }_2\circ f\circ {{\varphi }_1}^{-1}:{\varphi }_1(U_1)\to {\varphi }_2(U_2)$ is $C^{\mathrm{\infty }}$ at ${\varphi }_1(p)$.

---

2: Note

The definition of map $C^{\mathrm{\infty }}$ at point dictates only the existence of a chart around the point on the domain and a chart around the corresponding point on the codomain for which the coordinates function is $C^{\mathrm{\infty }}$ at the point image (not necessarily on the whole chart range), and it is not about any charts on the domain and the codomain.

---

3: Proof

There are a chart, $(U_p\subseteq M_1,{\varphi }_p)$, around $p$ and a chart, $(U_{f(p)}\subseteq M_2,{\varphi }_{f(p)})$, around $f(p)$ such that $f(U_p)\subseteq U_{f(p)}$ and ${\varphi }_{f(p)}\circ f\circ {{\varphi }_p}^{-1}:{\varphi }_p(U_p)\to {\varphi }_{f(p)}(U_{f(p)})$ is $C^{\mathrm{\infty }}$ at $\varphi (p)$.

$\tilde{f}{|}_{{\varphi }_1(U_1\cap U_p)}={\varphi }_2\circ f\circ {{\varphi }_1}^{-1}{|}_{{\varphi }_1(U_1\cap U_p)}={\varphi }_2\circ {{\varphi }_{f(p)}}^{-1}\circ {\varphi }_{f(p)}\circ f\circ {{\varphi }_p}^{-1}\circ {\varphi }_p\circ {{\varphi }_1}^{-1}{|}_{{\varphi }_1(U_1\cap U_p)}$ is $C^{\mathrm{\infty }}$ at ${\varphi }_1(p)$, because ${\varphi }_p\circ {{\varphi }_1}^{-1}$ is $C^{\mathrm{\infty }}$ at ${\varphi }_1(p)$, ${\varphi }_{f(p)}\circ f\circ {{\varphi }_p}^{-1}$ is $C^{\mathrm{\infty }}$ at ${\varphi }_p(p)$, and ${\varphi }_2\circ {{\varphi }_{f(p)}}^{-1}$ is $C^{\mathrm{\infty }}$ at ${\varphi }_{f(p)}(f(p))$.

So, $\tilde{f}$ is $C^{\mathrm{\infty }}$ at ${\varphi }_1(p)\in {\varphi }_1(U_1)$.

---

References

<The previous article in this series | The table of contents of this series | The next article in this series>