What is the most succinct way to define vector fields and orientation of submanifolds in this context?

Question

I am writing some notes on submanifolds of $\mathbb{R}^n$. I am reading several references in order to write short notes on this subject.

The main definitions that I am using are below:

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Definition 1: Let $M\subseteq \mathbb{R}^n$ be any subset. We say that $(V,\psi )$ is a $m$-parametrization of $M$ if the following propositions are true:

1. $V\subseteq\mathbb{R}^m$ is open and $\psi:V\to \mathbb{R}^n$ is a smooth immersion;
2. $\psi:V\to \psi [V]$ is a homeomorphism such that $\psi [V]=U\cap M$ in which $U\subseteq\mathbb{R}^n$ is open.

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Definition 2: Let $M\subseteq \mathbb{R}^n$ be any subset. We say that $M$ is a (smooth) submanifold (of $\mathbb{R}^n$) with dimension $m$ if for all $p\in M$ there's a $m$-parametrization $(V,\psi)$ of $M$ such that $p\in \psi [V]$.

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Definition 3: Let $M\subseteq\mathbb{R}^m$ and $N\subseteq\mathbb{R}^n$ be two submanifolds. Then

1. We say that $f:M\to N$ is a $C^\infty$-morphism in $p\in M$ if there're an open neighborhood $V\subseteq\mathbb{R}^m$ of $p$ and a smooth map $F:V\to\mathbb{R}^n$ such that $F|_{V\cap M}=f|_{V\cap M}$.
2. We say that $f:M\to N$ is a $C^\infty$-morphism if $f$ is a $C^\infty$-morphism in $p$ for all $p\in M$.

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Definition 4: Let $M\subseteq\mathbb{R}^n$ be a submanifold and $p\in M$. We define the tangent space of $M$ in $p$ as $$\color{red}{T_pM}:=\big\{\dot{\gamma}(0)\in\mathbb{R}^n:\gamma \in M^\mathbb{R}\wedge \gamma \in C^\infty\wedge \gamma (0)=p\big\}$$

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Definition 5: Let $M\subseteq\mathbb{R}^m$ and $N\subseteq\mathbb{R}^n$ be two submanifolds and $p\in M$. Suppose that $f:M\to N$ is a $C^\infty$-morphism. The differential of $f$ in $p$ is the map $\color{red}{(df)_p}:T_pM\to T_{f(p)}N$ such that $(df)_p(\dot{\gamma }(0))=(df\circ \gamma )_0(1)$ for all smooth map $\gamma :\mathbb{R}\to M$ satisfying $\gamma (0)=p$.

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My question is: Given the previous definitions, what is the best way (i.e. the most succinct way) to define vector fields and orientation of submanifolds?

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EDIT:

I have the following theorem in my mind: If $M\subseteq \mathbb{R}^n$ is submanifold with dimension $m$ and there're $v_1,\cdots,v_{n-m}:M\to \mathbb{R}^n$ linearly independent vector fields, then $M$ is oriented.

I would like some definition of vector fields and orientation such that the previous theorem is easy to prove (I don't want to use the notion of atlas).

Answer 1

A vector field in this case can be defined as a mapping $\xi:M\subset \Bbb{R}^m\to \Bbb{R}^m$ such that for every $p\in M$, $\xi(p)\in T_pM$. You can then talk about various regularity conditions: continuity, $C^r$, $C^{\infty}$ and so on.

For orientations on an orientable submanifold you can define it as a choice of an equivalence class of continuous nowhere-vanishing top-degree differential form on $M$ (the equivalence relation being $\alpha\sim\beta$ if and only if there exists an everywhere positive continuous function $f$ on $M$ such that $\alpha = f\cdot \beta$). Just so we're clear: orientability and orientation are different concepts. The first is the ability to carry an orientation, i.e whether or not there exist a continuous top-degree differential form on $M$, while the latter is a particular choice of such (an equivalence class of) a differential form.