Equivalence of Definitions of Tangent Vector
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.
To discuss this in more detail, feel free to use the talk page.
Once you have done so, you can remove this instance of {{MissingLinks}} from the code.
Put lemmas on their own pages. Improve structure.
Until this has been finished, please leave {{refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
Contents
Theorem
Let $M$ be a smooth manifold.
Let $m \in M$ be a point.
Let $V$ be an open neighborhood of $m$.
Let $C^\infty \left({V, \R}\right)$ be defined as the set of all smooth mappings $f: V \to \R$.
The following definitions of the concept of Tangent Vector are equivalent:
Definition 1
A tangent vector $X_m$ on $M$ at $m$ is a linear transformation:
- $X_m: C^\infty \left({V, \R}\right) \to \R$
which satisfies the Leibniz law:
- $\displaystyle X_m \left({f g}\right) = X_m \left({f}\right) \, g \left({m}\right) + f \left({m}\right) \, X_m \left({g}\right)$
Definition 2
Let $I$ be an open real interval with $0 \in I$.
Let $\gamma: I \to M$ be a smooth curve with $\gamma \left({0}\right) = m$.
Then a tangent vector $X_m$ at a point $m \in M$ is a mapping
- $X_m: C^\infty \left({V, \R}\right) \to \R$
defined by:
- $X_m \left({f} \right) := \dfrac {\mathrm d} {\mathrm d \tau} {\restriction_0} \, f \circ \gamma \left({\tau}\right)$
for all $f \in C^\infty \left({V, \R}\right)$.
Proof
Definition 2 implies Definition 1
Let $\lambda \in \R$ and $f, g \in C^\infty \left({V, \R}\right)$.
| \(\displaystyle X_m \left({f + \lambda g} \right)\) | \(=\) | \(\displaystyle \frac {\mathrm d} {\mathrm d \tau} {\restriction_0} \, \left({f + \lambda g}\right) \circ \gamma \left({\tau}\right)\) | Definition 2 | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \frac {\mathrm d} {\mathrm d \tau} {\restriction_0} \, \left({f \circ \gamma \left({\tau}\right)}\right) + \lambda \frac {\mathrm d} {\mathrm d \tau} {\restriction_0} \, \left({g \circ \gamma} \left({\tau}\right) \right)\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle X_m \left({f}\right) + \lambda \, X_m \left({g}\right)\) |
Thus $X_m$ is linear.
| \(\displaystyle X_m \left({f g} \right)\) | \(=\) | \(\displaystyle \frac {\mathrm d} {\mathrm d \tau} {\restriction_0} \left({f g}\right) \circ \gamma \left({\tau}\right)\) | Definition 2 | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \frac {\mathrm d} {\mathrm d \tau} {\restriction_0} f \circ \gamma \left({\tau}\right) \, g \circ \gamma \left({0}\right) + f \circ \gamma \left({0}\right) \, \frac {\mathrm d} {\mathrm d \tau} {\restriction_0} g \circ \gamma \left({\tau}\right)\) | Product Rule | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle X_m \left({f}\right) \, g \left({m}\right) + f \left({m}\right) \, X_m \left({g}\right)\) | Definition 2, $\gamma \left({0}\right) = m$ |
Hence $X_m$ satisfies the Leibniz law.
Thus $X_m$ satisfies Definition 1.
$\Box$
Definition 1 implies Definition 2
Lemma 1
Let $X_m$ be a tangent vector at $m \in M$ according to Definition 1.
Let $V$ be an open neighborhood of $M$.
Let $f \in C^\infty \left({V, \R}\right)$ be constant.
Then $X_m \left({f}\right) = 0$.
Proof of Lemma 1
Let $f \left({m}\right) = 0$.
Then, by constancy, $f = 0$ on $V$.
Hence, by linearity, $X_m \left({0}\right) = 0$.
Let $f \left({m}\right) \ne 0$.
| \(\displaystyle X_m \left({f}\right)\) | \(=\) | \(\displaystyle X_m \left({1 \, f}\right)\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle X_m \left({1}\right) \, f \left({m}\right) + X_m \left({f}\right)\) | Leibniz law | ||||||||||
| \(\displaystyle \iff\) | \(\) | \(\displaystyle \) | |||||||||||
| \(\displaystyle X_m \left({1}\right)\) | \(=\) | \(\displaystyle 0\) | $f \left({m}\right) \ne 0$ |
$f$ is constant, if and only if $\exists \lambda \in \R : f \left({V}\right) = \left\{ {\lambda} \right\}$ if and only if $f = \lambda$.
$\exists$ in the above? Should it not be $\forall$?
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this point in more detail, feel free to use the talk page.
If you are able to explain it, then when you have done so you can remove this instance of {{Explain}} from the code.
| \(\displaystyle \lambda X_m \left({1}\right)\) | \(=\) | \(\displaystyle X_m \left({\lambda}\right)\) | Definition of Linear Mapping | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle X_m \left({f}\right)\) | as $f = \lambda$ | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle 0\) | as $X_m \left({1}\right) = 0$ |
$\Box$
Let $X_m$ be a tangent vector at $m \in M$ according to Definition 1.
Denote $n := \dim M$.
Let $f \in C^\infty \left({V, \R}\right)$.
Let $(U, \kappa)$ be a chart with $\kappa \left( {m} \right) = 0$.
Let $\kappa^i$ be the $i$th coordinate function of the chart $(U, \kappa)$.
By, Taylor's Theorem/n Variables :
| \(\displaystyle f\) | \(=\) | \(\displaystyle f \circ \kappa^{-1} \circ \kappa\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \left({f \circ \kappa^{-1} }\right) \left({\kappa}\right)\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \left({f \circ \kappa^{-1} }\right) \left({0}\right) + \sum_{i \mathop = 1}^n \frac {\partial} {\left({f \circ \kappa^{-1} }\right)} {\partial{\kappa^i} } \left({0}\right) \, \kappa^i + \mathcal O_2 \left({\kappa}\right)\) |
Observe that $\displaystyle \left({f \circ \kappa^{-1} }\right) \left({0}\right) = \left({f \circ \kappa^{-1} }\right) \left( {\kappa \left({m}\right)}\right) = f \left({m}\right) $ is a constant mapping on $V$.
The above needs more than observation.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this point in more detail, feel free to use the talk page.
If you are able to explain it, then when you have done so you can remove this instance of {{Explain}} from the code.
Define $X^i := X_m \left({\kappa ^i}\right)$.
Then by linearity:
- $\displaystyle X_m \left({f}\right) = X_m \left({f \left({m}\right)}\right) + \sum_{i \mathop = 1}^n \frac {\partial \left({f \circ \kappa^{-1} }\right)} {\partial \kappa^i} \left({0}\right) \ X^i + X_m \left({\mathcal O_2 \left({\kappa}\right) }\right)$
Lemma 2
- $X_m \left({\mathcal O_2 \left({\kappa}\right)}\right) = 0$
Proof of Lemma 2
By Taylor's Theorem/n Variables, for each summand of $\mathcal O _2 \left({\kappa}\right)$ there exists $i \in \left\{ {1, \ldots, n} \right\}$ and an $h \in C^\infty \left({V, \R}\right)$ with $h \left({m}\right) = 0$ such that the summand is $\kappa^i h$.
| \(\displaystyle X_m \left({\kappa^i h}\right)\) | \(=\) | \(\displaystyle X_m \left({\kappa^i}\right) h \left({m}\right) + \kappa^i \left({m}\right) X_m \left({h}\right)\) | Definition 1 (Leibniz law) | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle X_m \left({\kappa^i}\right) 0 + 0 X_m \left({h}\right)\) | as $h \left({m}\right) = 0$, $\kappa^i \left({m}\right) = 0$ | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle 0\) |
Thus the sum $\mathcal O_2 \left({\kappa}\right)$ vanishes.
$\Box$
By Lemma 1 :
- $\displaystyle X_m \left({f \left({m}\right)} \right) = 0$
By Lemma 2 :
- $\displaystyle X_m \left({\mathcal O_2 \left({\kappa }\right)}\right) = 0$
Hence:
- $\displaystyle X_m \left({f}\right) = \sum_{i \mathop = 1}^n \frac {\partial \left({f \circ \kappa^{-1}}\right)} {\partial \kappa^i} \left({0}\right) \ X^i$
Let $\left\{{e_i}\right\}$ be a basis of $\R^n$ such that:
- $\displaystyle \kappa = \sum_{i \mathop = 1}^n \kappa^i e_i$
If you are familiar with this area of mathematics, you may be able to help improve $\mathsf{Pr} \infty \mathsf{fWiki}$ by determining the precise term which is to be used.
To discuss it in more detail, feel free to use the talk page.
When this has been done, {{Disambiguate}} should be removed from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
Choose a smooth curve $\gamma: I \to M$ with $0 \in I \subseteq \R$ such that $\gamma \left({0}\right) = m$ and:
- $\dfrac {\mathrm d \kappa^i \circ \gamma} {\mathrm d \tau} \left({0}\right) := X^i$
Then:
| \(\displaystyle X_m \left({f}\right)\) | \(=\) | \(\displaystyle \sum_{i \mathop = 1}^n \frac{\partial \left({f \circ \kappa^{-1} }\right)} {\partial \kappa^i} \left({\kappa \left({m}\right)}\right) \frac {\mathrm d \kappa^i \circ \gamma} {\mathrm d \tau} \left({0}\right)\) | as $\kappa \left({m}\right) = 0$ | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \sum_{i \mathop = 1}^n \frac {\partial \left({f \circ \kappa^{-1} }\right)} {\partial \kappa^i} \left({\kappa \circ \gamma \left({0}\right)}\right) \frac {\mathrm d \kappa^i \circ \gamma} {\mathrm d \tau} \left({0}\right)\) | as $m = \gamma \left({0}\right)$ | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \sum_{i \mathop = 1}^n \left. {\frac {\partial \left({f \circ \kappa^{-1} }\right)} {\partial \kappa^i} \left({\kappa \circ \gamma \left({\tau}\right)}\right) \frac {\mathrm d \kappa^i \circ \gamma} {\mathrm d \tau} \left({\tau}\right)} \right \rvert_{\tau \mathop = 0}\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \intlimits {\frac {\mathrm d \left({f \circ \kappa^{-1} \circ \kappa \circ \gamma}\right)} {\mathrm d \tau} \left({\tau}\right)} {\tau \mathop = 0} {}\) | Chain Rule for Derivatives | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \intlimits {\frac {\mathrm d \left({f \circ \gamma} \right)} {\mathrm d \tau} \left({\tau}\right)} {\tau \mathop = 0} {}\) | as $f \circ \kappa^{-1} \circ \kappa = f$ | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \frac {\mathrm d} {\mathrm d \tau} {\restriction_0} \, f \circ \gamma \left({\tau}\right)\) |
Hence $X_m$ is a tangent vector according to Definition 2.
This proves the assertion.
$\blacksquare$
Sources
- 2013: Gerd Rudolph and Matthias Schmidt: Differential Geometry and Mathematical Physics: $\S 1.4$: Proposition $1.4.7$
