---- - Let $M$ be a smooth manifold of dimension $k$; - Let $p \in M$; - Let $C_{}$ denote the set of smooth [[parameterized curve|parameterized curves]] on $M$ $\gamma: I \ni 0 \to M$ satisfying $\gamma(0)=p$. [[TODO]] update notation here based on the dog video > [!definition] Definition. ([[tangent space to a manifold]]) > Declare $\gamma_{1} \sim \gamma_{2}$ if and only if $(\varphi \circ \gamma_{1})'(0)=(\varphi \circ \gamma_{2})'(0)$ for any [[coordinate chart|chart]] $\varphi: U \ni p \to \mathbb{R}^{k}$ about $p$. The set $C /{\sim}$ of [[equivalence class|equivalence classes]] is called the **tangent space to $M$ at $p$**. > [!justification] > We need to: - Verify that $\sim$ is indeed an [[equivalence relation]]; - Verify the linear structure on $C / \sim$; - Verify that this definition is [[coordinate chart|coordinate-independent]]. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```