---- > [!definition] Definition. ([[regular curve]]) > A [[parameterized curve|parameterized differentiable curve]] is said to be **regular** if it satisfies the [[differentiable Euclidean submanifold (with or without boundary)|the differentiable Euclidean manifold rank condition]] as a [[coordinate patch]] on a $1$-manifold in $\mathbb{R}^{n}$. That is, if $\alpha'(t) \neq \b 0$ for all $t \in I$. ^3690b8 > [!intuition] Justification > If we think about the geometric image as the path 'traced out' by a point moving around the plane, to enforce $\alpha'(t) \neq \b 0$ is to enforce that this point never slows to a halt or reverses direction ([[intermediate value theorem|IVT]]). ^5e6325 ---- #### ---- #### 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 > ```