^^ [[Schutz - 2 - Differentiable Manifolds and Tensors|2. Differentiable Manifolds and Tensors]] <<[[Schutz - 2.4 - Global Considerations|2.4 Global Considerations]] | [[Schutz - 2.6 - Functions on M|2.6 Functions on M]]>> # Curves > **Curve** > A *curve* is a differentiable [[Schutz - 1.2 - Mappings#^6c8b58|injection]] from an open set of $\mathbb{R}$ [[Schutz - 1.2 - Mappings#^6c8b58|into]] $M$. More precisely, if the curve is defined on the interval $I \subset \mathbb{R}$, then $\gamma: I \rightarrow M$ is a $C^k$-curve if the component functions of $\gamma$ are $C^k$ differentiable. ^5c64cb > **Image of the curve** > One associates with each point $\lambda \in I$ a point in $M$, called the *image point* of $\lambda$. More broadly, the *image* of the curve is $\gamma[I] \subseteq M$. This curve is parameterized by $\lambda$ and the image $\gamma[I]$ gives us the ordinary notion of a curve. Two curves are different even if they have the same image in $M$, because they can are in principle parameterized differently, i.e. they assign different parameter values to each image point. ^6a34c8 ![[Pasted image 20210123191354.png]]