# Curve

Functions of the form $$f : I \rightarrow \mathbb{R}^n$$ where $$I \subseteq \mathbb{R}$$ are called curves. The co-domain does not necessarily have to be the $$n$$ dimensional Euclidean plane.

Curves are often parametrized such that we have $$f : t \rightarrow \mathbf{x}(t)$$, where $$t \in I$$. As a shorthand, the curve can be denoted as $$\mathbf{x}$$. Here we implicitly assume that the range of the function has the standard basis, that is, $$\mathbf{x} = [x^1, \cdots, x^m]^T = x^i \mathbf{e}_i$$ where the last term uses Einstein summation convention.

## Thoughts

Created: 2022-03-13 Sun 21:44

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