Curve

Topics: Calculus

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(definition)

We call a function $\sigma :[a,b]\to {\mathbb{R}}^n$ a curve.

Example 1

Let $\sigma :[0,1]\to {\mathbb{R}}^2$ be defined by $\sigma (t)=(t,t)$.

For instance, we have that two points in that curve are:

$$\sigma (0)=(0,0)$$

$$\sigma (1)=(1,1)$$

This curve looks like:

Example 2

Let $\sigma :[0,1]\to {\mathbb{R}}^2$ be given by $\sigma (t)=(t,t^2)$.

This curve looks like:

Example 3

Let $\sigma :[-\pi ,5\pi ]\to {\mathbb{R}}^2$ be given by $\sigma (\theta )=(\cos \theta ,\sin \theta )$.

This curve is a circle.