Chapter 10, Problem 1GYR

To determine

Define and provide details of parametrization.

Explanation of Solution

Description:

Parametrization of the curve consists of both equations and intervals of a curve together.

If x and y in the plane are given with a functions, that is $x=f\left(t\right)$ and $y=g\left(t\right)$ with an interval $I$ of $t$ values, then the points $\left(x,y\right)$ can be rewrite as $\left(f\left(t\right),g\left(t\right)\right)$.

Thus, the parametrization of the curve in xy-plane is described.

Since we are rewriting the variables in a single parameter t, that is $y=f\left(t\right)$. Therefore the function $y=f\left(x\right)$ always has a parametrization.

Thus, the $y=f\left(x\right)$ always have a parametrization.

The parametrization of a curve is not unique.

Example:

The parametrization for a circle is $\left(x,y\right)=\left(\cos t,\sin t\right)$ for $0\le t\le 2\pi$, similarly $\left(x,y\right)=\left(\cos 2t,\sin 2t\right)$ for $0\le t\le \pi$.

Thus, the parametrization curve is not unique.