- Inicio
- Vector analysis
- Parametrization of surfaces
- Ejercicios
Parametrization of surfaces
Find the parametrization of the surface delimited by the revolution of the parabole $z=x^2$, for $z$ between $1$ and $3$.
Since it is a revolution body, we can use the last example of surfaces, this way $$\varphi(x,\theta) =(x\cdot\cos\theta,x\cdot\sin\theta,x^2)$$
The domain will be $\theta\in[0,2\pi]$ and for $x$, it is necessary to bear in mind that $x=\sqrt{z}$, therefore if $z\in[1,3], \ x\in[\sqrt{1},\sqrt{3}]=[1,\sqrt{3}]$.
Another form to parametrize would be to take $z$ as a variable, then $$\varphi(z,\theta) =(\sqrt{z}\cdot\cos\theta,\sqrt{z}\cdot\sin\theta,z), \ \ z\in[1,3]$$
The parametrization of the surface is $\varphi(x,\theta) =(x\cdot\cos\theta,x\cdot\sin\theta,x^2), \ \ x\in[1,\sqrt{3}]$ or $\varphi(z,\theta) =(\sqrt{z}\cdot\cos\theta,\sqrt{z}\cdot\sin\theta,z), \ \ z\in[1,3]$