Find a parametric representation for the following surface. Write it as a vector. z^2 = 1 + x^2 +...


Find a parametric representation for the following surface. Write it as a vector. {eq}z^2 = 1 + x^2 + y^2{/eq} for {eq}1 \leq z \leq 10{/eq}

Parametric Surfaces

A surface given as {eq}\displaystyle z=f(x,y) {/eq} can be written parametrically with two parameters, as {eq}\displaystyle \mathbf{r}(u,v)=\langle x(u,v), y(u,v), z(u,v)\rangle, (u,v)\in\mathbf{R}^2. {/eq}

When finding a parametrization of a surface, we can use cylindrical coordinates if the surface involves spheres, cones or cylinders.

In cylindrical coordinates, we describe the region based on the polar angle (that the line passing through the origin and the projection of the point onto the xy-plane makes with the positive x-axis)

on the polar radius, (given by the distance from the projection of the point to the xy-plane to the origin) and the z-coordinate of the point in Cartesian coordinates.

The Cartesian - cylindrical conversion is given by the following formulae,

{eq}\displaystyle x=r\cos \theta, y=r\sin \theta, z=z. {/eq}

Answer and Explanation:

The surface {eq}\displaystyle z^2=1+x^2+y^2, 1\leq z\leq 10 \iff z=\sqrt{1+x^2+y^2} {/eq}

is the upper cone with the axis of symmetry, the z-axis and vertex at {eq}\displaystyle (0,0,1) {/eq} and bounded by the plane {eq}\displaystyle z=10. {/eq}

Therefore, we will use cylindrical coordinate to find a parametrization of the surface as a vector function of two parameters.

Because the projection of the cone onto the xy-plane is a full disk, centered at the origin, we will use sine and cosine functions for the x and y-variables.

{eq}\displaystyle \boxed{\mathbf{r}(u,v)=\langle u\cos v, u\sin v, \sqrt{1+u^2}\rangle, 0\leq u\leq 3\sqrt{11}, 0\leq v\leq 2\pi}. {/eq}

The bounds of u were found by finding u when {eq}\displaystyle z=1 \iff 1+u^2=1\implies u=0, \displaystyle z=10 \iff 1+u^2=100\implies u=3\sqrt{11}. {/eq}

Learn more about this topic:

Evaluating Parametric Equations: Process & Examples

from Precalculus: High School

Chapter 24 / Lesson 3

Related to this Question

Explore our homework questions and answers library