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Find a parametric representation for the surface. The part of the ellipsoid x^{2} + 2y^{2} +...

Question:

Find a parametric representation for the surface.

The part of the ellipsoid {eq}\displaystyle\; x^{2} + 2y^{2} + 3z^{2} = 1\; {/eq} that lies to the left of the {eq}\,xz {/eq}-plane

Finding a Parametric Representation for the Surface:

A parametric surface is a surface in the Euclidean space {eq}R^3 {/eq} which is defined by a parametric equation with two parameters {eq}\vec{r}:R^{2}\rightarrow R^{3}. {/eq} Parametric representation is a very general way to specify a surface, as well as implicit representation. Recall that a curve in space is given by parametric equations as a function of single parameter {eq}t {/eq} is, {eq}x=x\left ( t \right ) \ y=y\left ( t \right ) \ z=z\left ( t \right ). {/eq} In particular, it may be useful to parametrize such surface using two parameters possibly different from {eq}x {/eq} and {eq}y. {/eq} In particular, a surface given by the parametric equations {eq}x=x\left ( u,v \right ) \ y=y\left ( u,v \right ) \ z=z\left ( u,v \right ) {/eq} is referred to as a parametric surface and the two independent variable{eq}u {/eq} and {eq}v {/eq} as parameters. Parametric representation for the surface with three variables can be derived as, {eq}\vec{r}\left ( t \right )=x\left ( t \right )\vec{i}+y\left ( t \right )\vec{j}+z\left ( t \right )\vec{k}. {/eq}

Answer and Explanation:

From the given information, the part of the ellipsoid is, {eq}x^{2} + 2y^{2} + 3z^{2} = 1 . {/eq}

And from the given information, we know that the surface lies to the left of the {eq}xz {/eq}-plane. Hence {eq}z=0, {/eq} and the value of {eq}y {/eq} should be less than zero. Therefore, {eq}y<0. {/eq}

From the given equation,

{eq}x^{2} + 2y^{2} + 3z^{2} = 1 \\ 2y^{2}=1-3z^{2}-x^{2} \\ y^{2}=\frac{1-3z^{2}-x^{2}}{2} \\ y=\pm \sqrt{\frac{1-3z^{2}-x^{2}}{2}} {/eq}

Since the value of {eq}y {/eq} should be {eq}y<0. {/eq}

Therefore,

{eq}y=-\sqrt{\frac{1-3z^{2}-x^{2}}{2}} {/eq}

Now, consider the parameters for {eq}x=s \\ z=t {/eq}

Therefore, the parametric equation of the surface is,

{eq}r\left ( s,t \right )=< s\vec{i}-\sqrt{\frac{1-3t^{2}-s^{2}}{2}}\vec{j}+t\vec{k}> . {/eq}


Learn more about this topic:

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Evaluating Parametric Equations: Process & Examples

from Precalculus: High School

Chapter 24 / Lesson 3
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