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### Section 17.2 : Parametric Surfaces

5. The portion of the sphere of radius 6 with $$x \ge 0$$.

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Because we have a portion of a sphere we’ll start off with the spherical coordinates conversion formulas.

$x = \rho \sin \varphi \cos \theta \hspace{0.5in}y = \rho \sin \varphi \sin \theta \hspace{0.5in}z = \rho \cos \varphi$

However, we are actually on the surface of the sphere and so we know that $$\rho = 6$$. With this the conversion formulas become,

$x = 6\sin \varphi \cos \theta \hspace{0.5in}y = 6\sin \varphi \sin \theta \hspace{0.5in}z = 6\cos \varphi$ Show Step 2

The set of parametric equations that will give the full sphere is then,

$\vec r\left( {\theta ,\varphi } \right) = \left\langle {x,y,z} \right\rangle = \left\langle {6\sin \varphi \cos \theta ,6\sin \varphi \sin \theta ,6\cos \varphi } \right\rangle$

Remember that all we do is plug the conversion formulas for $$x$$, $$y$$, and $$z$$ into the $$x$$, $$y$$ and $$z$$ components of the vector $$\left\langle {x,y,z} \right\rangle$$ and we have a set of parametric equations. Also note that because the resulting vector equation is an equation in terms of $$\theta$$ and $$\varphi$$ those will also be the variables for our set of parametric equation.

Show Step 3

Finally, we need to deal with the fact that we don’t actually want the full sphere here. We only want the portion of the sphere for which $$x \ge 0$$.

We can restrict $$x$$ to this range if we restrict $$\theta$$ to the range $$- \frac{1}{2}\pi \le \theta \le \frac{1}{2}\pi$$ .

We’ve not put any restrictions on $$z$$ and so that means that we’ll take the full range of possible $$\varphi$$ or $$0 \le \varphi \le \pi$$. Recall that $$\varphi$$ is the angle a point in spherical coordinates makes with the positive $$z$$-axis and so that is the quantity we’d need to restrict if we’d wanted to restrict $$z$$ (for example $$z \le 0$$).

Putting all of this together gives the following set of parametric equations for the portion of the surface we are after.

$\require{bbox} \bbox[2pt,border:1px solid black]{{\vec r\left( {\theta ,\varphi } \right) = \left\langle {6\sin \varphi \cos \theta ,6\sin \varphi \sin \theta ,6\cos \varphi } \right\rangle \,\,\,\,\,\,\,\, - \frac{1}{2}\pi \le \theta \le \frac{1}{2}\pi \,\,\,\,,\,\,\,\,\,0 \le \varphi \le \pi }}$