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Section 6-2 : 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.

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