Question

Give an example of a function whose image is the unit sphere

${\mathit{x}}^{\mathbf{2}}\mathbf{+}{\mathit{y}}^{\mathbf{2}}\mathbf{+}{\mathit{z}}^{\mathbf{2}}\mathbf{=}\mathbf{1}$

inR³.

Short answer

f$\left[\begin{array}{c}\varphi \\ \theta \end{array}\right]=\left[\begin{array}{c}\sin \varphi \cos \theta \\ \sin \varphi \sin \theta \\ \cos \theta \end{array}\right]$is a function whose image is the unit sphere $x^2+y^2+z^2=1$ in ${\mathrm{ℝ}}^3{\mathrm{ℝ}}^3$

Step-by-step solution

Consider the parameters.

$$\begin{gathered} image\left(f\right)=\left\{f\left(x\right):xinX\right\} \\ =\left\{binY:b=f\left(x\right),forsomexinX\right\} \\ x=\sin \varphi \cos \theta \\ Let,Y=\sin \varphi \sin \theta \\ z=\cos \varphi \end{gathered}$$The image of a function consists of all the values the function takes in its target space. If f is a function from X to Y, then

$$\begin{gathered} x^2+y^2+z^2=1 \\ \Rightarrow {\left(\sin \varphi \cos \theta \right)}^2+{\left(\sin \varphi \sin \theta \right)}^2+{\left(\cos \varphi \right)}^2=1 \\ \Rightarrow {\sin }^2\varphi \left({\cos }^2\theta +{\sin }^2\theta \right)+{\cos }^2\varphi =1 \end{gathered}$$

Let, $$\begin{gathered} y=\sin \varphi \sin \theta \\ z=\cos \varphi \end{gathered}$$

Substitute these in the equation of sphere, thus,

$$\begin{gathered} x^2+y^2+z^2=1 \\ \Rightarrow {\left(\sin \varphi \cos \theta \right)}^2+{\left(\sin \varphi \sin \theta \right)}^2+{\left(\cos \varphi \right)}^2 \\ \Rightarrow {\sin }^2\varphi \left({\cos }^2\theta +{\sin }^2\theta \right)+{\cos }^2\varphi =1 \\ \Rightarrow {\sin }^2\varphi +{\cos }^2\varphi =1 \\ \therefore 1=1 \end{gathered}$$

Final answer.

$f\left[\begin{array}{c}\varphi \\ \theta \end{array}\right]=\left[\begin{array}{c}\sin \varphi \cos \theta \\ \sin \varphi \sin \theta \\ \cos \varphi \end{array}\right]$ is a parametrization of the ellipse $x^2+y^2+z^2=1$in${\mathrm{ℝ}}^3{\mathrm{ℝ}}^3$ .