x^2 + y^2 + z^2 = 2.
Not sure how to start this problem. If someone could point me in the right direction that would be great.
Thanks
Then you must be completely misunderstanding everything because any text I have ever seen defined spherical coordinates using $\displaystyle \rho= \sqrt{x^2+ y^2+ z^2$.
So what formulas does your text use to define "polar" and "spherical" coordinates? (The two you post are for spherical coordinates but since there are three coordinates, you should have three formulas, not 2.)
Since linalg123 hasn't got back to us, I will continue myself.
Every Calculus text I have ever seen has defined "polar coordinates" with the formulas
$\displaystyle x= r cos(\theta)$
amd
$\displaystyle y=r sin(\theta)$
Strictly speaking, "polar coordinates" is defined only in two dimensions but and immediate extension is "cylindrical coordinates" using 'z' as the third variable,
Spherical coordinates are defined by the formulas
$\displaystyle x= \rho cos(\theta)sin(\phi)$
$\displaystyle y= \rho sin(\theta)sin(\phi)$
$\displaystyle z= \rho cos(\theta)$
It is then easy to see that, in cylindrical coordinates, $\displaystyle x^2+ y^2+ z^2= r^2cos^2(\theta)+ r^2sin^2(\theta)+ z^2= r^2(cos^2(\theta)+ cos^2(\theta)+ z^2= r^2+ z^2$ so that $\displaystyle x^2+ y^2+ z^2= 2$ becomes $\displaystyle r^2+ z^2= 2$
And in spherical coordinates,
$\displaystyle x^2+ y^2+ z^2= \rho^2 cos^2(\theta)sin^2(\phi)+\rho^2 sin^2(\theta) sin^2(\phi)+ \rho^2 cos^2(\phi)$
$\displaystyle = \rho^2 sin^2(\phi)(cos^2(\theta)+ sin^2(\theta)+ \rho^2 cos^(\phi)= \rho^2 (sin^2(\phi)+ cos^2(\phi))= \rho^2$
So in spherical coordinates $\displaystyle x^2+ y^2+ z^2= 2$ becomes [tex]\rho^2= 2[/itex] or, since $\displaystyle \rho$, the distance from the origin to the point, is never negative,
$\displaystyle \rho= \sqrt{2}$.