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

- Mar 20th 2012, 07:57 PMlinalg123Write down the equation in spherical and cylindrical coordinates.
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 - Mar 21st 2012, 02:33 AMProve ItRe: Write down the equation in spherical and cylindrical coordinates.
- Mar 27th 2012, 08:49 PMlinalg123Re: Write down the equation in spherical and cylindrical coordinates.
i know that θ= tan^-1(y/x)

and ϕ= cos^-1 (z/r)

but how do i know what x,y,z are? - Mar 28th 2012, 04:47 AMlinalg123Re: Write down the equation in spherical and cylindrical coordinates.
any further help on this question would be much appreciated i can't find it in the textbook or on the internet anywhere.

- Mar 29th 2012, 09:28 AMHallsofIvyRe: Write down the equation in spherical and cylindrical coordinates.
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.) - Mar 30th 2012, 05:01 PMHallsofIvyRe: Write down the equation in spherical and cylindrical coordinates.
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,

$\displaystyle \rho= \sqrt{2}$.