If `x=r sinthetacosphi, y =r sin thetasinphiandz=rcostheta`, then `x^(2)+y^(2)+x^(2)` is independent of which of following ?

Options

A. r only
B. `r,phi`
C. `theta,phi`
D. `r,theta`

Answer

C

Explanation

As given , x =r `sinthetacosphi,y=rsinthetasinphi`. And `z=r costheta`
Now , `x^(2)+y^(2)+z^(2)=r^(2)sin^(2)thetacos^(2)phi+r^(2)sin^(2)phisin^(2)theta+r^(2)cos^(2)theta`
`=r^(2)sin^(2)theta(sin^(2)phi+cos^(2)phi)+r^(2)cos^(2)theta`
`=r^(2)sin^(2)theta+r^(2)cos^(2)theta`
`=r^(2)(sin^(2)theta+cos^(2)theta)`
`=r^(2)`
Thus , `x^(2)+y^(2)+z^(2)` is independent of `theta and phi`.