# Math Help - Normal to a sphere using jacobian

1. ## Normal to a sphere using jacobian

Hi. I need a bit of help with the following question:

Calculate the normal to the sphere of radius
R and centred at the origin, using the appropriate Jacobians and the spherical coordinates. Show that the normal is directed everywhere along the radial vector from the sphere centre, r = (x, y, z).

So initially I desrcibed the sphere using polar coordinates:

x = Rcos(phi)sin(phi) y = Rsin(phi)sin(phi)and z = Rcos(phi) ;

Then I got three jacobians:

Jx=d(y,z)/d(phi,z) Jy=d(z,x)/d(phi,z) Jz=d(x,y)/d(phi,z)

For this I got:
Jx=2Rsin(phi)cos(phi)
Jy=Rsin(phi)sin(phi)-Rcos(phi)cos(phi)
Jz=0

So N=2Rsin(phi)cos(phi) i + Rsin(phi)sin(phi)-Rcos(phi)cos(phi) j

Just wanted to know if I have done this coorectly, and how to do the final 'show that' part of the question.

Cheers.

2. No, what you have is NOT correct. For one thing, the normal will have 0 k component only in the xy-plane and you are supposed to have the normal at any point on the sphere.

In fact, since any line through the origin is a normal to this sphere, the normal at point (x,y,z) should be just a multiple of xi+ yj+ zk. That is what you are asked to verify in the last part.

I think your fundamental problem is your expression of the spherica coordinates:
"x = Rcos(phi)sin(phi) y = Rsin(phi)sin(phi)and z = Rcos(phi)".

You must have two parameters, $\phi$ and $\theta$:
$x = Rcos(\theta)sin(\phi)$
$y = Rsin(\theta)sin(\phi)$
and $z = Rcos(\phi)$
(Note: engineering notation reverses " $\theta$" and " $\phi$".)

Now, you want your Jacobians to be
$Jx=d(y,z)/d(\theta, \phi)$, $Jy=d(z,x)/d(\theta,\phi)$, and $Jz=d(x,y)/d(\theta,\phi)$.