Thread: Area vectors for 1-D sphere

1. Area vectors for 1-D sphere

I'm trying to find the volume and surface area of a 1-D dimensional sphere, i.e. retaining only the radial dependence.

I know that the volume element for a 3-D sphere would be

$\displaystyle dV = r^2\sin\theta{d}\theta{d}\phi{d}{r}$

If it's one-dimensional would it just be ? Or would it just be ?

With regards to the surface area vector in 3-D it is

so in 1-D would it be

or would it just be ?

Essentially what I'm trying to do is a Finite Volume Method for a 1-D sphere and I want to find the surface area vectors, and volume for my Finite Volume Cells.

2. You are referring to a "three-dimensional sphere" as the usual "sphere" (strictly speaking a "ball"- "sphere" refers to the surface of the ball). The "two-dimensional" sphere then would be the disk whose surface is a circle, and the "one dimensional sphere" would be an interval whose "surface" is the two endpoints of the interval. The "volume element" would be just "dx" and there is no "surface area element" because the surface of a "one-dimensional sphere" is not continuous.