Difference between dS and ds̄

[TranscendArcu]

Difference between "dS" and "ds̄"

http://img268.imageshack.us/img268/126/skjermbilde20111201kl71.png

I often see theorems such as the one above in my coursework, but I do not actually understand the difference between "dS" and "ds̄". I presume, because the latter has a "hat," that the latter also has something to do with vectors. So what's the difference between these two differential distances?

 

[Mark44]

http://img268.imageshack.us/img268/126/skjermbilde20111201kl71.png

I often see theorems such as the one above in my coursework, but I do not actually understand the difference between "dS" and "ds̄". I presume, because the latter has a "hat," that the latter also has something to do with vectors. So what's the difference between these two differential distances?

The dS on the left is a vector, an increment of some vector S (looks sort of like the integral for work). The one on the right has dS as a scalar. The integrand here is the dot product of two vectors, which results in a scalar.

 

[Ray Vickson]

The dS on the left is a vector, an increment of some vector S (looks sort of like the integral for work). The one on the right has dS as a scalar. The integrand here is the dot product of two vectors, which results in a scalar.

$d\vec{S}$ is the vector whose direction is $\vec{n}$ and whose magnitude is $dS$, so, in fact, we have $$d\vec{S} = \vec{n} dS.$$

RGV