George Green's Paper: Explaining the minus sign in area element

[Karthiksrao]

Hello,

In Green's famous paper : "An essay on the Application of mathematical Analysis to the theories of Electricity and Magnetism" - which is freely available online here :
http://arxiv.org/abs/0807.0088

in article 3, page 16, last paragraph, while deriving his now famous Green's theorem, he gives one particular form for bringing in area element into the triple integral.

He says : dy dz = -(dx/dw) dq

where, dq is the area element; dw is the small line perpendicular to the surface

Can anybody explain how this is got ?
The minus sign is probably because he is at this stage considering the larger values of 'x' , so that the direction of dw will be in the opposite direction to dx

Thanks a lot!

 

[fresh_42]

This is simply the condition being perpendicular to the surface:
$$\begin{bmatrix}dx\\dy\,dz \end{bmatrix} \perp \begin{bmatrix}d\sigma''\\d\omega \end{bmatrix} \Longleftrightarrow \left\langle \begin{bmatrix}dx\\dy\,dz \end{bmatrix} , \begin{bmatrix}d\sigma''\\d\omega \end{bmatrix}\right\rangle =d\omega \,(dy\,dz) + dx \,(d\sigma'')=0$$