Deriving Vector Area of a Surface S Using the Cross Product

[rbwang1225]

Define

$\textbf{a}\equiv\int_{S}{d\textbf{a}}$

How do I show that $\textbf{a}=\frac{1}{2}\oint{\textbf{r}\times d\textbf{l}}$

Actually, this is the problem of the EM book of Griffiths, but I don't understand his hint.

Any help would be appreciated.

 

[henry_m]

One trick is to dot the latter expression with an arbitrary constant vector, and then use cyclicity of the triple product. You should be able to cast it into a form where it is easy to use Stokes' Theorem.

 

[rbwang1225]

O.K., I got the result.

But is there any other more physical or geometrical way to derive that?

Thanks!