Demonstrate divergence and rotational.

Question

Show that $$DIV(A)=\lim_{\Delta s\rightarrow0}\frac{\displaystyle\int\int_{\Delta v}A\cdot nds}{\Delta v}$$and, $$ROT(A)\cdot n=\lim_{\Delta s\rightarrow 0}\frac{\displaystyle\oint_{C}A\cdot dr}{\Delta s}$$ Is there a demonstration for these results? One suggestion they gave me is to use the mean value theorem for integrals.

Any suggestions?

Answer 1

The equations $$ DIV(A)=\lim_{\Delta s\rightarrow0}\frac{\displaystyle\int\int_{\Delta v}A\cdot nds}{\Delta v}$$and $$ROT(A)\cdot n=\lim_{\Delta s\rightarrow 0}\frac{\displaystyle\oint_{C}A\cdot dr}{\Delta s}$$

are NOT consequences of the ordinary definition of the divergence and rotational of a vector field: they are more general since they require only the integrability of the vector field $A$ and no differentiability nor continuity conditions. Thus the standard definition of those vector operators follows easily from them, since considering

$$DIV(A)=\dfrac{\partial A_x}{\partial x}+\dfrac{\partial A_y}{\partial y}+\dfrac{\partial A_z}{\partial z} \tag1$$

$$ROT(A)= \left| {\begin{array}{*{20}{c}}{\vec i}&{\vec j}&{\vec k}\\{\displaystyle \frac{\partial }{{\partial x}}}&{\displaystyle \frac{\partial }{{\partial y}}}&{\displaystyle \frac{\partial }{{\partial z}}}\\A_x&A_y&A_z\end{array}} \right|\tag2$$

1. If you use Gauss divergence theorem, $(1)$ can be shown easily. Try this yourself.
2. If you use Stokes curl theorem, $(2)$ can be shown easily. Try this also yourself.