1. ## Flux

$\int \int_S F.N.ds$

$N = \frac{\frac{d(r)}{d(u)} X \frac{d(r)}{d(v)}}{|\frac{d(r)}{d(u)} X \frac{d(r)}{d(v)}|}$

$ds = |\frac{d(r)}{d(u)} X \frac{d(r)}{d(v)}|dudv$

Flux:

$\int \int_S F. \frac{d(r)}{d(u)} X \frac{d(r)}{d(v)} dudv$

2. that is correct

$\int\int F.n dS = \int\int F.\left(\frac{\partial r}{\partial u} \times \frac{\partial r}{\partial v} \right) dA$

3. I always write it like the following:

Let $\Phi (u,v)$ be a parametrization of the surface $S$

then flux = $\int \int_{S} F \cdot dS = \int \int_{D} F \cdot (T_{u} \times T_{v}) \ du \ dv$

where $T_{u} = \frac {\partial \Phi}{\partial u}$

$T_{v}= \frac {\partial \Phi}{\partial v}$

and $D$ is an elementary region in the uv-plane

There are other ways to write it, but I always stick with this definition.

4. Originally Posted by Random Variable
I always write it like the following:

Let $\Phi (u,v)$ be a parametrization of the surface $S$

then flux = $\int \int_{S} F \cdot dS = \int \int_{D} F \cdot (T_{u} \times T_{v}) \ du \ dv$

where $T_{u} = \frac {\partial \Phi}{\partial u}$

$T_{v}= \frac {\partial \Phi}{\partial v}$

and $D$ is an elementary region in the uv-plane

There are other ways to write it, but I always stick with this definition.
Ok. Thanks