how to convert $dS$ into $dxdy$

Question

Flux of $\boldsymbol{\vec{F}=-\hat{i}+2\hat{j}+3\hat{k}}$ across the surface $S : \boldsymbol{z=0,0\leq x\leq 2,0\leq y\leq 3}$ in direction of $\boldsymbol{\hat{k}}$ is equal to?

So we have to calculate $\int F.\hat{n} dS$,but i dont know how to convert $dS$ to $dxdy$. What is the general method? Any help appreciated.

Answer 1

General method:

1. Parametrize the surface as $$ \vec x(s,t) = (x_1(s,t),x_2(s,t),x_3(s,t)).$$
2. The surface element is given by $$ \left| \frac{\partial \vec x}{\partial s} \times \frac{\partial \vec x}{\partial t}\right|dsdt$$ and you integrate over the parameters $(s,t).$ The $\times$ is a cross product and $|\cdot|$ is the length of the vector.

Your question:

Your surface is just a rectangle in the $x-y$ plane. $dS = dxdy.$

Connection between them:

Let $\vec x(s,t) = (s,t,0)$ for $0<s<2$ and $0<t<3.$ Verify this is a correct parametrization of the surface.

Plugging in the above formula you'll find the surface element is $dsdt$. Then just call $s$ and $t$ by their more natural names $x$ and $y.$