What are surface elements?

Question

would somebody please help me out with surface integrating? While calculating flux, there is the integrand $F\cdot n$ however in the definition of surface integration I am reading about $|| r_{u} \times r_{v}||$. Thanks very much. I do not remember why I keep having to change $d\mathbf{S}$ into $\mathbf{n} dS$.

Answer 1

I would say $d\mathbf{S}$ is the vector element of surface area and $dS$ is the scalar element of surface area. So $d\mathbf{S} = \mathbf{N} dS$ where $\mathbf N$ is the unit normal to the surface.

Which normal? The one chosen according to the rules you are using! Technically, you are integrating over an oriented surface, and $\mathbf{N}$ is determined by the orientation.

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added
Note if $\mathbf r(u,v)$ is a parametric description of the surface, then at each point of the surface, $\frac{\partial\mathbf r}{\partial u}$ and $\frac{\partial\mathbf r}{\partial v}$ are tangents to the surface, $\frac{\partial\mathbf r}{\partial u}\times \frac{\partial\mathbf r}{\partial v}$ is a normal to the surface. The norm $\|\frac{\partial\mathbf r}{\partial u}\times \frac{\partial\mathbf r}{\partial v}\|$ is the length of that normal vector. Then think of $dS = \|\frac{\partial\mathbf r}{\partial u}\times \frac{\partial\mathbf r}{\partial v}\|\;du\;dv$, where $\|\frac{\partial\mathbf r}{\partial u}\times \frac{\partial\mathbf r}{\partial v}\|$ is the area of a paralellogram with sides $\frac{\partial\mathbf r}{\partial u}$ and $\frac{\partial\mathbf r}{\partial v}$. And $d\mathbf S = (\frac{\partial\mathbf r}{\partial u}\times \frac{\partial\mathbf r}{\partial v})\;du\;dv$ is a vector with magnitude $dS$ and perpendicular to the surface.