Demystifying Parameterization and Surface Integrals

[LCKurtz]

Introduction
This article will attempt to take the mystery out of setting up surface integrals. It will explain the basic ideas underlying surface integration and show you how to parameterize surfaces to set up the corresponding integrals efficiently. You will need to have had or be taking $3D$ calculus covering multiple integrals to get the most out of this presentation. The focus is on how to set up the integrals but not so much on techniques of evaluating them.
Vector functions giving space curves
Readers will be familiar with the vector function $$\vec R(t)= \langle x(t),y(t) \rangle = \langle 2\cos t, 2\sin t\rangle,~0\le t \le 2\pi$$which maps the $t$ interval $I=[0,2\pi]$ onto a circle of radius $2$ in the $x,y$ plane. Here, the value of the single independent variable $t$ determines the location of a point on the circle. I will call the interval $I$ the “parameter space”.
Similarly, a vector function of the form$$\vec R(t) = \langle x(t),y(t),z(t)\rangle,~a...

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[Orodruin]

Some very nice examples, but I am missing a discussion on the directed surface element $d\vec S = \vec n \, dS$, where $\vec n$ is a unit normal. This is very important for flux integrals in general and for the divergence theorem in particular:
$$
\int_\Omega (\nabla\cdot \vec v) dV = \oint_{\partial\Omega} \vec v \cdot d\vec S.
$$
The discussion would also fit right in before $dS = |\vec R_u \times \vec R_v|du\, dv$. Since the normal is orthogonal to both $\vec R_u$ and $\vec R_v$, the directed surface element is just $d\vec S = (\vec R_u \times \vec R_v) du\, dv$ without the absolute value around the cross product. At least to me this feels like a more natural quantity than $dS$ and it is how I introduce the surface element when I teach vector analysis.

 

[LCKurtz]

Some very nice examples, but I am missing a discussion on the directed surface element $d\vec S = \vec n \, dS$, where $\vec n$ is a unit normal. This is very important for flux integrals in general and for the divergence theorem in particular:
$$
\int_\Omega (\nabla\cdot \vec v) dV = \oint_{\partial\Omega} \vec v \cdot d\vec S.
$$
The discussion would also fit right in before $dS = |\vec R_u \times \vec R_v|du\, dv$. Since the normal is orthogonal to both $\vec R_u$ and $\vec R_v$, the directed surface element is just $d\vec S = (\vec R_u \times \vec R_v) du\, dv$ without the absolute value around the cross product. At least to me this feels like a more natural quantity than $dS$ and it is how I introduce the surface element when I teach vector analysis.

Hi Orodruin. I debated with myself about that and decided to leave the focus on the parameterization of surfaces. When I taught the course I spent the first lecture on parameterization, and on another day discussed flux integrals as an application. I do agree that the formula $\hat ndS = \pm \vec R_u \times \vec R_v~dudv$ is important and useful. Maybe I will eventually do a follow-up about that.

 

[jambaugh]

I have wished we had a definition of a "partial differential" (and introduce the idea to my students) in the sense that given our position vector $\mathbf{r}(u,v)$ as a function of our two parameters, it is convenient to define the two "partial differentials" as the differential of the function treating all but one variable as constant:
$$d_u\mathbf{r} = \frac{\partial \mathbf{r}}{\partial u} du, \quad d_u\mathbf{r} = \frac{\partial \mathbf{r}}{\partial u} dv$$
and thence:
$$d\mathbf{S} = d_u\mathbf{r}\times d_v\mathbf{r}$$

That's my wish as far as notation goes. When I teach Calc III, I teach my students the triple product first, as a "determinant" function of three vectors in 3 dimensions and then explain the cross product as the mapping from two vectors to the vector which dots with the third to have the same effect. (Hinting strongly at dual vectors and Riesz rep theorem). It makes the geometric interpretation more natural since the "determinant" is the orientation signed volume of the parallelopiped formed by the three vectors. When we then get to surface integrals they've (most of em) internalized this definition and the vector surface area element makes more sense (or so I hope).