I have some short questions on some lingering confusing concepts, specific to surface integrals:
a) Is the surface integral in the Divergence and Stokes's Theorem the same thing?
Both require a unit-normal vector and a surface area component, $\vec n$dS, one is clearly a "flux integral", but is the one in Stoke's Theorem also a flux integral?
b) In both theorems, the surface integrals require unit-normal vectors in the integrand. But I remember working through some surface integrals where one shouldn't normalize the normal vector obtained — I don't remember the example concretely now. Why does this happen? Why keep the magnitude of the normal vector…sometimes? I also have used a formula in the past, but not often: $\vec n$ = $<-f_x,-f_y,1$> – why doesn't this formula show up more often in surface integral problems? Is this formula for a normal vector $\vec n$ = $<-f_x,-f_y,1$>…normalized / required to be normalized?
c) And finally, why do some surface integrals have this factor in its $ds$ surface area component, $\large||\frac{d\vec v}{d\theta}\times\frac{d\vec v} {d\phi}||$,say, for integrating over a surface that is a sphere. Does this factor just play the role of a "Jacobian", but for 3-D surfaces? And then perhaps in flat surfaces in 3-D space, e.g., a plane disk in space, such a "Jacobian" factor is computed differently? We know the Jacobian to be the determinant of the matrix of partial derivatives, e.g., yielding the variable "r" in polar and spherical coordinates integration. But $\large||\frac{d\vec v}{d\theta}\times\frac{d\vec v} {d\phi}||$ is the magnitude of a determinant (cross-product of partial derivative vectors), though.
Thanks so much in advance,
Best Answer
(a) A surface integral is a surface integral. The name "flux" comes from the obvious physical interpretation: When ${\bf v}$ is a flow field then $\int_S{\bf v}\cdot d\vec\omega$ is the amount of fluid traversing $S$ per unit of time in the chosen positive direction. For this reason other surface integrals are called "flux" even if there is nothing flowing to be seen: We talk about the "flux of heat" in the theory of heat conduction, or about the flux of ${\rm curl}\,{\bf F}$ through $S$ in Stokes' theorem.
(b) A surface integral does not involve a unit normal per se. Therefore a flux can be computed without taking any square roots: Assume that $S$ is given by a parametrization $$S:\quad Q\to{\mathbb R}^3,\qquad (u,v)\mapsto{\bf x}(u,v)$$ which induces via $${\bf n}:={{\bf x}_u\times{\bf x}_v\over |{\bf x}_u\times{\bf x}_v|}$$ the intended orientation. Then we have $$\int_S{\bf v}\cdot d\vec\omega=\int_Q{\bf v}\bigl({\bf x}(u,v)\bigr)\cdot\bigl({\bf x}_u(u,v)\times{\bf x}_v(u,v)\bigr)\>{\rm d}(u,v)\ .$$ Here we see the "vectorial surface element" $$d\vec\omega:=\bigl({\bf x}_u\times{\bf x}_v\bigr)\>{\rm d}(u,v)\qquad\bigl(={\bf n}\>{\rm d}\omega\bigr)\ ,$$ whereby ${\rm d}\omega$ denotes the "scalar surface element" $${\rm d}\omega:=\bigl|{\bf x}_u\times{\bf x}_v\bigr|\>{\rm d}(u,v)\ .$$ The latter, involving a nasty square root, is only needed when we actually want to compute the surface area, or have to compute the total mass of an "infinitesimally thin" sheet with variable surface density, or a similar quantity related to surface area.
(c) What you call a "Jacobian factor" is the cross product ${\bf x}_u\times{\bf x}_v$. This construction $\times$, possible only in ${\mathbb R}^3$, allows to write a "bilinear skew symmetric function" $$B: \quad ({\bf x},{\bf y})\mapsto B({\bf x},{\bf y})\in{\mathbb R}$$ of two vector variables as a scalar product in the following way: There is a vector ${\bf b}$ representing $B$ such that $$B({\bf x},{\bf y})={\bf b}\cdot({\bf x}\times{\bf y})$$ holds identically in ${\bf x}$ and ${\bf y}$. The curl vector occurring in Stokes' theorem is a typical such ${\bf b}$. What I'm saying is this: In order to really understand what's going on here one has to go into multilinear algebra.