surface integral and vector surface integral

Question

What is the difference between this surface integral and this vector surface integral? When can I use the first surface integral instead of the vector surface integral and when cannot?

Answer 1

In the first place get a modern advanced calculus text. What I see in your links is somewhere between "Bronstein-Semendjajew" and "Schaum's Outline". Both were already well seasoned when I studied maths in the 1950s.

The difference between the two kinds of integrals is the following: In the first integral a tiny piece of surface (a "surface element") just has some tiny area, whereas in the second its spacial orientation, resp., the direction of the positive normal, plays a rôle as well.

Which kind of integral you have to set up depends on the concrete differential-geometric or physical situation at stake. Here are some examples, whereby I'm referring to a parametric representation $$(u,v)\mapsto{\bf x}(u,v)$$ of the surface $S$ in question:

If you have to compute the total surface area of $S$ then it's just the scalar surface element $${\rm d}\omega=|{\bf x}_u\times{\bf x}_v|\>{\rm d}(u,v)\ .$$ If the surface is made of some material with density (weight per unit area) $\sigma$, and you need the total potential energy of $S$ with respect to sea level $x_3=0$ then you have to integrate $$\sigma\> x_3\>{\rm d}\omega=\sigma \>x_3(u,v)\>|{\bf x}_u\times{\bf x}_v|\>{\rm d}(u,v)\ .$$ If there is a fluid field ${\bf x}\mapsto{\bf v}({\bf x})$ given, and you want to know how much fluid passes through the "virtual" surface $S$ per second then it plays a rôle whether the field hits the different parts of $S$ orthogonally or tangentially. Physical considerations then show that you have to integrate $${\bf v}\cdot d\vec\omega={\bf v}\bigl({\bf x}(u,v)\bigr)\cdot\bigl({\bf x}_u(u,v)\times{\bf x}_v(u,v)\bigr)\>{\rm d}(u,v)$$ over $S$. The last equation is sometimes written as $${\bf v}\cdot d\vec\omega={\bf v}\cdot {\bf n}\>{\rm d}\omega={\bf v}\bigl({\bf x}(u,v)\bigr)\cdot{\bf n}(u,v)\>|{\bf x}_u\times{\bf x}_v|\>{\rm d}(u,v)\ ,$$ whereby the latter form contains superfluous squareroots.