What the hell are surface integrals?

[Howers]

I still can't quite see what I'm doing using surface integrals. I'd like an intuitive definition, something like Reinmann sums are to integrals. Believe me, I've seen enough theory with them and I'd rather have a feel for them before I go over the proofs.

Here is what I think:

You enclose a surface around a vector function. This surface is divided into tiny little squares. At the tangent point of each little square, you erect a normal vector. You then dot this normal vector with the vector that the vector function returns at that same point. You take the sum of these and get some scalar that tells you who knows what... the "strength or perpendicularness (bear with) with the given surface"?

Am I even remotely close?

I would request that no one make refrence to flux or line integrals, as my book already does that (miserbly I might add).

Thank you.

 

[ice109]

that's one type of surface integral, a divergence integral. there's also a curl integral which is the sum of the cross product all over the surface. you can also do cute things using gauss' theorem and stoke's theorem ( i don't remember which and how ) to calculate the area of the surface

 

[awvvu]

The surface integral of a vector function dotted with the normal vectors to the surface gives the net "flow" in/out of a surface (or, apologies... flux).

So, imagine a small piece of the surface. The dot product tells you the little flow that the vector field makes in/out of that small piece. When you sum over every piece, you get the total flow in/out. You only need to integrate over the surface because that is the only point where "stuff" enters or leaves.

There's a thing called the Divergence theorem, which states that the surface integral of a vector function is equal to the volume integral of the divergence of that function. Since the divergence intuitively means how much a vector field is "spreading out", summing up all the divergence from every single piece of the volume also tells you how much stuff "flows" in/out of a surface.

I'm basing my explanation off the book Div, Grad, Curl, and all That. It's informal and appeals to intuition a lot. I'm getting a good understanding of vector calculus from it. (Hopefully I didn't screw up this explanation since I'm just learning it myself too)

https://www.amazon.com/dp/0393925161/?tag=pfamazon01-20

 

[lurflurf]

physical: Say you have some stuff and it is moving (or not). Perhaps the stuff is in R^n (or some more complicated space). Say we have some surface S. The surface integral tells us the rate at which stuff is moving across the surface. Say the stuff is fish and the surface is a net the surface integral would be the rate at which we collect (or lose if negative) fish.

Mathematical: Say we work in R^3 as in many elementary calculus books. We do single, double, and triple integrals. We also do line, surface and volume integrals. So all we have done is allow integrals over lower spaces. So a line integral is a single integral allowed to move in three space. a Surface integral is a double integral allowed to move in three space. A volume integral is a triple integral as it has filled the space. You can imagine a double integral where you have bent the plane into some shape. This approach is often used to define or compute the sureface integral. We parametritize space
x(s,t),y(s,t),z(s,t)
S={(x(s,t),y(s,t),z(s,t))|0=<s,t=<1}
Surface Integral[f(v),v in S]
=Surface Integral[f((x,y,z)),(x,y,z) in S]
=double integreal[f((x(s,t),y(s,t),z(s,t))),s&t in [0,1])