What the hell are surface integrals?

• Howers
In summary, a surface integral is a type of integral that allows us to calculate the net "flow" in/out of a surface using a vector function and its normal vectors. This is similar to how Riemann sums are used to calculate integrals. There are different types of surface integrals, such as the divergence integral and curl integral, which can also be calculated using the Divergence theorem. Overall, surface integrals allow us to extend integrals to lower dimensions, such as lines and surfaces, by parametrizing the space and integrating over it.
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.

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

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

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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])

1. What is the purpose of surface integrals?

Surface integrals are used to calculate the amount of a vector field passing through a given surface. This can help in understanding the flow or flux of a physical quantity, such as fluid or electricity, through a surface.

2. How is a surface integral different from a regular integral?

A regular integral is used to find the area under a curve in a two-dimensional plane. A surface integral, on the other hand, deals with calculating the flux through a three-dimensional surface. It involves integrating a vector field over a surface rather than a function over an interval.

3. What are the types of surface integrals?

There are two types of surface integrals: the surface integral of the first kind and the surface integral of the second kind. The surface integral of the first kind is used to find the flux of a vector field through a surface, while the surface integral of the second kind is used to find the surface area of a given surface.

4. How is a surface integral calculated?

A surface integral is calculated by first parameterizing the surface using two variables, typically denoted by u and v. Then, the surface is divided into small rectangles and the flux through each rectangle is calculated using the dot product of the vector field and the unit normal vector at that point. Finally, all the flux values are added together using a double integral.

5. What are the applications of surface integrals in real life?

Surface integrals have numerous applications in various fields such as physics, engineering, and mathematics. They are used to calculate the flow of fluids through pipes, the flux of electric fields through surfaces, and the surface area of a three-dimensional object, among others. They are also used in computer graphics to render three-dimensional objects.

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