# General question about surface integrals

• kevinf
In summary, the conversation revolves around evaluating a surface integral with a given vector and surface. The group discusses finding dS and using cylindrical coordinates for the integral. They also clarify that the surface is a cylinder and the normal direction for dS is always horizontal.
kevinf
hi in my engineering mathematics class, we are going over surface integrals again. i have some general question about this subject. sorry for not using the template.

say that i have a problem that goes like this.

"evaluate $$\int(v*dS)$$ (where the * means dot) where v= (3y,2x$$^{2}$$,z$$^{3}$$) and S is the surface of the cylinder x$$^{2}$$ + y$$^{2}$$=1, 0<z<1. "

to find dS, is it always the unit vector of z=6-2x-2y (gradient of z divide by magnitude of z) multiply by dx and dy? i am not sure if i am right on the dx and dy though.

also some of the problems' solutions use cylindrical coordinates, which i understand, but some of them only use r d$$\varphi$$ dz. if i remember correclty from my calculus classes, i thought the cylindrical coordinates also included dr?

Last edited:
hi kevinf!
kevinf said:
also some of the problems' solutions use cylindrical coordinates, which i understand, but some of them only use r d$$\varphi$$ dz. if i remember correclty from my calculus classes, i thought the cylindrical coordinates also included dr?

Yes, but the integral is a surface integral, so there's only two ds …

r is constant over the surface.
"evaluate $$\int(v*dS)$$ (where the * means dot) where v= (3y,2x$$^{2}$$,z$$^{3}$$) and S is the surface of the cylinder x$$^{2}$$ + y$$^{3}$$, 0<z<1. "

sorry, i don't understand what the cylinder is

sorry it was a typo. i corrected it in the original post.

ahh!

in that case, i don't understand …
kevinf said:
to find dS, is it always the unit vector of z=6-2x-2y (gradient of z divide by magnitude of z) multiply by dx and dy? i am not sure if i am right on the dx and dy though.

… dS is in the normal direction, which will always be horizontal (no z)

(and wouldn't it be easier to use cylindrical coordinates?

## 1. What is a surface integral?

A surface integral is a type of integral that is used to calculate the area of a surface. It involves summing up infinitesimal areas on a surface and is typically denoted by ∫∫S f(x,y) dS. It is an important concept in mathematics and physics, particularly in the study of vector fields and flux.

## 2. How is a surface integral different from a line integral?

While both surface and line integrals involve summing up infinitesimal elements, the main difference lies in the type of elements being summed. In a surface integral, infinitesimal areas on a surface are summed up, whereas in a line integral, infinitesimal lengths on a curve are summed up. Additionally, surface integrals involve double integrals while line integrals involve single integrals.

## 3. What are some real-world applications of surface integrals?

Surface integrals have a wide range of applications in various fields. In physics, they are used to calculate the flux of a vector field through a surface and to calculate the surface area of a curved object. In engineering, they are used to calculate the moment of inertia of a three-dimensional object. They are also commonly used in fluid dynamics to calculate the flow rate of a fluid through a surface.

## 4. How do you set up a surface integral?

To set up a surface integral, you first need to define the bounds of the double integral, which will depend on the shape and orientation of the surface. Next, you need to determine the integrand, which is the function being integrated over the surface. This can be a scalar function or a vector function. Finally, you need to choose an appropriate coordinate system and set up the integral using the appropriate formula.

## 5. Are there any special cases of surface integrals?

Yes, there are a few special cases of surface integrals that are commonly used. One is the surface integral of a constant function, which results in the area of the surface. Another is the surface integral of a vector field, which is used to calculate the flux through a surface. There is also the surface integral of a surface density, which is used to calculate the mass or charge of a three-dimensional object.

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