Calculating DS - Multivariable calculus help

[Beautiful]

I'm having trouble calculating the dS vector. I know there are multiple ways to find dS but can anyone explain them to me. Or redirect me to a site that can help me with this. I've looked in my book and I've found some info on it but I want additional sources that could maybe explain them a little better.
Thank you so much! I really appreciate it.

 

[Beautiful]

I think one of my biggest issues with finding the dS vector has been that I've done it in so many ways so far and getting confused and in turn making too many mistakes. So I'm basically looking for a overview of all the ways it can be calculated (If that sort of thing exists).

Thank you

 

[Matterwave]

I'm guessing by dS you mean the infinitessimal surface area vector? As it is an infinitessimal vector, it doesn't really have a calculated value since it should go to 0 in the appropriate limit. So I think you will have to be more specific on what you're asking.

 

[Beautiful]

Yeah I think you are referring to something different. It arises when we start dealing with surface integrals. I think it's usually the perpendicular vector to a surface. And there are special cases that allow for shortcuts when working through complex problems.

 

[Beautiful]

The attachment is of a example problem. This would be the type of dS vector I am referring to.

 

[Matterwave]

Are you referring to an integral like this:

$$\iint_S \vec{A}\cdot d\vec{S}$$

I am referring to the $d\vec{S}$ found in that expression. It's an infinitesimal vector, like dx when we are integrating:

$$\int f(x)dx$$

As $d\vec{S}$ is an infinitesimal vector, there's no way to really calculate a "value" for this vector itself, but there are certainly ways to help you evaluate the surface integral itself. So perhaps you are asking about ways to evaluate vector surface integrals in general?

 

[Beautiful]

Hmm apparently so. Would "ways to evaluate vector surface integrals in general" include problems involving The flux, Stokes' thm, divergence thm...etc

 

[Matterwave]

Certainly. In your previous picture, for example, it seems to me that one wants to use the 2-dimensional divergence theorem to convert that surface integral into a line integral, which might be easier:

$$\iint_S (\nabla\cdot\vec{F})dA=\oint_C \vec{F}\cdot\hat{n} dl$$

But I'm confused on why the picture has a vector sign on the dS. In that case the divergence of a vector field is a scalar field, and should just be integrated over a surface, not a surface vector...hmmmm... maybe it's a typo?

 

[Beautiful]

Possibly, I found a random example online. Just to confirm we are referring to the same type of dS vector.

Well here is another example of a question (which I have the solution to). From my understanding there is a special case that allows you to quickly identify the dS as:
dS=R("region") dzdΘ
And R would be equal to 2 in this case

Examples like these give me problems. I have no idea where this special case came from. Is there a list somewhere where I can find all the possible ways the dS vector can be found?

 

[Beautiful]

Thanks for the help Matterwave. I really appreciate it

 

[Matterwave]

I don't think you will find such a list of possible ways to reduce a surface integral into a nicer form. There are just too many potential integrals. Mostly you will have to work off of experience.

There's always nice reductions that occur if the vector field you are integrating over has symmetries with respect to the surface. For example, if the vector field is radially directed, and the surface is a sphere, then the vector field will always be parallel (or anti-parallel) with the dS vector. In this case, one can remove the dot product. Alternatively, if the vector field is always perpendicular with the surface, you know the surface integral is 0. If the surface is a closed surface, then you can probably use the divergence theorem to turn it into a volume integral, or if the surface is not a closed surface, perhaps use the divergence theorem to turn it into a line integral.

All of this basically comes with experience and practicing problems. Even for regular integrals, you just need some experience in how to evaluate them.

Actually the vast majority of integrals will be impossible to do analytically using elementary methods, but the integrals you see from courses or lectures should be able to be evaluated.

As far as your attached thumbnail goes, there's no need for vectors in that problem, so you wouldn't be dealing with a dS vector, you would be dealing with a dA infinitesimal area.

In this case, the volume would be 3*A where A is the area of the circle. Since the density doesn't depend on z, you can basically turn the volume integral into a surface integral over A multiplied by 3.

If you are asking why is dA=rdzdθ, then this is the area element in polar coordinates.

For those, you can certainly look up, for example:

http://en.wikipedia.org/wiki/Cylindrical_coordinates
http://en.wikipedia.org/wiki/Spherical_coordinates