Homework Help: Surface integrals

alexvenk · Feb 11, 2015

1. The problem statement, all variables and given/known data
Its more of a general issue of understanding than a specific problem

I have to evaluate a few surface integrals and im not sure about the geometric significance of what im evaluating or even of what to evaluate. Examples.

If n is the unit normal to the surface S evaluate Integral of: r . n dS over a) a unit cube. b) a sphere

Not sure what im evaluating or what the significance of it is. If someone could explain the general steps to take ill try and apply it to the specific problems

2. Relevant equations

3. The attempt at a solution

haruspex · Feb 11, 2015

Let's consider r.n over the cube. In particular, at a face parallel to the YZ plane. Can you think of a geometric interpretation of the dot product r.n?
Regardless of any intepretation, are you able to do the integrals?

alexvenk · Feb 11, 2015

haruspex said: ↑

Let's consider r.n over the cube. In particular, at a face parallel to the YZ plane. Can you think of a geometric interpretation of the dot product r.n?
Regardless of any intepretation, are you able to do the integrals?

If i know what integral to do, i can do integration, its just a matter of knowing what I'm integrating. Is r a position vector? r.n gives us the projection of vector r onto the normal of the surface.

haruspex · Feb 11, 2015

alexvenk said: ↑

If i know what integral to do, i can do integration, its just a matter of knowing what I'm integrating. Is r a position vector? r.n gives us the projection of vector r onto the normal of the surface.

Because you had written a dot between the r and n, and I know n is a vector even though you did not show it as one typographically, I assumed r was also a vector - presumably the position vector. And yes, their dot product will be the component of r in the normal direction. So for a face parallel to the YZ plane, the x component. Note that this is constant for that face.
dS is also a vector usually, though because of the n vector already there I'm not sure whether the question setter has already substituted n dS, dS being a scalar area element, for the vector dS. I'd need to see the typography of the original.
If dS is still a vector then clearly the integral produces a vector.

alexvenk · Feb 11, 2015

yeah ok it makes sense that all of the sides of a cube will have a normal that is just in the directions of one of the axis

Here is the question:
http://imgur.com/jqJTF5h

im struggling with pretty much all the questions there, but we are specifically talking about 4 here.

haruspex · Feb 11, 2015

alexvenk said: ↑

yeah ok it makes sense that all of the sides of a cube will have a normal that is just in the directions of one of the axis

Here is the question:
http://imgur.com/jqJTF5h

im struggling with pretty much all the questions there, but we are specifically talking about 4 here.

That helps. Seeing how questions 4, 5 and 6 are printed, I'm reasonably sure that the author is using ##\vec{dS}## and ##\vec n dS## interchangeably with the same meaning. Can you see how to write the integral in Cartesian form for the x=0 face?

alexvenk · Feb 11, 2015

haruspex said: ↑

That helps. Seeing how questions 4, 5 and 6 are printed, I'm reasonably sure that the author is using ##\vec{dS}## and ##\vec n dS## interchangeably with the same meaning. Can you see how to write the integral in Cartesian form for the x=0 face?

Im not entirely certain but ill give it a shot:

http://imgur.com/umVYFS3

haruspex · Feb 11, 2015

alexvenk said: ↑

Im not entirely certain but ill give it a shot:

http://imgur.com/umVYFS3

Your Cartesian substitution for dS is wrong. This is not a volume integral.

alexvenk · Feb 11, 2015

haruspex said: ↑

Your Cartesian substitution for dS is wrong. This is not a volume integral.

ahh yeah i was wondering whether that would be right or not, itll be dydx then, which evaluates at 0 right? do i then sum all the sides?

haruspex · Feb 11, 2015

alexvenk said: ↑

ahh yeah i was wondering whether that would be right or not, itll be dydx then, which evaluates at 0 right? do i then sum all the sides?

No, not dydx. This is orthogonal to the x axis.

alexvenk · Feb 11, 2015

haruspex said: ↑

No, not dydx. This is orthogonal to the x axis.

oh, i see, so dydz / dzdy then? does order matter?
Thanks a bunch for your help and patience!
edit:typo