Surface Integrals: Understanding & Examples

[alexvenk]

Homework Statement

Its more of a general issue of understanding than a specific problem

I have to evaluate a few surface integrals and I am not sure about the geometric significance of what I am 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 I am 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

Homework Equations

The Attempt at a Solution

 

[haruspex]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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