divergence theorem problems

[PG1995]

Hi

Could you please help me with these queries? Thank you.

Regards
PG

 

[PG1995]

Hi

I hope someone is not in process of replying. I think I have made some progress and would update the questions soon. Anyway, if you have composed replies to the queries, then there is no problem because I'm still struggling with some of the gaps in my understanding. Thank you.

Regards
PG

 

[steveB]

EDIT: I was in the process, but this is what I wrote. Please add another post after this one to modify your question.

For Q1, remember that D is the electric field displacement vector, and there is no symbol for derivative anywhere in sight. The vector has components and each component is a function of x, y and z. Hence, the notation simply means evaluate the vector component at x=0 or y=0 ... etc.

For Q2, now that you understand Q1, you probably can see that those values are just a result of the substitution mentioned.

For Q3, this will also make sense once the correct substitutions are made. However, you an pretty much get to this step by just thinking in your head. You are integrating over the 6 sides of a cube. The top and bottom surfaces of the cube give zero because there is no z-component to the field (Dz=0). The Dy component does not change on the two sides that are normal to Dy, hence the vectors are equal and point in on one surface and point out on the other surface, which means they cancel out. For the two remaining sides, one is zero because the component is zero on the x=0 surface. This leaves only one surface to be calculated.

 

[PG1995]

Thanks a lot, Steve.

EDIT: I was in the process, but this is what I wrote. Please add another post after this one to modify your question.

No problem at all. As I wrote I was still struggling with some gaps in my understanding so your help is really appreciated.

How do I translate the given problem into a graphical form?

I think MrAl can be quite a help with this because in the past he was able to help me visualize divergence using pictures.

Regards
PG

 

[steveB]

I'm not sure what you mean by graphical form.

You could do 3D surface plots for each of the 6 surfaces of the cube you are calculating over. You are interested in the field that is normal to each of the surfaces, but since this is a rectangular coordinate vector and you are integrating over a cube, it's trivial to identify the component needed for the surface you are concerned about.

Once you can visualize the component functions on those surfaces, then you can visualize the full 3D picture in your head. Three surfaces have zero fields normal to them. Two other surfaces have x^2 and -x^2 field dependence over them, and they cancel. And, the remaining side which is not zero has a 2y dependence over it.

I don't know if this is what you are thinking about. Am I missing your question?

 

[PG1995]

Thank you.

Am I missing your question?

No, I don't think so. My question or whatever I was trying wasn't clear enough. For some reason I'm feeling really hard to phrase my questions today. Will update the question soon.

Regards
PG