Problem on divergence and curl

[dextercioby]

I can define the flux but not with a scalar product.Or not with the unit vector...

So the question is:must the manifold be smooth or not?

Daniel.

 

[pmb_phy]

I can define the flux but not with a scalar product.Or not with the unit vector...

So the question is:must the manifold be smooth or not?

Daniel.

Choose a criteria to which you wish to define what you're looking for. If you can find the flux then is that adequate for the purpose which you are interested in?

Pete

 

[dextercioby]

No,Pete,u missunderstood me...I have hothing against the idea & the definition of a flux of a vector field...It's just that in the case of a nonsmooth manifold (like the cube) it doesn't make any sense,unless u consider the cube without its corners and sides...

Daniel.

 

[pmb_phy]

No,Pete,u missunderstood me...I have hothing against the idea & the definition of a flux of a vector field...It's just that in the case of a nonsmooth manifold (like the cube) it doesn't make any sense,unless u consider the cube without its corners and sides...

Daniel.

No, Daniel, I did understand you. My response was intended to say that when you used the term "must" that one has to have a definition in hand before answering your question. So you must first choose a definition for the divergence and then you can address your question.

Let me give you an example; suppose we choose the definition for divergence above (flux per unit volume) and also demand that the surface normal exist at all points on the surface for the diverence to be defined. Then the answer to your question is yes, it must. However, suppose that we define the divergence as the sum that I gave above (i.e. as expressed in Cartesian form - div E = parial E/parial x + etc... oops! I made a mistake in that definition - I gave the gradient ). Then the answer too your question is no. And I've seen a ton of places/texts which define the divergence in this fashion. E.g. Kaplan's "Advanced Calculus" text.

Pete

 

[dextercioby]

Okay,i didn't look at your formula...

The divergence has indeed a differential involving definition...The part with the flux comes just as an application to the Gauss-Ostrogradski formula and,by considering the scalar product in the surface integral,it restrains the applicability.


Daniel.

 

[reilly]

When it's important to worry about edges, or points with surface integrals, then the best idea is to take the limit of a problem you can solve -- like a sphere -> cube. You can convince yourself that the contributions to flux from lines or points is zero -- unless there are some very singular field characteristics. The area of a line is rather small, so it takes a big field to drive some flux through a line. Not to worry, cubes are just fine.

regards,
Reilly Atkinson

 

[heman]

if I'm looking at a graph of a vector field, how do I recognize positive divergence or negative curl, for example?

 

[heman]

Hi,
I was revising my Electrodynamic notes and i came up with some queries!

Why can't an Electric Field rotate or act like a whirlpool!
I know Mathematically it is Zero but what will be the physical explanation of this??

and one more thing while deriving the differential form of Gauss Law,we shrink the body to differential element ,obviously its volume decreases but what happens to the charge!
Does all the charge concenterate in that small differential element or we cut the body and keep on removing its contents till we reach differential element!