Understanding Divergence and Curl on a 3D Surface

[Phymath]

Can anyone please explain what Diverange and Curl actually physically represent on a 3d surface, i know what the operators are, but what do they actually mean?
Thanks all

 

[speeding electron]

Do you mean you would like an idea of what they are from an intuitive physical point of view? Well...

The divergence is basically the surface integral of a vector function out of an infinitesimally small box, or other small closed shape. We take the limit of this integral divided by the shape's volume, as the volume tends to zero. It can be looked at as how much the vectors of the function in a small region are pointing out from a point, that is how much they diverge, meaning go in different directions.
For example if at a point the arrows used to represent the function are all pointing in the same direction, they are not diverging, and the divergence is zero. Looking at it from the point of view of the flux out of a small surface, the flux into the surface is canceled out by the flux out of it on the other side.
However, if say the arrows are changing from going forward to going back on themselves, they are diverging, since they are at starting to go in different directions. In terms of flux, either the flux on one side of the shape is not as much as the opposing flux on the other ( i.e. the arrows are not as long on one side as on the other), and the fluxes don't cancel, or the arrows are in different directions on opposite sides of the shape, and all the flux is either in or out of the shape. Then there is a non-zero divergence.

Now the curl...this has the basic idea of how much a vector function, or the arrows representing it twist round on each other in a small region of space. It is not to do with surfaces so much as lines and line integrals. It is the limit of the ratio of the line integral of the vector function around a small closed curve to the smallest area that can cover the curve.
Again, if all the arrows are pointing in the same direction, the integral along one side of the loop/curve will cancel out the integral from the other side. But if say the arrows loop back round on themselves towards the other side of the curve, there will be a net integral, and so a non-zero curl.

 

[Phymath]

u blew my mind up man...there's some peices of it there...and there...jk'n so basically div is how much the vector function "diverges" in a small area, and curl is how much the vector function "curls" back on it self

 

[speeding electron]

Yep...pretty self-explanatory when you look back on it...I'm not sure whether to take the fact that I 'blew your mind up' as a compliment...anyway you got the idea: div is going out in different directions, curl is twisting back round, curling, rotating, whatever...

 

[mathwonk]

another way to look at it is via the basic theorems using these terms, i.e. green's theorem, gauss's theorem, and the divergence theorem.

e.g. if you look at greens thm i believe it says that the integral of Adx + Bdy around a closed path, equals the integral of the curl of (A,B) over the inside of the path.

But look at the expression Adx + Bdy, integrated in terms of a parametrization x(t),y(t) of the path. It becomes [A dx/dt + B dy/dt] dt which is the dot product of the vector field (A,B) with the velocity vector (dx/dt, dy/dt), i.e. the tangent vector to the path.

Now this dot product measures how much the vector field is tangent to the path. So this quantity is largest when the vector field remains tangent to the path all the way around the path, i.e. when it rotates around the inside of the path, as the path does.

Since greens thm says this same quantity is obtained by integrating "curl (A,B)" over the interior of the path, then "curl (A,B)" must be measuring also the same thing, i.e. how much the vector field curls around inside the path.

I guess I do not understand this perfectly myself, but I think of it like that.


In the same way, the divergence theorem says that when you integrate the dot product of the vector field (A,B,C) against the outward normal vector to the surface, integrated over the surface, you get the same answer as when you integrate the quantity "divergence of (A,B,C)" over the interior of the surface.

Since the first integral measures how much the vector field points out of the surface, and averages that over the surface, it computes how much of the field is flowing out of the surface. since the thm says the integral of the "divergence" measures the same thing, hence the divergence must measure those sources inside the surface from which the material forming the field is flowing.

Speeding electron has a better intuitive grip on it than I do, but this is sort of the opposite of his explanation. I.e. he went from the large to the small, i.e. he took a derivative to define the curl, or the divergence, and I am saying what happens when you integrate those quantities back again.

Does this help? I ahd an old book called something like partial differential equations of physics that made this stuff really clear.