Physical meaning of Curl and Div operators

hello,
i want a good and plain physical meaning of Curl and Div operators.

Re: microwave

Divergence:
Physicallyt he divergenceo fa vectorquantityr epresenttsh e rate of change
of the field strength in the direction of the field. If the divergence of a vector field is positive at a point 'P' then something is diverging from a small volume surrounding that point and that point is acting as a source. If it is negative, then something is converging into a small volume surounding that point and that point is acting as a sink. If the divergence is zero at a point 'P', then the rate at which something entering a small volume surrounding that point is equal to the rate at which it is leaving that volume.

The physical significance of the divergence of a vector field is the rate at which "density" exits a given region of space. The definition of the divergence therefore follows naturally by noting that, in the absence of the creation or destruction of matter, the density within a region of space can change only by having it flow into or out of the region. By measuring the net flux of content passing through a surface surrounding the region of space, it is therefore immediately possible to say how the density of the interior has changed. This property is fundamental in physics, where it goes by the name "principle of continuity."

OR

CURL

Physically the curl of a vector field represents the rate of changeo f field
strength in a direction at right angles to the field and is a measure of rotation
of some thing in a small volume surrounding a particular point.

*******

Divergence & Curl as Explained by Fabrice P. Laussy
They apply to vector-valued fields.


The divergence of such a field is, at any point, a scalar, such that if
you multiply this scalar by the volume of a little neighborhood of the
point, the result is how much of the field is flowing out of the volume
(so this is, out of its surface really, because what is inside cannot get
out other than by the surface, and nothing can get in at a distance
neither: it all has to pass through the surface).


The divergence (and the curl too) are limiting process. So when we say
"little volume", this work best and best for volumes more and more
little. It's actually true for volumes of 0 sizes. But this should not
worry you. It's the same with derivatives in R. Here we are talking of
derivatives too, differential forms really, so it's no surprise the limit
crops up, the same as in R.


The divergence is thus:


(div E)(x) = lim (Flux of E through S) / (Volume of S)


I suppose you know what the flux is. It is the surface integral of the
normal component of a vector. The limit is when the Surface S tends to 0
area, while its volume inside tends to 0 volume. What (more and more)
little area you take is not important. It doesn't depend on it. There we
are talking about the "conceptual" meaning of it, so it's good it doesn't
depend on the shape involved. Now if you should compute the divergence
*this* way, you are probably wanting to use an easy surface: a plane, a
sphere, and so on...


You notice I wrote (div E)(x). The divergence is a *local* concept
defined for each point in space. So you give it a vector-valued field
(E), and it gives you back another scalar-valued field, its divergence,
or div E. The definition is for a point, though, the one you are taking
the surface around, to see how much at this place is flowing (in or out,
depending of the sign). I wrote x but this is really (r, theta, phi) or
(X, Y, Z) in R^3.


The divergence is thus the amount of "divergence" of a field. This is
why, in electromagnetism, we have div B = 0. That means, anywhere you
are, everything which come from one direction towards a point must leave
in another direction. There is no, otherwise stated, magnetic charges:
things which could give birth to a magnetic flux. Another law is the
divergence of E, the electrif field, is 4pi rho, with rho the density of
charges (a limiting process too: the amount of charge in a tiny volume V
over the volume of V when all this vanish to 0). So in empty space, E
like B, cannot diverge, it just pass. But when there's charges, it's
flowing out: electric charges create an electric field. That is the
qualitative meaning of the law (Gauss Law). The exact or quantitative
meaning is specified by 4pi, sign of quantity, etc... That tells you for
instance the field is newtonian etc, etc... It's just to be worked out.


Now for the curl.


It demands vector-valued fields too but returns vectors, not
scalars. Here's its definition.


The curl of E at a point x is the direction (or vector) k such that when
you take a little contour around x which normal is k, and you multiply
the curl at x by the area inside the contour, you have the circulation of
E around this contour. The same applies, it's true only at the limit. So


k cdot (curl E)(x) = lim (Circulation of E around C) / Surface of C


cdot means scalar product. C is the contour, also, its actual form
doesn't bother. Could be a circle or a square or anything weirdo. Its
normal is k. So you see the curl measures how much is the field going in
one direction instead of another. If it's zero, that means that for *all*
contours, E is not essentially pointing globally in one direction rather
than an other. It's changing from place to place, but on the overall, it
balances out. If it's not zero, means there is a net impulse in one
direction.


People like to say the curl measures how much he field circulates. Well,
sure, but that's a tricky definition. First you have to remember this is
local. It's not because a field is "obviously" circulating as a whole
(like the one in a whirlpool) that it actually has a non-nul curl. You
can cook-up examples (not that cooked up actually, they are really used
in fluid mechanics) that are actually turning around but which curl is
0. Other examples do exist of curls of fields which vectors are all in
the same direction, while not having a 0 curl. You can picture out the
meaning of curl with fluid dynamics. When you put a paddle in a field of
velocities of a fluid (that is, in the fluid), with its axis stuck at the
point you are willing to know the value of the curl at, if the curl is
non-0, the paddles will gain angular velocity. This is because the fluid
pushes harder on one side than on the other. It may well push in the same
direction, but if it pushes harder with one, it will make it move. So
that's the meaning of curl. Curl of V is in fluid mechanics omega/2 or
something, with omega the angular speed of the paddle. In
electromagnetism, at first, you have the Curl of E to be 0 (this is in
stationnary case). That means you can move your charge up and down, back
and forth in E, the energy you'll have to give it to move it in higher
potential regions, it will be given back to you when you go the other
way, same path or not, doesn't matter (remember, our definitions are
independant of what surface, or what contour you take). So when all
fields are at rest, you can move "conservatively" around: what you give
one way, you recover exactly the other way. This is because of 0 curl.
Then things start to move, and the Curl of E is minus the time derivative
of B (over c, the speed of light---it's one of the Maxwell equation). So
that means you cannot move your charge without loss as before. Either you
will have to give energy so as to complete your tour, or you may also
acquire some, depending on signs. Means E can make work when B is
moving. It is not "conservative".


That's the way we do use differential forms in physics, at least at
first. You will need (or may want) to study tougher theory which make
better sense of these operators. They are many. Differential forms and
distributions theories are two instances. Divergence of a field in
distribution theory is the divergence as we just saw plus a delta
function attached to each surface of discontinuity. Distributions
simplify considerably all these matters. There is a whole lot of algebra
too you must learn, like Div(a B) = a Div (B) + grad (a) cdot B..., that
is associated with differential operators. This part is rather
simple. Doesn't mean it's all there is to it however. The theory is
rather deep.


Oh, also, from the definition we just saw, you can derive special
definitions when you apply these operators in, say, a space with
cartesian coordinates. The div is then the sum of partial
derivatives. That's fine for practical purpose. If you're peculiarly good
at making sense of things, you can even draw back the "physical"
definition from this definition in term of components. But don't let you
deceived and believe Div or Curl are just that: formulaes. They're not.
They are marvelous tools which combine in several fashion, with great
theorems like Ostrogradsky or Stokes for instance (the integral of our
definition). The best advice I can give you is to take some genuine
2D-fields ((x, y) is nice for Div) and (0, 1-y^2) is nice for Curl), and
work it out, and see if you can get the feeling that the value of the
divergence is indeed the stuff you see it says: it diverges there by
so-much, here there is no circulation, here there is, etc, etc...


Hope this helps.