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    Physical meaning of Curl and Div operators

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

    •   Alt16th April 2006, 11:56



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    Re: microwave

    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."



    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

    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.

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