From: "Fabrice P. Laussy" Subject: Re: Physical Meanings of Divergence and Curl.. Date: Fri, 25 Aug 2000 21:37:04 +0200 Newsgroups: sci.math Summary: [missing] Vijaya Chandran wrote: > Hi, > > I was going thro an introductionary text on Electromagnetics when I > came across the concepts of Divergence and Curl. I really couldn't get > the actual physical meanings of these concepts. Can someone give a > physical and intutive explanation of what Divergence and Curl means ? > > Thanks in advance, > Vijaya Chandran. They are easy but not trivial, so it's good you ask. However, you should have asked to sci.physics. They are making sense for the mathematician too, but if a pure mathematician gives you *his* definition, you won't look at electromagnetism the same as before. Actually, many physicists don't know the mathematician's definition. Here's an explanation a physicist could give you: 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. F.P.L. ============================================================================== From: Len Smiley Subject: Re: Physical Meanings of Divergence and Curl.. Date: Sun, 27 Aug 2000 04:36:12 GMT Newsgroups: sci.math In article <39A693C4.4E1C68F8@rsl.ukans.edu>, Vijaya Chandran wrote: > Hi, > > I was going thro an introductionary text on Electromagnetics when I > came across the concepts of Divergence and Curl. I really couldn't get > the actual physical meanings of these concepts. Can someone give a > physical and intutive explanation of what Divergence and Curl means ? > > Thanks in advance, > Vijaya Chandran. > > Hi. Let me add a little colloquy to M. Laussy's post, which has, I'm sure, been useful to you. If you picture the paddle-wheel of Fig. III-18 in "Div, Grad, Curl" by Schey (as mentioned by Prof. Edgar), you have to imagine that it is infinitesimal. All that really means is that further shrinking won't change its sought behavior (Unfortunately, this is data-dependent). Now remember what question you're asking the paddle-wheel: "What is the curl of the given vector field at the concurrence point of the paddles?" Now be careful: the answer is a 3D vector, which we think of as a "physical vector" or "arrow" originating at the point. This answer vector's direction is interpreted (or obtained, depending on your point of view) as the direction vector of the SPINDLE of the paddlewheel which spins it the best (we think of the concurrency point of the paddles in the axis of this spindle as magically fixed, and the spindle able to be redirected with two degrees of freedom [a sphere, of course]). "Best" depends on the choice of handedness. "Worst" is then the opposite direction. The magnitude of the curl vector is proportional to the angular speed of the paddle. You might object - what if there's a tie? Well, there can't be (infinitesimally) if the vector field being measured is differentiable enough for the curl to be useful. This may help to emphasize that the curl is a vector. Len Sent via Deja.com http://www.deja.com/ Before you buy.