Purpose of each of the operators , divergence, gradient and curl?

[Pseudo Statistic]

purpose of each of the "operators", divergence, gradient and curl?

Hi.
Can anybody give me a reasonably simple explanation of what the purpose of each of the "operators", divergence, gradient and curl? (I've been looking but I never found something simple to understand)
I know how to evaluate them mathematically, but I really don't know what they mean or what I accomplish...
Can anybody shed some light on the matter?
Also, how can they be applied to E&M?
Thanks loads.

 

[robphy]

These operators are used to write Maxwell's Equations for electromagnetism, first written as a system of partial differential equations, in a more compact vectorial way by Heaviside.
http://en.wikipedia.org/wiki/Maxwell's_equations .
Underlying the use of these operators is the Stokes Theorem
http://en.wikipedia.org/wiki/Stokes_theorem .

 

[mezarashi]

Stokes Theorem is very important when one wants to transform equations from differential form to their integral form. (as you know Maxwell's equations exist in both forms)

Divergence is a scalar that tells us about the rate at which "stuff" flows out of a given volume. My physical interpretation of it is that it is the law of continuity or better known as Gauss's Law. You can use divergence of a vector field to see how much the field is flowing outwards and use that to determine its contents or density. A divergence of zero would indicate that there is no outward flux and so the net effect seen is that there appears to be nothing inside.

The gradient of a function is a vector function tells us about the rate of change of the function. At any given point the direction given by the gradient tells us about the direction of maximum change. The magnitude of this particular vector then is the magnitude of the change. If you place a ball at any point on an x-y plane with varying height and is defined by h(x,y), then the ball will start falling in the direction described by its gradient.

The curl is probably the most difficult to generalize physically. I feel as if it were created just for magnetic fields. It describes magnetic fields so perfectly , and the "opposite" of the curl, the divergence of any magnetic field is always zero. The best way to think of it would to think of curl as the measure of the rotation-ness of the contents of the field. This is where you get the concept of rotational and irrotational fields.

 

[arildno]

The curl is probably the most difficult to generalize physically. I feel as if it were created just for magnetic fields. It describes magnetic fields so perfectly , and the "opposite" of the curl, the divergence of any magnetic field is always zero. The best way to think of it would to think of curl as the measure of the rotation-ness of the contents of the field. This is where you get the concept of rotational and irrotational fields.

I would say it was "invented" for fluid mechanics.
The curl of the velocity field, the vorticity vector, is twice the local angular velocity vector; i.e, how fast (and, in what plane) an infinitesemal portion of the fluid is rotating about the point in question.

 

[Pseudo Statistic]

Stokes Theorem is very important when one wants to transform equations from differential form to their integral form. (as you know Maxwell's equations exist in both forms)

Divergence is a scalar that tells us about the rate at which "stuff" flows out of a given volume. My physical interpretation of it is that it is the law of continuity or better known as Gauss's Law. You can use divergence of a vector field to see how much the field is flowing outwards and use that to determine its contents or density. A divergence of zero would indicate that there is no outward flux and so the net effect seen is that there appears to be nothing inside.

The gradient of a function is a vector function tells us about the rate of change of the function. At any given point the direction given by the gradient tells us about the direction of maximum change. The magnitude of this particular vector then is the magnitude of the change. If you place a ball at any point on an x-y plane with varying height and is defined by h(x,y), then the ball will start falling in the direction described by its gradient.

The curl is probably the most difficult to generalize physically. I feel as if it were created just for magnetic fields. It describes magnetic fields so perfectly , and the "opposite" of the curl, the divergence of any magnetic field is always zero. The best way to think of it would to think of curl as the measure of the rotation-ness of the contents of the field. This is where you get the concept of rotational and irrotational fields.

Thanks loads man... makes a whole lot more sense to me now.
Now I have one other question which is close, but not quite the topic-- how can one interpret line integrals? (Those integrals with the circle in the middle of the summa sign... I think those are line integrals over a closed path)
And how are they related to grad/div/curl?
Thanks.

 

[rockyshephear]

Does the divergence (which is a scalar) have units. Or it is usual to here a statement like this. "The divergence of the fluid at infintisimal point x,y,z is 6.3? If no units, and I say to you "The divergence of something is 10", what if I'm talking about the divergence of stars in a universe. Is that 10 different than if I'm talking about fluid molecule in a fluid flow? What is useful at all about his information?
Thanks

 

[HallsofIvy]

No "function" has any units other than those assigned for a specific application. The units on div F depend on the units on F itself as well as the units assigned to the x, y, and z variables. Assuming that in an application, x, y, and z are length measurements, then div F will have the units of F itself, divided by a length unit.

 

[rockyshephear]

Ah. So divergence is answering a question with another question. It's not an 'answer' per se. Like the derivative of something is an equation. OK, I'll buy that. Then, obviously because the divergence is not a number like 10, and since it would be something line x^2 + 2y + z^4, am I assuming you fill in the x,y,z coordinates of the infinitesimal sphere where you are curious about the flow of something, to get a final answer that is a number? Are these numbers only good in comparison. ie We have more flow going on at this point then that point because the divergence or flow thru my sphere surface is greater than the flow thru that point. Who cares? I'd like an example of why this is important to anyone? That's what I'm missing. A real world example.

I have yet to see a real world problem solved with divergence as the result that means anything to a real world problem. I'm sure it is ultra difficult to give an example, one that solves a problem, but I'll ask anyway.

 

[Office_Shredder]

Ah. So divergence is answering a question with another question. It's not an 'answer' per se. Like the derivative of something is an equation. OK, I'll buy that. Then, obviously because the divergence is not a number like 10, and since it would be something line x^2 + 2y + z^4, am I assuming you fill in the x,y,z coordinates of the infinitesimal sphere where you are curious about the flow of something, to get a final answer that is a number? Are these numbers only good in comparison. ie We have more flow going on at this point then that point because the divergence or flow thru my sphere surface is greater than the flow thru that point. Who cares? I'd like an example of why this is important to anyone? That's what I'm missing. A real world example.

I have yet to see a real world problem solved with divergence as the result that means anything to a real world problem. I'm sure it is ultra difficult to give an example, one that solves a problem, but I'll ask anyway.

One example: You can use divergence (and a theorem about how it relates to integrals) to derive the water pressure at a given depth

In general, divergence answers the question about how much stuff is leaving a given volume. If you have a function f that describes the quantity of stuff at every point, and a volume V, you can integrate the divergence of f over the surface of the volume to calculate how much stuff is going in or out. This is useful for things like electricity, where you want to calculate the flow of charge, fluid dynamics, where you want to calculate the flow of the fluid, and other applications

 

[rockyshephear]

How about a real world example where the problem is stated, we do the math using divergence and come up with a real world answer...like "The water pressure is X at 100 foot depth."
Thanks

 

[niloy]

can anyone give me some idea about velocity potential and perturbation velocity potential?

 

[Phyisab****]

I believe I learned this from Griffith's EM. They are actually pretty much what they sound like. Imagine throwing a fist full of ping pong balls into a vector field, which you will be making these operations on. Divergence is related to how quickly the balls will spread out from each other. Curl is exactly what it sounds like. I will quote griffiths here. "Float a small paddlewheel; if it starts to rotate, then you placed it at a point of nonzero curl." Gradient is the simplest of all. Imagine you are on a mountain. The gradient points exactly up hill.

 

[niloy]

thank you mr.Phyisab**** .this is the best explation i have heard about these things