Laplacian VS gradient of divergence

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  • #1
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i don't really understand the difference :(

2V versus ∇ (∇ . V) ???

can anyone give me a simple example to showcase the application difference?

thanks!
 

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  • #2
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The laplacian acts on a scalar function and returns a scalar function. It is the divergence of the gradient. The gradient of the divergence would act on a vector function and return a vector function.

If you have a scalar function that gives the elevation at different points on a mountain, the gradient tells you which way is the steepest at any point on the mountain. Its magnitude tells you how steep the slope is at that point.

The divergence, on the other hand, tells you if the vectors are converging or diverging at a specific point. In electrostatics, for example, field lines radiate from positive charges. They are diverging, so the divergence is positive. If you had a vector function that gave you the velocity of a fluid at a point, a positive divergence would tell you that there is a source, and a negative divergence would tell you that there is a "sink".

I might have it backwards, but I think this is correct.
 
  • #3
654
2
The laplacian acts on a scalar function and returns a scalar function. It is the divergence of the gradient. The gradient of the divergence would act on a vector function and return a vector function.

If you have a scalar function that gives the elevation at different points on a mountain, the gradient tells you which way is the steepest at any point on the mountain. Its magnitude tells you how steep the slope is at that point.

The divergence, on the other hand, tells you if the vectors are converging or diverging at a specific point. In electrostatics, for example, field lines radiate from positive charges. They are diverging, so the divergence is positive. If you had a vector function that gave you the velocity of a fluid at a point, a positive divergence would tell you that there is a source, and a negative divergence would tell you that there is a "sink".

I might have it backwards, but I think this is correct.
yea, i kinda know what divergeance and gradient mean on their own

but when combined it becomes complicated :(

gradient of a divergence????

divergence of a gradient???

so the difference is that laplacian returns a scalar value

gradient of divergence returns a vector value

is there some sort of visualization i can make use of?
 
  • #4
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Well that's a little trickier. Here's wikipedia's explanation of the laplacian: "The Laplacian Δƒ(p) of a function ƒ at a point p, up to a constant depending on the dimension, is the rate at which the average value of ƒ over spheres centered at p, deviates from ƒ(p) as the radius of the sphere grows."

As for the gradient of the divergence, I guess it points to where the divergence is increasing the most. So taking the example of electrostatics, the divergence is the charge. The gradient of the divergence points to the steepest change in charge in the positive direction.

Not really sure I understand it, but that's the best I've got.
 
  • #5
654
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Well that's a little trickier. Here's wikipedia's explanation of the laplacian: "The Laplacian Δƒ(p) of a function ƒ at a point p, up to a constant depending on the dimension, is the rate at which the average value of ƒ over spheres centered at p, deviates from ƒ(p) as the radius of the sphere grows."

As for the gradient of the divergence, I guess it points to where the divergence is increasing the most. So taking the example of electrostatics, the divergence is the charge. The gradient of the divergence points to the steepest change in charge in the positive direction.

Not really sure I understand it, but that's the best I've got.
ah i sort of understand the 2nd part about the gradient of divergence, you mean it tells you where the change of divergence is the greatest? i see.

but i don't understand something, whats the difference between a divergence of magnitude 10 vs 1?
from this picture, how will the arrows move be like for 10 , and how will it be like for 1?
[PLAIN]http://www.math.umn.edu/~nykamp/m2374/readings/divcurl/divcurl0x.png [Broken]

but the first part about laplacian is :(, i will have to read it again, thanks
 
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  • #6
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The higher the magnitude of the divergence, the more they diverge. . .

Let's say that those arrows are created by the vector function -- since I don't know what function you actually used--: (and now you must forgive me for lack of knowing TeX!)
f = (ax, ay)
The divergence of f is
div f = a + a = 2a

A divergence of magnitude 1 would mean that a =1/2, so the arrows have a smaller magnitude. A divergence of magnitude 10 means that a = 5, so the arrows expand rapidly.

This makes sense with the graphical interpretation: if the divergence is 0, then there should be no source, so in this case it is 0 everywhere. (There are many examples of functions with zero divergence that are non-zero, like any non-compressible fluid flow in a tank with a constant volume of fluid, or a magnetic field where there is no changing electric field nearby.)

If you had a tiny box and measured the net flux in or out of the box at a point, that gives you the divergence. This is actually the http://en.wikipedia.org/wiki/Divergence#Definition_of_divergence" of divergence.

To summarize and get back to your question, a higher divergence in that picture means that the arrows would have a higher magnitude.
 
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  • #7
654
2
The higher the magnitude of the divergence, the more they diverge. . .

Let's say that those arrows are created by the vector function -- since I don't know what function you actually used--: (and now you must forgive me for lack of knowing TeX!)
f = (ax, ay)
The divergence of f is
div f = a + a = 2a

A divergence of magnitude 1 would mean that a =1/2, so the arrows have a smaller magnitude. A divergence of magnitude 10 means that a = 5, so the arrows expand rapidly.

This makes sense with the graphical interpretation: if the divergence is 0, then there should be no source, so in this case it is 0 everywhere. (There are many examples of functions with zero divergence that are non-zero, like any non-compressible fluid flow in a tank with a constant volume of fluid, or a magnetic field where there is no changing electric field nearby.)

If you had a tiny box and measured the net flux in or out of the box at a point, that gives you the divergence. This is actually the http://en.wikipedia.org/wiki/Divergence#Definition_of_divergence" of divergence.

To summarize and get back to your question, a higher divergence in that picture means that the arrows would have a higher magnitude.
oh , so higher divergence just means a longer arrow? the direction of the arrows won't change?

so in the case of a positive charge,

gauss law says del . E = p / e0

so basically its saying the divergence of the electric field of that charge is equal to the volume charge density p divide by e0 ?

so if my p increases, my divergence increase. so the field lines become longeR?? weird.

how should a larger divergence of Efield lines look?
 
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  • #8
654
2
i just looked at my textbook and it showed this picture having +ve divergence

-> --> ---> ---->
-> --> ---> ---->
-> --> ---> ---->
-> --> ---> ---->

but how come? this are straight lines?
does the changing lengths of the arrows means changing magnitude of divergence and hence it is a "source" point?

so basically, even if arrows don't spread out of a point, it can still be +ve divergence like that example above?
 
  • #9
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so basically, even if arrows don't spread out of a point, it can still be +ve divergence like that example above?
_____
->| -->| ---> ---->
->| -->| ---> ---->
->| -->| ---> ---->
->| -->| ---> ---->
====

^^ Pretend that is a box around the arrows.

Lets look at how much is coming in vs going out of the box. Let -> indicate a unit vector. (So --> has magnitude 2, ---> has magnitude 3, etc.) Into the box, there's 4 coming in (they are all perpendicular to the wall, so no need to take a dot product). Likewise, there are 12 going out. Therefore, the integral of the divergence in that box is positive (if it were 12 going in and 4 going out, there'd be a negative divergence)

We can now explain the definition of divergence: it's how much is going out minus how much is coming in through a surface of infinitesimal volume around a point. This is written out mathematically on that wikipedia page. Furthermore, it leads to the divergence theorem and Guass' law, and, more indirectly, Stokes' theorem.

So yes, your comment is correct. Furthermore, you can even have lines "spreading out" and have a negative divergence, if they decrease sharply in magnitude --in this case, the point they come out of usually has an undefined divergence.
 
  • #10
olivermsun
Science Advisor
1,244
118
i just looked at my textbook and it showed this picture having +ve divergence

-> --> ---> ---->
-> --> ---> ---->
-> --> ---> ---->
-> --> ---> ---->

but how come? this are straight lines?
does the changing lengths of the arrows means changing magnitude of divergence and hence it is a "source" point?
Just to add to some of the other explanations: suppose those arrows are the velocity vectors of cars in traffic. It means the cars should be spreading apart as traffic flows to the right. There need not be a "source" of mass (cars) at all, although if you are looking at density of cars then the places where there is a divergence could be interpreted as sinks. Substitute gas molecules or electrons for cars, etc.
 
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  • #11
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ah i see thanks everyone!
 

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