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I'm studying vector analysis and it is hard for me to understand what divergence of a vector field really is. I know that $divF=\nabla\cdot F$ but I don't understand what kind of quantity it gives and what it is used for. Could anyone give me a good example of what it is used for (in real physics)?

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It is just $$\partial F_x/\partial x + \partial F_y/\partial y + \partial F_z/\partial z $$ and measures whether the field is a source or sink at a given place. A basic introduction is here:

http://en.wikipedia.org/wiki/Divergence

and the most important relationship that gives the divergence an "intuitively comprehensible" meaning is Gauss' theorem

http://en.wikipedia.org/wiki/Gauss_theorem

The flux of the field $F$ over the surface $S$ of a small volume - the integral $\oint F\cdot dS$ where $dS$ is a vector normal to the infinitesimal area $dS$, with the same magnitude - may be written as ${\rm div} F$ times $dV$. The latter is the volume of the interior bounded by the area $S$. If the volume is non-infinitesimal, one has to replace ${\rm div}(F)dV$ by the integral of this.

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Well I knew about ∂Fx/∂x+∂Fy/∂y+∂Fz/∂z, I just needed some explanation about what it means. Thanks for the links anyway! –  snickers Jun 13 '11 at 11:21
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@snickers, the Gauss theorem is the key to the intuition about it; it measures the amount of vector flux going into an infinitesimal sphere around the point. With into it means; the normal component of the field to the surface –  lurscher Jun 13 '11 at 22:43
    
@lurscher, I think I understand now, thanks ! So is it right to say that a magnetic field has no divergence because the same amount of vector flux of this field goes both in and out any volume (because there is no magnetic monopole)? –  snickers Jun 14 '11 at 20:40
    
yeah thats pretty much the idea –  lurscher Jun 14 '11 at 20:42
    
A later comment suggests "div grad curl and all that" by h.m.schey (copied fromthe cover which is all lower case) I found that a bit hard going, and prefer "A Student's Guide to Vectors and Tensors" by Daniel Fleutsch, which has nice explanatory diagrams and text on page 47. As an aside, to see how much he cares for his readers, see theguardian.com/books/2009/feb/11/… –  Harry Weston Nov 28 '13 at 12:25
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Divergence of a vector field at any point gives the gain of flux through a infinitesimal volume. A vector field specifies a vector to every point in the coordinate space. If you take a infinitesimal volume at any such point, the sum of the dot product of the vector field and area vector ( area with its direction normal to the surface) through all the faces of the infinitesimal volume is the divergence of the vector field. The sum is in fact the total gain of flux. The flux going out will be positive and coming in to the volume will be positive.

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I like the following:

Consider, for example, centrally symmetric field in the space, defined by the formula

$$\overrightarrow A=f(r)\overrightarrow r$$ Now, the flux through a sphere of radius r centered at the origin is

$$q(r)=4\pi r^2f(r)$$ Thus the number of vector lines originating in a thin layer between two such spheres is

$$\rm dq(r)=4\pi \rm d[r^2f(r)]=4\pi[2rf(r)+r^2f'(r)]\rm dr$$ Hence, this layer has per unit volume

$$ \rm div \overrightarrow A =\frac{\rm d q}{4\pi r^2}=\frac{2f(r)}{r}+f'(r)$$ vector lines.

In particular, the centrally symmetric field with no sources outside the origin is characterized by the fact that

$$ \rm div \overrightarrow A = \frac{2f(r)}{r}+f'(r)=0 $$ Solution of this differential equation:

$$f(r)=\frac{\rm const}{r^2}$$ We came to Newton's law.

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To get a realistic feel of the divergence of a vector field. Consider water flowing in a pipe. We can associate every point inside the pipe with a vector field that gives the magnitude and direction of the flowing water. Taking the divergence of the the velocity vector at any point would mean to calculate how much of water accumulates (outgoing - incoming) into the infinitesimal volume around that point. If the flow is steady with constant velocity then the water flowing into and out of the volume would be equal and thus divergence zero which mathematics says is correct because the divergence of constant vector is zero. Had there been a source of water which constantly produced water then all water would flow out of the volume and thus would the divergence be positive and negative for a sink inside the volume.

Hope this helped..

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I learnt to understand it as a measure of ' the rate of change of magnitude of a vector field.' So that if a field 'has' divergence, $Div F$ is non zero, then the magnitude of vectors within some region will be changing.

A physical example I encountered is a can of compressed air that has been recently opened (alright this is a little artificail but it serves a purpose!). Just after opening the air near the back of the can can't escape as there is air in the way and so the velocity is low in that region. Near the opening the air can readily escpe and so the velocity in that region is fast. Thus measuring the divergence from the back to the front gives a non zero value and hence we have 'divergence'.

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Thanks for the example –  snickers Jun 13 '11 at 11:24
    
Hi @snickers, It may be problematic that you've ticked this Answer, depending on what you read into it. Pay attention to Luboš's mention of Gauss' theorem. –  Peter Morgan Jun 13 '11 at 12:29
    
for the record I agree with the comments about checking the mathematical detail! A good physicist has a grasp of both the physical and mathematical descriptions of reality! –  Nic Jun 13 '11 at 13:27
    
@snickers, can i suggest a book called 'Div, grad, Curl and all that.' –  Nic Jun 13 '11 at 13:29
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This answer is just not true. The vector field ${\bf F}=x^2\hat{\bf y}$ has zero divergence, but the magnitudes of vectors are changing. Conversely, the vector field ${\bf F}=(\cos x)\hat{\bf x}+(\sin x)\hat{\bf y}$ has nonzero divergence almost everywhere but constant magnitude everywhere. –  Ted Bunn Jun 13 '11 at 15:56
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The most obvious example from physics is the Maxwell equation $$ \nabla\cdot \mathbf{E} = 4\pi\,\rho $$ which simply states that the electric field $\mathbf{E}$ "comes out of" any charged particle (where there is a finite charge density $\rho$), and does not have any source at places where $\rho$ is zero.

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