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: and the most important relationship that gives the divergence an "intuitively comprehensible" meaning is 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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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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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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