Divergence

The divergence ∇⋅F\nabla\cdot\mathbf{F} of a differentiable vector function F\mathbf{F} is a Scalar field that represents the volume density of the outgoing flux of F\mathbf{F} from an infinitesimal volume around a given point. In simpler terms, it represents how F\mathbf{F} "emanates" from a point. Positive divergence means the function emanates outwards from a source, negative divergence means the function compresses inwards towards a sink.

It is linked to the flux by the divergence theorem.

In Cartesian coordinates, the divergence can be written as

∇⋅F=(∂∂x+∂∂y+∂∂z)⋅(Fx,Fy,Fz)=∂Fx∂x+∂Fy∂y+∂Fz∂z=∑i=13∂Fi∂i\nabla\cdot\mathbf{F}=\left( \frac{ \partial }{ \partial x } +\frac{ \partial }{ \partial y } + \frac{ \partial }{ \partial z } \right)\cdot(F_{x},F_{y},F_{z})=\frac{ \partial F_{x} }{ \partial x } +\frac{ \partial F_{y} }{ \partial y } +\frac{ \partial F_{z} }{ \partial z }=\sum_{i=1}^{3} \frac{ \partial F_{i} }{ \partial i }

where i=x,y,zi=x,y,z denotes the axis in the last expression.

Useful results

Given a generic position vector r=rr^\mathbf{r}=r \hat{\mathbf{r}}, the following useful results hold:

∇⋅r=3,∇⋅(rr)=2r,∇⋅(rr2)=1r2,∇⋅(rr3)=4πδ3(r),∇⋅(rr4)=−1r4\nabla\cdot\mathbf{r}=3,\quad \nabla\cdot\left( \frac{\mathbf{r}}{r} \right)=\frac{2}{r},\quad \nabla\cdot\left( \frac{\mathbf{r}}{r^{2}} \right)=\frac{1}{r^{2}},\quad \nabla\cdot\left( \frac{\mathbf{r}}{r^{3}} \right)=4\pi \delta^{3}(\mathbf{r}),\quad \nabla\cdot\left( \frac{\mathbf{r}}{r^{4}} \right)=- \frac{1}{r^{4}}

where δ3(r)\delta ^{3}(\mathbf{r}) is the three-dimensional Dirac delta. For proof of ∇⋅(r/r3)=4πδ3(r)\nabla\cdot(\mathbf{r}/r^{3})=4\pi \delta^{3}(\mathbf{r}) see Electric field > Divergence.

In curvilinear coordinates

Say the function F\mathbf{F} has the generic form

F=Fuu^+Fvv^+Fww^\mathbf{F}=F_{u}\mathbf{\hat{u}}+F_{v}\mathbf{\hat{v}}+F_{w}\mathbf{\hat{w}}

We want to evaluate the flux

∮SF⋅da\oint_{S} \mathbf{F}\cdot d\mathbf{a}

where SS is the surface bounding an infinitesimal volume created by moving in each direction by an amount (du,dv,dw)(du,dv,dw) respectively. Since (u^,v^,w^)(\hat{\mathbf{u}},\hat{\mathbf{v}},\mathbf{\hat{w}}) is a orthonormal basis, the volume formed is a rectangular solid (in the infinitesimal limit), and the sides have lengths (dlu,dlv,dlw)=(f du,g dv,h dw)(dl_{u},dl_{v},dl_{w})=(f\ du,g\ dv,h\ dw), with ff, gg and hh some functions that depend on the coordinate system of choice (see Curvilinear coordinates functions). Since we are working with a rectangular solid, we can break the surface down into rectangular faces glued onto each other.

The "front" surface i

da=−(gh) dvdwu^d\mathbf{a}=-(gh)\ dvdw\mathbf{\hat{u}}

so that

F⋅da=−(ghFu) dvdw\mathbf{F}\cdot d\mathbf{a}=-(ghF_{u})\ dvdw

The "back" surface is identical, but with the sign flipped and evaluated on u+duu+du instead of dudu. For any differentiable function, we have

F(u+du)−F(u)=dFduduF(u+du)-F(u)=\frac{dF}{du}du

the front and back surface are in total

[∂∂u(ghFu)] dudvdw=1fgh∂∂u(ghFu) dτ\left[ \frac{ \partial }{ \partial u } (ghF_{u}) \right]\ dudvdw=\frac{1}{fgh}\frac{ \partial }{ \partial u } (ghF_{u})\ d\tau

since dτ=dludlvdlwd\tau=dl_{u}dl_{v}dl_{w}. The "left" and "right" surfaces, using the same logic, give

1fgh∂∂v(fhFv) dτ\frac{1}{fgh}\frac{ \partial }{ \partial v } (fhF_{v})\ d\tau

and the "top" and "bottom" give

1fgh∂∂w(fgFw) dτ\frac{1}{fgh}\frac{ \partial }{ \partial w } (fgF_{w})\ d\tau

Thus, the whole integral gives

∮SF⋅da=1fgh[∂∂u(ghFu)+∂∂v(fhFv)+∂∂w(fgFw)] dτ\oint_{S}\mathbf{F}\cdot d\mathbf{a}=\frac{1}{fgh}\left[ \frac{ \partial }{ \partial u } (ghF_{u})+\frac{ \partial }{ \partial v } (fhF_{v})+\frac{ \partial }{ \partial w } (fgF_{w}) \right]\ d\tau

The coefficient of dτd\tau defines the divergence in curvilinear coordinates:

∇⋅F=1fgh[∂∂u(ghFu)+∂∂v(fhFv)+∂∂w(fgFw)]\nabla\cdot\mathbf{F}=\frac{1}{fgh}\left[ \frac{ \partial }{ \partial u } (ghF_{u})+\frac{ \partial }{ \partial v } (fhF_{v})+\frac{ \partial }{ \partial w } (fgF_{w}) \right]

and therefore we get

∮SF⋅da=(∇⋅F) dτ\oint_{S}\mathbf{F}\cdot d\mathbf{a}=(\nabla\cdot\mathbf{F})\ d\tau

which is almost the divergence theorem. The key difference is that this has only been applied onto an infinitesimal volume, not an arbitrary one. But the extension is easy: any (finite) volume can be broken down into infinitesimals, but the issue that needs to be fixed is that simply summing all the contributions may not give the correct results. Of course, the divergence theorem integrates over the external bounding surface, whereas a simple sum integrates over the surfaces of all infinitesimal chunks, including internal ones. Fortunately, each internal surface occurs at the boundary of another adjacent surface (they are all back-to-back), and since dad\mathbf{a} always points outwards, the normal vector is mirrored for adjacent pairs, which means that they cancel out. This occurs everywhere surfaces touch, which is to say everywhere except on the outside where there are no more surfaces. Thus, the only surfaces that don't cancel are though outside, which when summed over produce the total bounding are of a volume. Thus, we can write

∮SF⋅da=∫V(∇⋅F) dτ\oint_{S}\mathbf{F}\cdot d\mathbf{a}=\int_{V}(\nabla\cdot\mathbf{F})\ d\tau

which is the actual divergence theorem.