Wikipedia:Reference desk/Archives/Mathematics/2022 March 31

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March 31

Equivalence of definitions of divergence

The wikipedia article on divergence lists two definitions of divergence:

$\left.\operatorname {div} \mathbf {F} \right|_{\mathbf {x_{0}} }=\lim _{V\to 0}{\frac {1}{|V|}}$ $\scriptstyle S(V)$ $\mathbf {F} \cdot \mathbf {\hat {n}} \,dS$ which is the coordinate free definition,

and the coordinates based definitions (such as),

$\operatorname {div} \mathbf {F} =\nabla \cdot \mathbf {F} =\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)\cdot (F_{x},F_{y},F_{z})={\frac {\partial F_{x}}{\partial x}}+{\frac {\partial F_{y}}{\partial y}}+{\frac {\partial F_{z}}{\partial z}}.$

How do we prove that both of them imply each other? I really require a proper reference, but a sketch will also do. - Abdul Muhsy talk 14:47, 31 March 2022 (UTC)

See Divergence theorem.  --Lambiam 19:27, 31 March 2022 (UTC)

Thanks. I am aware of it but am not sure how it answers my question-- Abdul Muhsy talk 08:26, 1 April 2022 (UTC)

The link connecting the two is the identity

$f(\mathbf {x_{0}} )=\lim _{V\to \emptyset }{\frac {1}{|V|}}\iiint _{V}f\,\mathrm {d} V$

(in which the shrinking closed region $V$ keeps enclosing $\mathbf {x_{0}}$) applied to $f=\mathbf {\nabla } \cdot \mathbf {F} =\operatorname {div} \mathbf {F} .$  --Lambiam 09:04, 1 April 2022 (UTC)

@Lambiam: This should follow from the continuity of f, provided that the region V has diameter →0. It doesn't seem to be true otherwise though. Consider the region V(ϵ) bounded by x=−ϵ, and x=1+ϵ, y=−ϵ, y=ϵ, z=−ϵ, z=ϵ, and let f(x, y, z) = x. Then

${\frac {1}{|V(\epsilon )|}}\iiint _{V(\epsilon )}f\,\mathrm {d} V$

is the average value of f on V(ϵ), 1/2. The volume of V(ϵ), 4ϵ²(1+2ϵ)→0 as ϵ→0. But V(ϵ) encloses (0,0,0) and f=0 at this point. The article just says "as V shrinks to zero", which seems rather vague and does not state that the diameter of V is meant to have a limit of 0. --RDBury (talk) 06:13, 2 April 2022 (UTC)

I agree the article is too loose about the shrinking. I considered writing $\lim _{V\to \{\mathbf {x_{0}} \}}$, implying $\bigcap _{\epsilon }V_{\epsilon }=\{\mathbf {x_{0}} \}$. Do you think this suffices to resolve the issue in the article?  --Lambiam 09:39, 2 April 2022 (UTC)

Looking more closely at the formulation at Divergence § Definition, it is a bit messed-up. First "a closed volume V enclosing x₀" is introduced, and next we are supposed to consider "the volume of V, as V shrinks to zero". So is this the volume of a volume? (The problem is present in its core already in Divergence theorem, which uses the term "volume" for a subset of space, instead of for a scalar quantity.) The text should also require that the surface of these closed (in fact compact) subsets is piecewise smooth – how else can we have normals?  --Lambiam 14:00, 2 April 2022 (UTC)

I'm not sure that adding that the intersections of the "regions" is a single point would be enough. In any case it would be much easier to prove that limit converges to the value of the function if the diameters, and therefore the maximum distance from the point, converges to 0. I'd say simply use a ball of radius ϵ except that I think part of the the point of coordinate free formula is you can easily derive the definition in various coordinate systems. For example for the rectangular coordinate system it's easiest to assume that the region is a cube with sides 2ϵ. You're right that the boundaries should be piecewise smooth for the definition of divergence. It shouldn't matter for the limit above being equal to the value of the function though. But assuming a smooth boundary won't be restrictive in practice; the last I heard no one was using coordinate systems with fractal-like coordinate surfaces.

Another potential issue is the limit above seems to depend critically on the assumption that f is continuous, which would seem to require that F be continuously differentiable. But the non-coordinate free definitions only require that F be differentiable. There are certainly examples of functions which are differentiable but not continuously differentiable, which leaves open the possibility that the divergence of a sufficiently pathological F could depend on the coordinate system chosen. I suppose for typical engineering applications no one cares about pathological flows, but if you start talking about turbulent flows then it might be a different story. RDBury (talk) 17:53, 2 April 2022 (UTC)

Application of the divergence theorem already requires $\mathbf {F}$ to be continuously differentiable, and the section Divergence § Definition in coordinates makes the same requirement, although its need there is unclear (to me). Is something lost if we add this requirement also to {{Section link}}: required section parameter(s) missing? As to the incredible shrinking "volume" $V$, parametrizing it by its scalar volume $v=\mu (V),$ my understanding of "shrinking" is that $v<v'$ implies $V(v)\subset V(v')$; it shrinks "in place" (not like a toy balloon that is released and flies off). In combination with the limit intersection being a point, I think this implies the diameter goes (weakly monotonically) to zero as well.  --Lambiam 20:25, 2 April 2022 (UTC)