Question Number 157199 by Fresnel last updated on 20/Oct/21
$${What}\:{is}\:{the}\:{general}\:{expression}\:{of}\:{the}\:{divergence}\left({div}\overset{\dashrightarrow} {{V}}\right) \\ $$
Answered by TheHoneyCat last updated on 21/Oct/21
![It depends on your level: 1) If (to you) V^→ is a tridimentionnal vector V^→ =V_x e_x ^→ +V_y e_y ^→ +V_z e_z ^→ then: div V^→ =(∂V_x /∂x)+(∂V_y /∂y)+(∂V_z /∂z) 2) V^→ is a function from R^n to R^n with: denoting V_i its coordinate−functions (ie the function V_i from R^n to R such that ∀i∈[∣1,n∣] V^→ .e_i ^→ =V_i ) Then: div V^→ =Σ_(i=1) ^n (∂V_i /∂x_i ) 3) If V^→ is a differentiable map of R^n →R^n (or any set isomorphic to it) its divergence is defined as above it might not look like there is a difference from 2) but the fact that it is differentable assures that the divergence does not depend on the choice of coordinates, hence it is a general operator on the set of differentible functions, and no longer a formula that refers to “actual” coordinates](https://www.tinkutara.com/question/Q157279.png)
$$\mathrm{It}\:\mathrm{depends}\:\mathrm{on}\:\mathrm{your}\:\mathrm{level}: \\ $$$$\left.\mathrm{1}\right)\:\mathrm{If}\:\left({to}\:{you}\right)\:\:\overset{\rightarrow} {\mathrm{V}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{tridimentionnal}\:\mathrm{vector} \\ $$$$\overset{\rightarrow} {\mathrm{V}}={V}_{{x}} \overset{\rightarrow} {{e}}_{{x}} +{V}_{{y}} \overset{\rightarrow} {{e}}_{{y}} +{V}_{{z}} \overset{\rightarrow} {{e}}_{{z}} \\ $$$$\mathrm{then}:\:\mathrm{div}\:\overset{\rightarrow} {{V}}=\frac{\partial{V}_{{x}} }{\partial{x}}+\frac{\partial{V}_{{y}} }{\partial{y}}+\frac{\partial{V}_{{z}} }{\partial{z}} \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\:\overset{\rightarrow} {{V}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{function}\:\mathrm{from}\:\mathbb{R}^{{n}} \:\mathrm{to}\:\mathbb{R}^{{n}} \\ $$$$\mathrm{with}:\:\mathrm{denoting}\:{V}_{{i}} \:\mathrm{its}\:\mathrm{coordinate}−\mathrm{functions} \\ $$$$\left({ie}\:{the}\:{function}\:{V}_{{i}} \:{from}\:\mathbb{R}^{{n}} \:{to}\:\mathbb{R}\:{such}\:{that}\right. \\ $$$$\left.\forall{i}\in\left[\mid\mathrm{1},{n}\mid\right]\:\overset{\rightarrow} {{V}}.\overset{\rightarrow} {{e}}_{{i}} ={V}_{{i}} \:\right) \\ $$$$\mathrm{Then}:\:\mathrm{div}\:\overset{\rightarrow} {{V}}=\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\partial{V}_{{i}} }{\partial{x}_{{i}} } \\ $$$$ \\ $$$$\left.\mathrm{3}\right)\:\mathrm{If}\:\overset{\rightarrow} {{V}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{differentiable}\:\mathrm{map}\:\mathrm{of}\:\mathbb{R}^{{n}} \rightarrow\mathbb{R}^{{n}} \\ $$$$\left(\mathrm{or}\:\mathrm{any}\:\mathrm{set}\:\mathrm{isomorphic}\:\mathrm{to}\:\mathrm{it}\right) \\ $$$$\mathrm{its}\:\mathrm{divergence}\:\mathrm{is}\:\mathrm{defined}\:\mathrm{as}\:\mathrm{above} \\ $$$${it}\:{might}\:{not}\:{look}\:{like}\:{there}\:{is}\:{a}\:{difference} \\ $$$$\left.{from}\:\mathrm{2}\right)\:{but}\:{the}\:{fact}\:{that}\:{it}\:{is}\:{differentable} \\ $$$${assures}\:{that}\:{the}\:{divergence}\:{does}\:{not}\:{depend} \\ $$$${on}\:{the}\:{choice}\:{of}\:{coordinates},\:{hence}\:{it}\:{is}\:{a}\: \\ $$$${general}\:{operator}\:{on}\:{the}\:{set}\:{of}\:{differentible} \\ $$$${functions},\:{and}\:{no}\:{longer}\:{a}\:{formula}\:{that}\:{refers} \\ $$$${to}\:“{actual}''\:{coordinates} \\ $$
Commented by TheHoneyCat last updated on 21/Oct/21
$$ \\ $$$$\mathrm{Some}\:\mathrm{people}\:\mathrm{define} \\ $$$$\overset{\rightarrow} {\bigtriangledown}\::=\:\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\partial}{\partial{x}_{{i}} }\overset{\rightarrow} {{e}}_{{i}} \\ $$$$\mathrm{and}\:\mathrm{say}\:\mathrm{that}\:\mathrm{div}\:\overset{\rightarrow} {{F}}:=\overset{\rightarrow} {\bigtriangledown}.\overset{\rightarrow} {{F}} \\ $$$$\left.\mathrm{this}\:\mathrm{would}\:\mathrm{be}\:\mathrm{equivalent}\:\mathrm{to}\:\mathrm{2}\right) \\ $$$$\mathrm{but}\:\mathrm{a}\:\mathrm{tiny}\:\mathrm{bit}\:\mathrm{more}\:“{powerful}''\:\mathrm{from}\:\mathrm{a} \\ $$$$\mathrm{computationnal}\:\mathrm{point}\:\mathrm{of}\:\mathrm{vue},\:\mathrm{for}\:\mathrm{you}\:\mathrm{can} \\ $$$$\mathrm{learn}\:\mathrm{its}\:“{expression}\:{in}\:{other}\:{coordinates}'' \\ $$$$\mathrm{even}\:\mathrm{some}\:\mathrm{improper}\:\mathrm{ones}. \\ $$