Question and Answers Forum

All Questions      Topic List

Electrostatics Questions

Previous in All Question      Next in All Question      

Previous in Electrostatics      Next in Electrostatics      

Question Number 157199 by Fresnel last updated on 20/Oct/21

What is the general expression of the divergence(divV^⇢ )

$${What}\:{is}\:{the}\:{general}\:{expression}\:{of}\:{the}\:{divergence}\left({div}\overset{\dashrightarrow} {{V}}\right) \\ $$

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

$$\mathrm{It}\:\mathrm{depends}\:\mathrm{on}\:\mathrm{your}\:\mathrm{level}: \\ $$$$\left.\mathrm{1}\right)\:\mathrm{If}\:\left({to}\:{you}\right)\:\:\overset{\rightarrow} {\mathrm{V}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{tridimentionnal}\:\mathrm{vector} \\ $$$$\overset{\rightarrow} {\mathrm{V}}={V}_{{x}} \overset{\rightarrow} {{e}}_{{x}} +{V}_{{y}} \overset{\rightarrow} {{e}}_{{y}} +{V}_{{z}} \overset{\rightarrow} {{e}}_{{z}} \\ $$$$\mathrm{then}:\:\mathrm{div}\:\overset{\rightarrow} {{V}}=\frac{\partial{V}_{{x}} }{\partial{x}}+\frac{\partial{V}_{{y}} }{\partial{y}}+\frac{\partial{V}_{{z}} }{\partial{z}} \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\:\overset{\rightarrow} {{V}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{function}\:\mathrm{from}\:\mathbb{R}^{{n}} \:\mathrm{to}\:\mathbb{R}^{{n}} \\ $$$$\mathrm{with}:\:\mathrm{denoting}\:{V}_{{i}} \:\mathrm{its}\:\mathrm{coordinate}−\mathrm{functions} \\ $$$$\left({ie}\:{the}\:{function}\:{V}_{{i}} \:{from}\:\mathbb{R}^{{n}} \:{to}\:\mathbb{R}\:{such}\:{that}\right. \\ $$$$\left.\forall{i}\in\left[\mid\mathrm{1},{n}\mid\right]\:\overset{\rightarrow} {{V}}.\overset{\rightarrow} {{e}}_{{i}} ={V}_{{i}} \:\right) \\ $$$$\mathrm{Then}:\:\mathrm{div}\:\overset{\rightarrow} {{V}}=\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\partial{V}_{{i}} }{\partial{x}_{{i}} } \\ $$$$ \\ $$$$\left.\mathrm{3}\right)\:\mathrm{If}\:\overset{\rightarrow} {{V}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{differentiable}\:\mathrm{map}\:\mathrm{of}\:\mathbb{R}^{{n}} \rightarrow\mathbb{R}^{{n}} \\ $$$$\left(\mathrm{or}\:\mathrm{any}\:\mathrm{set}\:\mathrm{isomorphic}\:\mathrm{to}\:\mathrm{it}\right) \\ $$$$\mathrm{its}\:\mathrm{divergence}\:\mathrm{is}\:\mathrm{defined}\:\mathrm{as}\:\mathrm{above} \\ $$$${it}\:{might}\:{not}\:{look}\:{like}\:{there}\:{is}\:{a}\:{difference} \\ $$$$\left.{from}\:\mathrm{2}\right)\:{but}\:{the}\:{fact}\:{that}\:{it}\:{is}\:{differentable} \\ $$$${assures}\:{that}\:{the}\:{divergence}\:{does}\:{not}\:{depend} \\ $$$${on}\:{the}\:{choice}\:{of}\:{coordinates},\:{hence}\:{it}\:{is}\:{a}\: \\ $$$${general}\:{operator}\:{on}\:{the}\:{set}\:{of}\:{differentible} \\ $$$${functions},\:{and}\:{no}\:{longer}\:{a}\:{formula}\:{that}\:{refers} \\ $$$${to}\:``{actual}''\:{coordinates} \\ $$

Commented by TheHoneyCat last updated on 21/Oct/21

  Some people define  ▽^→  := Σ_(i=1) ^n (∂/∂x_i )e_i ^→   and say that div F^→ :=▽^→ .F^→   this would be equivalent to 2)  but a tiny bit more “powerful” from a  computationnal point of vue, for you can  learn its “expression in other coordinates”  even some improper ones.

$$ \\ $$$$\mathrm{Some}\:\mathrm{people}\:\mathrm{define} \\ $$$$\overset{\rightarrow} {\bigtriangledown}\::=\:\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\partial}{\partial{x}_{{i}} }\overset{\rightarrow} {{e}}_{{i}} \\ $$$$\mathrm{and}\:\mathrm{say}\:\mathrm{that}\:\mathrm{div}\:\overset{\rightarrow} {{F}}:=\overset{\rightarrow} {\bigtriangledown}.\overset{\rightarrow} {{F}} \\ $$$$\left.\mathrm{this}\:\mathrm{would}\:\mathrm{be}\:\mathrm{equivalent}\:\mathrm{to}\:\mathrm{2}\right) \\ $$$$\mathrm{but}\:\mathrm{a}\:\mathrm{tiny}\:\mathrm{bit}\:\mathrm{more}\:``{powerful}''\:\mathrm{from}\:\mathrm{a} \\ $$$$\mathrm{computationnal}\:\mathrm{point}\:\mathrm{of}\:\mathrm{vue},\:\mathrm{for}\:\mathrm{you}\:\mathrm{can} \\ $$$$\mathrm{learn}\:\mathrm{its}\:``{expression}\:{in}\:{other}\:{coordinates}'' \\ $$$$\mathrm{even}\:\mathrm{some}\:\mathrm{improper}\:\mathrm{ones}. \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com