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

$$Whatisthegeneralexpressionofthedivergence\left(div\stackrel{⇢}{V}\right)$$

Answered by TheHoneyCat last updated on 21/Oct/21

$$\mathrm{It}\mathrm{depends}\mathrm{on}\mathrm{your}\mathrm{level}:$$$$1)\mathrm{If}\left(toyou\right)\overrightarrow{\mathrm{V}}\mathrm{is}\mathrm{a}\mathrm{tridimentionnal}\mathrm{vector}$$$$\overrightarrow{\mathrm{V}}=V_x{\overrightarrow{e}}_x+V_y{\overrightarrow{e}}_y+V_z{\overrightarrow{e}}_z$$$$\mathrm{then}:\mathrm{div}\overrightarrow{V}=\frac{\partial V_x}{\partial x}+\frac{\partial V_y}{\partial y}+\frac{\partial V_z}{\partial z}$$$$$$$$2)\overrightarrow{V}\mathrm{is}\mathrm{a}\mathrm{function}\mathrm{from}{\mathbb{R}}^n\mathrm{to}{\mathbb{R}}^n$$$$\mathrm{with}:\mathrm{denoting}{V}_i\mathrm{its}\mathrm{coordinate}-\mathrm{functions}$$$$(iethefunction{V}_{i}from{\mathbb{R}}^{n}to\mathbb{R}suchthat$$$$\mathrm{\forall }i\in [\mid 1,n\mid ]\overrightarrow{V}.{\overrightarrow{e}}_i=V_i)$$$$\mathrm{Then}:\mathrm{div}\overrightarrow{V}=\underset{i=1}{\sum ^n}\frac{\partial V_i}{\partial x_i}$$$$$$$$3)\mathrm{If}\overrightarrow{V}\mathrm{is}\mathrm{a}\mathrm{differentiable}\mathrm{map}\mathrm{of}{\mathbb{R}}^n\to {\mathbb{R}}^n$$$$\left(\mathrm{or}\mathrm{any}\mathrm{set}\mathrm{isomorphic}\mathrm{to}\mathrm{it}\right)$$$$\mathrm{its}\mathrm{divergence}\mathrm{is}\mathrm{defined}\mathrm{as}\mathrm{above}$$$$itmightnotlooklikethereisadifference$$$$from2)butthefactthatitisdifferentable$$$$assuresthatthedivergencedoesnotdepend$$$$onthechoiceofcoordinates,henceitisa$$$$generaloperatoronthesetofdifferentible$$$$functions,andnolongeraformulathatrefers$$$$to‘‘{actual}^{″}coordinates$$

Commented by TheHoneyCat last updated on 21/Oct/21

$$$$$$\mathrm{Some}\mathrm{people}\mathrm{define}$$$$\overrightarrow{▽}:=\underset{i=1}{\sum ^n}\frac{\partial }{\partial x_i}{\overrightarrow{e}}_i$$$$\mathrm{and}\mathrm{say}\mathrm{that}\mathrm{div}\overrightarrow{F}:=\overrightarrow{▽}.\overrightarrow{F}$$$$\mathrm{this}\mathrm{would}\mathrm{be}\mathrm{equivalent}\mathrm{to}2)$$$$\mathrm{but}\mathrm{a}\mathrm{tiny}\mathrm{bit}\mathrm{more}‘‘{powerful}^{″}\mathrm{from}\mathrm{a}$$$$\mathrm{computationnal}\mathrm{point}\mathrm{of}\mathrm{vue},\mathrm{for}\mathrm{you}\mathrm{can}$$$$\mathrm{learn}\mathrm{its}‘‘expressioninother{coordinates}^{″}$$$$\mathrm{even}\mathrm{some}\mathrm{improper}\mathrm{ones}.$$

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