Analisi vettoriale/Formule più importanti dell'analisi vettoriale

$\frac{\delta \varphi }{\delta s}=\frac{{\varphi }_s-{\varphi }_0}{ds}(E.1)$

(la derivata dello scalare $\varphi$ rispetto alla direzione $𝐬$)

$grad\varphi =\frac{\delta \varphi }{\delta n}𝐧(E.3)$

$\mathrm{\nabla }\varphi =grad\varphi =𝐢\frac{\delta \varphi }{\delta x}+𝐣\frac{\delta \varphi }{\delta y}+𝐤\frac{\delta \varphi }{\delta z}(E.6)$

$gra{d}_s\varphi =\frac{\delta \varphi }{\delta s}(E.4)$

$grad\varphi =\frac{\delta \varphi (\mathrm{\Phi })}{\delta \mathrm{\Phi }}grad\mathrm{\Phi }E.7)$

$gra{d}_{q}R=-\frac{𝐑}{R}=-gra{d}_{a}R(E.8)$

$gra{d}_q(\frac{1}{R})=\frac{𝐑}{R^3}=-gra{d}_a(\frac{1}{R})(E.10)$

$gra{d}_a(𝐛𝐑)=𝐛(𝐛=costante)(E.11)$

${\oint }_s{a}_{n}dS=\underset{V}{\int }div𝐚dV(teoremadiGauss)(E.17)$

$div𝐚=\underset{\mathrm{\Delta }V\to 0}{\lim }\frac{\oint a_{n}dS}{\mathrm{\Delta }V}(E.18)$

$\mathrm{\nabla }𝐚=div𝐚=\frac{\delta a_x}{\delta x}+\frac{\delta a_y}{\delta y}+\frac{\delta a_z}{\delta z}(E.15)$

$div𝐚=a_{2n}-a_{1n}(E.29)$

$\oint a_{s}ds=\underset{L}{\int }ro{t}_n𝐚dS(teoremadiStokes)(E.27)$

$ro{t}_n𝐚=\underset{ds\to 0}{\lim }\frac{\oint a_{s}ds}{dS}(E.29)$

$[\mathrm{\nabla }𝐚]=rot𝐚=\left|\begin{array}{ccc}𝐢& 𝐣& 𝐤\\ \frac{\delta }{\delta x}& \frac{\delta }{\delta y}& \frac{\delta }{\delta z}\\ a_x& a_y& a_z\end{array}\right|(E.25)$

$rot𝐚=[𝐧\cdot ({𝐚}_{\mathrm{𝟐}}\mathbf{-}{𝐚}_{\mathrm{𝟏}}\mathbf{)}]$

${\oint }_{S}ro{t}_n𝐚ds=0(E.28)$

$\frac{\delta 𝐛}{\delta a}=\underset{\mathrm{\Delta }a\to 0}{\lim }\frac{{𝐛}^{\prime }-𝐛}{\mathrm{\Delta }a}(E.33)$

Derivate seconde

$divgrad\varphi ={\mathrm{\nabla }}^2\varphi =\frac{{\delta }^2\varphi }{\delta x^2}+\frac{{\delta }^2\varphi }{\delta y^2}+\frac{{\delta }^2\varphi }{\delta z^2}(E.40)$

${\mathrm{\nabla }}^{2}f(R)=\frac{{\delta }^{2}f}{\delta R^2}+\frac{2}{R}\frac{\delta f}{\delta R}(E.21)$

${\mathrm{\nabla }}^2(\frac{1}{R})=0$

Derivate di prodotto

$grad(\varphi \mathrm{\Phi })=\mathrm{\Phi }grad\varphi +\varphi grad\mathrm{\Phi }$

$div(\varphi 𝐚)=\varphi div𝐚+𝐚grad\varphi$

$rot(\varphi 𝐚)=\varphi rot𝐚+[grad\varphi \cdot 𝐚]$

$div[𝐚𝐛]=𝐛rot𝐚-𝐚rot𝐛(E.44)$

$grab(𝐚𝐛)=(𝐛\mathrm{\nabla })𝐚+(𝐚\mathrm{\nabla })𝐛+[𝐛rot𝐚]+[𝐚rot𝐛](E.45)$

$rot[𝐚𝐛]=(𝐛\mathrm{\nabla })𝐚+(𝐚\mathrm{\nabla })𝐛+𝐚div𝐛-𝐛div𝐚(E.46)$

$\frac{1}{2}\mathrm{\nabla }{a}^2=(𝐚\mathrm{\nabla })𝐚+[𝐚rot𝐚](E.47)$

Teorema di Green

$\int [\mathrm{\Phi }{\mathrm{\nabla }}^2\varphi +(\mathrm{\nabla }\varphi )(\mathrm{\nabla }\mathrm{\Phi })]dV=\oint \mathrm{\Phi }\frac{\delta \varphi }{\delta n}dS(E.52)$

$\int (\mathrm{\Phi }{\mathrm{\nabla }}^2\varphi -\varphi {\mathrm{\nabla }}^2\mathrm{\Phi })dV=\oint (\mathrm{\Phi }\frac{\delta \varphi }{\delta n}-\varphi \frac{\delta \mathrm{\Phi }}{\delta n})dS(E.53)$

${\oint }_L\varphi d𝐬=\underset{S}{\int }[𝐧\cdot grad\varphi ]dS(E.54)$

$\underset{V}{\int }rot𝐚dV={\oint }_S[𝐧𝐚]dS={\oint }_S[d𝐒\mathbf{\cdot }𝐚](E.56)$