Analisi vettoriale/Formule più importanti dell'analisi vettoriale

${\delta \phi \over \delta s}={\phi _{s}-\phi _{0} \over ds}\ \ \ \ (E.1)$

(la derivata dello scalare $\phi$ rispetto alla direzione $\mathbf {s}$)

$grad\ \phi ={\delta \phi \over \delta n}\mathbf {n} \ \ \ \ (E.3)$

$\nabla \ \phi =grad\ \phi =\mathbf {i} {\delta \phi \over \delta x}+\mathbf {j} {\delta \phi \over \delta y}+\mathbf {k} {\delta \phi \over \delta z}\ \ \ \ (E.6)$

$grad_{s}\ \phi ={\delta \phi \over \delta s}\ \ \ \ (E.4)$

$grad\ \phi ={\delta \phi (\Phi ) \over \delta \Phi }grad\ \Phi \ \ \ \ E.7)$

$grad_{q}R=-{\mathbf {R} \over R}=-grad_{a}R\ \ \ \ (E.8)$

$grad_{q}({1 \over R})={\mathbf {R} \over R^{3}}=-grad_{a}({1 \over R})\ \ \ \ (E.10)$

$grad_{a}(\mathbf {bR} )=\mathbf {b} (\mathbf {b} =costante)\ \ \ \ (E.11)$

$\oint _{s}a_{n}\ dS=\int _{V}div\ \mathbf {a} \ \ dV\ \ (teorema\ di\ Gauss)\ \ \ \ (E.17)$

$div\ \mathbf {a} =\lim _{\Delta V\to 0}{\oint a_{n}dS \over \Delta \ V}\ \ \ \ (E.18)$

$\nabla \ \mathbf {a} =div\ \mathbf {a} ={\delta a_{x} \over \delta x}+{\delta a_{y} \over \delta y}+{\delta a_{z} \over \delta z}\ \ \ \ (E.15)$

$div\ \mathbf {a} =a_{2n}-a_{1n}\ \ \ \ (E.29)$

$\oint a_{s}ds=\int _{L}rot_{n}\mathbf {a} \ dS\ \ \ (teorema\ di\ Stokes)\ \ \ \ (E.27)$

$rot_{n}\mathbf {a} =\lim _{ds\to 0}{\oint a_{s}\ ds \over dS}\ \ \ \ (E.29)$

$[\nabla \ \mathbf {a} ]=rot\ \mathbf {a} ={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\{\delta \over \delta x}&{\delta \over \delta y}&{\delta \over \delta z}\\a_{x}&a_{y}&a_{z}\end{vmatrix}}\ \ \ \ \ (E.25)$

$rot\ \mathbf {a} =[\mathbf {n} \cdot (\mathbf {a_{2}-a_{1})} ]$

$\oint _{S}rot_{n}\ \mathbf {a} \ ds=0\ \ \ \ (E.28)$

${\delta \mathbf {b} \over \delta a}=\lim _{\Delta a\to 0}{\mathbf {b'} -\mathbf {b} \over \Delta a}\ \ \ \ (E.33)$

Derivate seconde

$div\ \ grad\ \phi =\nabla ^{2}\phi ={\delta ^{2}\phi \over \delta x^{2}}+{\delta ^{2}\phi \over \delta y^{2}}+{\delta ^{2}\phi \over \delta z^{2}}\ \ \ \ (E.40)$

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

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

Derivate di prodotto

$grad(\phi \ \Phi )=\Phi \ grad\ \phi +\phi \ grad\ \Phi$

$div(\phi \ \mathbf {a} )=\phi \ div\ \mathbf {a} +\mathbf {a} \ grad\ \phi$

$rot(\phi \ \mathbf {a} )=\phi \ rot\ \mathbf {a} +[grad\ \phi \cdot \mathbf {a} ]$

$div[\mathbf {ab} ]=\mathbf {b} \ rot\ \mathbf {a} -\mathbf {a} \ rot\ \mathbf {b} \ \ \ \ (E.44)$

$grab(\mathbf {ab} )=(\mathbf {b} \nabla )\mathbf {a} +(\mathbf {a} \nabla )\mathbf {b} +[\mathbf {b} \ rot\ \mathbf {a} ]+[\mathbf {a} \ rot\ \mathbf {b} ]\ \ \ \ (E.45)$

$rot[\mathbf {ab} ]=(\mathbf {b} \nabla )\mathbf {a} +(\mathbf {a} \nabla )\mathbf {b} +\mathbf {a} \ div\ \mathbf {b} -\mathbf {b} \ div\ \mathbf {a} \ \ \ \ (E.46)$

${1 \over 2}\nabla a^{2}=(\mathbf {a\nabla } )\mathbf {a} +[\mathbf {a} \ rot\ \mathbf {a} ]\ \ \ \ (E.47)$

Teorema di Green

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

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

$\oint _{L}\phi \ d\mathbf {s} =\int _{S}[\mathbf {n} \cdot grad\ \phi ]dS\ \ \ \ (E.54)$

$\int _{V}rot\ \mathbf {a} \ dV=\oint _{S}[\mathbf {na} ]dS=\oint _{S}[d\mathbf {S\cdot a} ]\ \ \ \ (E.56)$