Visualizza wikitesto di Analisi vettoriale/Formule più importanti dell'analisi vettoriale

{{Analisi vettoriale}}

::::<math>{\delta \phi\over \delta s}={\phi_s-\phi_0\over ds}\ \ \ \ (E.1)</math>
(la derivata dello scalare <math>\phi</math> rispetto alla direzione <math>\mathbf{s}</math>)
::::<math>grad\ \phi={\delta \phi\over \delta n}\mathbf{n}\ \ \ \ (E.3)</math>
::::<math>\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)</math>
::::<math>grad_s\ \phi={\delta \phi\over \delta s}\ \ \ \ (E.4)</math>
::::<math>grad\ \phi={\delta \phi (\Phi)\over \delta \Phi}grad\ \Phi\ \ \ \ E.7)</math>
::::<math>grad_qR=-{\mathbf{R}\over R}=-grad_a R\ \ \ \ (E.8)</math>
::::<math>grad_q({1\over R})={\mathbf{R}\over R^3}=-grad_a({1\over R})\ \ \ \ (E.10)</math>
::::<math>grad_a (\mathbf{bR})=\mathbf{b} (\mathbf{b} = costante)\ \ \ \ (E.11)</math>
::::<math>\oint_s a_n\ dS=\int_V div\ \mathbf{a}\ \ dV\ \ (teorema\ di\ Gauss)\ \ \ \ (E.17)</math>
::::<math>div\ \mathbf{a}=\lim_{\Delta V \to 0}{\oint a_n dS\over \Delta\ V}\ \ \ \ (E.18)</math>
::::<math>\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)</math>
::::<math>div\ \mathbf{a}=a_{2n}-a_{1n}\ \ \ \ (E.29)</math> 
::::<math>\oint a_s ds=\int_L rot_n \mathbf{a}\ dS\ \ \ (teorema\ di\ Stokes)\ \ \ \ (E.27)</math>
::::<math>rot_n\mathbf{a}= \lim_{ds \to 0}{\oint a_s\ ds\over dS}\ \ \ \ (E.29)</math>
::::<math>[\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)</math>
::::<math>rot\ \mathbf{a}=[\mathbf{n}\cdot(\mathbf{a_2-a_1)}]</math>
::::<math>\oint_S rot_n\ \mathbf{a}\ ds=0\ \ \ \ (E.28)</math>
::::<math>{\delta \mathbf{b}\over \delta a}=\lim_{\Delta a \to 0}{\mathbf{b'}-\mathbf{b}\over \Delta a}\ \ \ \ (E.33)</math>

;Derivate seconde:

::::<math>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)</math>
::::<math>\nabla^2f(R)={\delta^2f\over \delta R^2}+{2\over R}{\delta f\over \delta R}\ \ \ \ (E.21)</math>
::::<math>\nabla^2({1\over R})=0</math>

;Derivate di prodotto:

::::<math>grad ( \phi\ \Phi)=\Phi\ grad\ \phi+\phi\ grad\ \Phi </math>
::::<math>div (\phi\ \mathbf{a})=\phi\ div\ \mathbf{a}+\mathbf{a}\ grad\ \phi</math> 
::::<math>rot (\phi\ \mathbf{a})=\phi\ rot\ \mathbf{a}+[grad\ \phi\cdot\mathbf{a}]</math>
::::<math>div [\mathbf{ab}]=\mathbf{b}\ rot\ \mathbf{a}-\mathbf{a}\ rot\ \mathbf{b}\ \ \ \ (E.44)</math>
::::<math>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)</math>
::::<math>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)</math>
::::<math>{1\over 2}\nabla a^2=(\mathbf{a\nabla})\mathbf{a}+[\mathbf{a}\ rot\ \mathbf{a}]\ \ \ \ (E.47)</math>

;Teorema di Green:

::::<math>\int[\Phi\nabla^2\phi+(\nabla\phi)(\nabla\Phi)]dV=\oint\Phi{\delta \phi\over \delta n}dS\ \ \ \ (E.52)</math>
::::<math>\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)</math>
::::<math>\oint_L\phi\ d\mathbf{s}=\int_S[\mathbf{n}\cdot grad\ \phi]dS\ \ \ \ (E.54)</math>
::::<math>\int_V rot\ \mathbf{a}\ dV=\oint_S[\mathbf{na}]dS=\oint_S[d\mathbf{S\cdot a}]\ \ \ \ (E.56)</math>

[[Categoria:Analisi vettoriale|Formule più importanti dell'analisi vettoriale]]
{{Avanzamento|100%|14 agosto 2017}}