Identitete vektorskega računa

Sledi seznam pomembnih matematičnih identitet, ki vključujejo odvode in integrale v vektorskem računu.

Operatorski zapis

Gradient

Za funkcijo $f(x,y,z)\!\,$ v trirazsežnih kartezičnih koordinatnih spremenljivkah je gradient vektorsko polje:

$\operatorname {grad} (f)\equiv \nabla f\equiv \mathbf {\vec {\nabla }} f={\begin{pmatrix}\displaystyle {\frac {\partial }{\partial x}},\ {\frac {\partial }{\partial y}},\ {\frac {\partial }{\partial z}}\end{pmatrix}}f={\frac {\partial f}{\partial x}}\mathbf {\hat {i}} +{\frac {\partial f}{\partial y}}\mathbf {\hat {j}} +{\frac {\partial f}{\partial z}}\mathbf {\hat {k}} \!\,$

kjer so $\mathbf {\hat {i}} \!\,$, $\mathbf {\hat {j}} \!\,$ in $\mathbf {\hat {k}} \!\,$ standardni enotski vektorji za osi $x\!\,$, $y\!\,$ in $z\!\,$. Bolj splošno, za funkcijo $n\!\,$ spremenljivk $\psi (x_{1},\ldots ,x_{n})\!\,$, imenovano tudi skalarno polje, je gradient vektorsko polje:

$\mathbf {\vec {\nabla }} \psi ={\begin{pmatrix}\displaystyle {\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{n}}}\end{pmatrix}}\psi ={\frac {\partial \psi }{\partial x_{1}}}\mathbf {\hat {e}} _{1}+\dots +{\frac {\partial \psi }{\partial x_{n}}}\mathbf {\hat {e}} _{n}\!\,,$

kjer so $\mathbf {\hat {e}} _{i}\,(i=1,2,\ldots ,n)\!\,$ med seboj ortogonalni enotski vektorji.

Kot pove že ime, je gradient sorazmeren z najhitrejšo (pozitivno) spremembo funkcije in kaže v smeri te spremembe.

Za vektorsko polje $\mathbf {\vec {A}} =\left(A_{1},\ldots ,A_{n}\right)\!\,$, imenovano tudi tenzorsko polje prvega reda, je gradient ali totalni odvod Jakobijeva matrika $n\times n\!\,$:

$\mathbf {\vec {J}} _{\mathbf {\vec {A}} }=\operatorname {d} \!\mathbf {\vec {A}} =(\mathbf {\vec {\nabla }} \!\mathbf {\vec {A}} )^{\top }=\left({\frac {\partial A_{i}}{\partial x_{j}}}\right)_{\!ij}\!\,.$

Za tenzorsko polje $\mathbf {T} \!\,$ poljubnega reda $k\!\,$, je gradient $\operatorname {grad} (\mathbf {T} )=\operatorname {d} \!\mathbf {T} =(\nabla \mathbf {T} )^{\top }\!\,$ tenzorsko polje reda $k+1\!\,$.

Za tenzorsko polje $\mathbf {T} \!\,$ reda $k>0\!\,$, je tenzorsko polje $\nabla \mathbf {T} \!\,$ reda $k+1\!\,$ definirano kot rekurzivna relacija:

$(\nabla \mathbf {T} )\cdot \mathbf {\vec {c}} =\nabla (\mathbf {T} \cdot \mathbf {\vec {c}} )\!\,,$

kjer je $\mathbf {\vec {c}} \!\,$ poljubni konstantni vektor.

Divergenca

V kartezičnih koordinatah je divergenca zvezno odvedljivega vektorskega polja $\mathbf {\vec {F}} =F_{x}\mathbf {\hat {i}} +F_{y}\mathbf {\hat {j}} +F_{z}\mathbf {\hat {k}} \!\,$ skalarna funkcija:

${\begin{aligned}\operatorname {div} \mathbf {\vec {F}} \equiv \nabla \cdot \mathbf {\vec {F}} &={\begin{pmatrix}{\dfrac {\partial }{\partial x}},\ {\dfrac {\partial }{\partial y}},\ {\dfrac {\partial }{\partial z}}\end{pmatrix}}\cdot {\begin{pmatrix}F_{x},\ F_{y},\ F_{z}\end{pmatrix}}\\[1ex]&={\frac {\partial F_{x}}{\partial x}}+{\frac {\partial F_{y}}{\partial y}}+{\frac {\partial F_{z}}{\partial z}}\!\,.\end{aligned}}$

Kot pove že ime, je divergenca (krajevna) mera stopnje razhajanja vektorjev v polju.

Divergenca tenzorskega polja $\mathbf {T} \!\,$ neničelnega reda $k\!\,$ je zapisana kot $\operatorname {div} (\mathbf {T} )\equiv \nabla \cdot \mathbf {T} \!\,$, kar je kontrakcija tenzorskega polja reda $k-1\!\,$. Natančneje, divergenca vektorja je skalar. Divergenco tenzorskega polja višjega reda se lahko najde z razstavitvijo tenzorskega polja v vsoto zunanjih produktov in rabo identitete:

$\nabla \cdot \left(\mathbf {A} \otimes \mathbf {T} \right)=\mathbf {T} (\nabla \cdot \mathbf {A} )+(\mathbf {A} \cdot \nabla )\mathbf {T} \!\,,$

kjer je $\mathbf {A} \cdot \nabla \!\,$ smerni odvod v smeri $\mathbf {A} \!\,$, pomnožen z njegovo velikostjo. Natančneje, za zunanji produkt dveh vektorjev:

$\nabla \cdot \left(\mathbf {\vec {A}} \mathbf {\vec {B}} ^{\top }\right)=\mathbf {\vec {B}} (\nabla \cdot \mathbf {\vec {A}} )+(\mathbf {\vec {A}} \cdot \nabla )\mathbf {\vec {B}} \!\,.$

Za tenzorsko polje $\mathbf {T} \!\,$ reda $k>1\!\,$ je tenzorsko polje $\nabla \cdot \mathbf {T} \!\,$ reda $k-1\!\,$ definirano z rekurzivno relacijo:

$(\nabla \cdot \mathbf {T} )\cdot \mathbf {\vec {c}} =\nabla \cdot (\mathbf {T} \cdot \mathbf {\vec {c}} )\!\,$

kjer je $\mathbf {\vec {c}}$ poljubni konstantni vektor.

Rotor

V kartezičnih koordinatah je rotor vektorskega polja $\mathbf {\vec {F}} =F_{x}\mathbf {\hat {i}} +F_{y}\mathbf {\hat {j}} +F_{z}\mathbf {\hat {k}} \!\,$ vektorsko polje:

${\begin{aligned}\operatorname {curl} \mathbf {F} \equiv =\nabla \times \mathbf {\vec {F}} &={\begin{pmatrix}\displaystyle {\frac {\partial }{\partial x}},\ {\frac {\partial }{\partial y}},\ {\frac {\partial }{\partial z}}\end{pmatrix}}\times {\begin{pmatrix}F_{x},\ F_{y},\ F_{z}\end{pmatrix}}\\[1em]&={\begin{vmatrix}\mathbf {\hat {i}} &\mathbf {\hat {j}} &\mathbf {\hat {k}} \\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\F_{x}&F_{y}&F_{z}\end{vmatrix}}\\[1em]&=\left({\frac {\partial F_{z}}{\partial y}}-{\frac {\partial F_{y}}{\partial z}}\right)\mathbf {\hat {i}} +\left({\frac {\partial F_{x}}{\partial z}}-{\frac {\partial F_{z}}{\partial x}}\right)\mathbf {\hat {j}} +\left({\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}}\right)\mathbf {\hat {k}} \!\,,\end{aligned}}$

kjer so $\mathbf {\hat {i}} \!\,$, $\mathbf {\hat {j}} \!\,$ in $\mathbf {\hat {k}} \!\,$ enotski vektorji za osi $x\!\,$, $y\!\,$ in $z\!\,$.

Kot pove že ime, je rotor mera, za koliko bližnji vektorji težijo v krožni smeri.

V Einsteinovem zapisu ima vektorsko polje $\mathbf {\vec {F}} ={\begin{pmatrix}F_{1},\ F_{2},\ F_{3}\end{pmatrix}}\!\,$ rotor, podan kot:

$\nabla \times \mathbf {\vec {F}} =\varepsilon ^{ijk}\mathbf {\hat {e}} _{i}{\frac {\partial F_{k}}{\partial x_{j}}}\!\,,$

kjer je $\varepsilon =\pm 1\!\,$ ali $0\!\,$ Levi-Civitajev simbol parnosti (permutacije).

Za tenzorsko polje $\mathbf {T} \!\,$ reda $k>1\!\,$ je tenzorsko polje $\nabla \times \mathbf {T} \!\,$ reda $k\!\,$ definirano z rekurzivno relacijo:

$(\nabla \times \mathbf {T} )\cdot \mathbf {\vec {c}} =\nabla \times (\mathbf {T} \cdot \mathbf {\vec {c}} )\!\,,$

kjer je $\mathbf {\vec {c}} \!\,$ poljubni konstantni vektor.

Tenzorsko polje reda večjega od ena se lahko razstavi v vsoto zunanjih produktov, nato pa se lahko rabi naslednjo identiteto:

$\nabla \times \left(\mathbf {A} \otimes \mathbf {T} \right)=(\nabla \times \mathbf {A} )\otimes \mathbf {T} -\mathbf {A} \times (\nabla \mathbf {T} )\!\,.$

Še posebej za zunanji produkt dveh vektorjev:

$\nabla \times \left(\mathbf {\vec {A}} \mathbf {\vec {B}} ^{\top }\right)=(\nabla \times \mathbf {\vec {A}} )\mathbf {\vec {B}} ^{\top }-\mathbf {\vec {A}} \times (\nabla \mathbf {\vec {B}} )\!\,.$

Laplaceov operator

V kartezični koordinatah je Laplaceov operator funkcije $f(x,y,z)\!\,$:

$\Delta f\equiv \nabla ^{2}\!f\equiv (\nabla \cdot \nabla )f={\frac {\partial ^{2}\!f}{\partial x^{2}}}+{\frac {\partial ^{2}\!f}{\partial y^{2}}}+{\frac {\partial ^{2}\!f}{\partial z^{2}}}\!\,.$

Laplaceov opeartor je mera, koliko se funkcija spreminja na majhni sferi s središčem v točki.

Ko je Laplaceov operator enak 0, se funkcija imenuje harmonična funkcija. To pomeni:

$\Delta f=0\!\,.$

Za tenzorsko polje $\mathbf {T} \!\,$ se Laplaceov operator v splošnem zapiše kot:

$\Delta \mathbf {T} =\nabla ^{2}\mathbf {T} =(\nabla \cdot \nabla )\mathbf {T} \!\,$

in je tenzorsko polje istega reda.

Za tenzorsko polje $\mathbf {T} \!\,$ reda $k>0\!\,$ je tenzorsko polje $\nabla ^{2}\mathbf {T} \!\,$ reda $k\!\,$ definirano z rekurzivno relacijo:

$\left(\nabla ^{2}\mathbf {T} \right)\cdot \mathbf {\vec {c}} =\nabla ^{2}(\mathbf {T} \cdot \mathbf {\vec {c}} )\!\,,$

kjer je $\mathbf {\vec {c}} \!\,$ poljubni konstantni vektor.

Posebni zapisi

V Feynmanovem podpisnem zapisu:

$\nabla _{\mathbf {B} }\!\left(\mathbf {A{\cdot }B} \right)=\mathbf {A} {\times }\!\left(\nabla {\times }\mathbf {B} \right)+\left(\mathbf {A} {\cdot }\nabla \right)\mathbf {B} \!\,,$

kjer zapis $\nabla _{\mathbf {B} }\!\,$ pomeni, da podpisani gradient deluje samo na faktorju $\mathbf {B} \!\,$.

Bolj splošen, a podoben je Hestenesov zapis z nadpisno piko v geometrijski algebri. Zgornja identiteta se nato izrazi kot:

${\dot {\nabla }}\left(\mathbf {\vec {A}} {\cdot }{\dot {\mathbf {\vec {B}} }}\right)=\mathbf {\vec {A}} {\times }\!\left(\nabla {\times }\mathbf {\vec {B}} \right)+\left(\mathbf {\vec {A}} {\cdot }\nabla \right)\mathbf {\vec {B}} \!\,,$

kjer nadpisne pike definirajo obseg vektorskega odvoda. Vektor s piko, v tem primeru $\mathbf {\vec {B}} \!\,$, se odvaja, medtem ko $\mathbf {\vec {A}} \!\,$ (brez pike) ostane konstanten.

Uporabnost Feynmanovega podpisnega zapisa je v njegovi rabi pri izpeljavi identitet vektorskih in tenzorskih odvodov, kot v naslednjem primeru, ki rabi algebrsko identiteto $\mathbf {\vec {C}} \cdot (\mathbf {\vec {A}} \times \mathbf {\vec {B}} )=(\mathbf {\vec {C}} \times \mathbf {\vec {A}} )\cdot \mathbf {\vec {B}} \!\,$:

${\begin{aligned}\nabla \cdot (\mathbf {\vec {A}} \times \mathbf {\vec {B}} )&=\nabla _{\mathbf {\vec {A}} }\cdot (\mathbf {\vec {A}} \times \mathbf {\vec {B}} )+\nabla _{\mathbf {\vec {B}} }\cdot (\mathbf {\vec {A}} \times \mathbf {\vec {B}} )\\[2pt]&=(\nabla _{\mathbf {\vec {A}} }\times \mathbf {\vec {A}} )\cdot \mathbf {\vec {B}} +(\nabla _{\mathbf {\vec {B}} }\times \mathbf {\vec {A}} )\cdot \mathbf {\vec {B}} \\[2pt]&=(\nabla _{\mathbf {\vec {A}} }\times \mathbf {\vec {A}} )\cdot \mathbf {\vec {B}} -(\mathbf {\vec {A}} \times \nabla _{\mathbf {\vec {B}} })\cdot \mathbf {\vec {B}} \\[2pt]&=(\nabla _{\mathbf {\vec {A}} }\times \mathbf {\vec {A}} )\cdot \mathbf {\vec {B}} -\mathbf {\vec {A}} \cdot (\nabla _{\mathbf {\vec {B}} }\times \mathbf {\vec {B}} )\\[2pt]&=(\nabla \times \mathbf {\vec {A}} )\cdot \mathbf {\vec {B}} -\mathbf {\vec {A}} \cdot (\nabla \times \mathbf {\vec {B}} )\!\,.\end{aligned}}$

Alternativna metoda je raba kartezičnih komponent operatorja nabla na naslednji način (z implicitnim seštevanjem po indeksu $i\!\,$):

${\begin{aligned}\nabla \cdot (\mathbf {\vec {A}} \times \mathbf {\vec {B}} )&=\mathbf {\hat {e}} _{i}\partial _{i}\cdot (\mathbf {\vec {A}} \times \mathbf {\vec {B}} )\\[2pt]&=\mathbf {\hat {e}} _{i}\cdot \partial _{i}(\mathbf {\vec {A}} \times \mathbf {\vec {B}} )\\[2pt]&=\mathbf {\hat {e}} _{i}\cdot (\partial _{i}\mathbf {\vec {A}} \times \mathbf {\vec {B}} +\mathbf {\vec {A}} \times \partial _{i}\mathbf {\vec {B}} )\\[2pt]&=\mathbf {\hat {e}} _{i}\cdot (\partial _{i}\mathbf {\vec {A}} \times \mathbf {\vec {B}} )+\mathbf {\hat {e}} _{i}\cdot (\mathbf {\vec {A}} \times \partial _{i}\mathbf {\vec {B}} )\\[2pt]&=(\mathbf {\hat {e}} _{i}\times \partial _{i}\mathbf {\vec {A}} )\cdot \mathbf {\vec {B}} +(\mathbf {\hat {e}} _{i}\times \mathbf {\vec {A}} )\cdot \partial _{i}\mathbf {\vec {B}} \\[2pt]&=(\mathbf {\hat {e}} _{i}\times \partial _{i}\mathbf {\vec {A}} )\cdot \mathbf {\vec {B}} -(\mathbf {\vec {A}} \times \mathbf {\hat {e}} _{i})\cdot \partial _{i}\mathbf {\vec {B}} \\[2pt]&=(\mathbf {\hat {e}} _{i}\times \partial _{i}\mathbf {\vec {A}} )\cdot \mathbf {\vec {B}} -\mathbf {\vec {A}} \cdot (\mathbf {\hat {e}} _{i}\times \partial _{i}\mathbf {\vec {B}} )\\[2pt]&=(\mathbf {\hat {e}} _{i}\partial _{i}\times \mathbf {\vec {A}} )\cdot \mathbf {\vec {B}} -\mathbf {\vec {A}} \cdot (\mathbf {\hat {e}} _{i}\partial _{i}\times \mathbf {\vec {B}} )\\[2pt]&=(\nabla \times \mathbf {\vec {A}} )\cdot \mathbf {\vec {B}} -\mathbf {\vec {A}} \cdot (\nabla \times \mathbf {\vec {B}} )\!\,.\end{aligned}}$

Druga metoda za izpeljavo identitet vektorskih in tenzorskih odvodov je zamenjava vseh pojavitev vektorja v algebrski identiteti z operatorjem nabla, pod pogojem, da se nobena spremenljivka ne pojavi hkrati znotraj in zunaj obsega operatorja ali hkrati znotraj obsega enega operatorja v izrazu in zunaj obsega drugega operatorja v istem izrazu (tj. operatorji morajo biti ugnezdeni). Veljavnost tega pravila izhaja iz veljavnosti Feynmanove metode, saj se lahko vedno nadomesti podpisani operator nabla in nato takoj izpusti podpisani indeks pod pogojem pravila. Na primer, iz identitete $\mathbf {\vec {A}} \cdot (\mathbf {\vec {B}} \times \mathbf {\vec {C}} )=(\mathbf {\vec {A}} \times \mathbf {\vec {B}} )\cdot \mathbf {\vec {C}} \!\,$ se lahko izpelje $\mathbf {\vec {A}} \cdot (\nabla \times \mathbf {\vec {C}} )=(\mathbf {\vec {A}} \times \nabla )\cdot \mathbf {\vec {C}} \!\,$, ne pa $\nabla \cdot (\mathbf {\vec {B}} \times \mathbf {\vec {C}} )=(\nabla \times \mathbf {\vec {B}} )\cdot \mathbf {\vec {C}} \!\,$, niti se iz $\mathbf {\vec {A}} \cdot (\mathbf {\vec {B}} \times \mathbf {\vec {A}} )=0\!\,$ ne da izpeljati $\mathbf {\vec {A}} \cdot (\nabla \times \mathbf {\vec {A}} )=0\!\,$. Po drugi strani pa podpisani operator nabla deluje na vseh pojavitvah podpisanega indeksa v členu, tako da $\mathbf {\vec {A}} \cdot (\nabla _{\mathbf {\vec {A}} }\times \mathbf {\vec {A}} )=\nabla _{\mathbf {\vec {A}} }\cdot (\mathbf {\vec {A}} \times \mathbf {\vec {A}} )=\nabla \cdot (\mathbf {\vec {A}} \times \mathbf {\vec {A}} )=0\!\,$. Prav tako se iz $\mathbf {\vec {A}} \times (\mathbf {\vec {A}} \times \mathbf {\vec {C}} )=\mathbf {\vec {A}} (\mathbf {\vec {A}} \cdot \mathbf {\vec {C}} )-(\mathbf {\vec {A}} \cdot \mathbf {\vec {A}} )\mathbf {\vec {C}} \!\,$ lahko izpelje $\nabla \times (\nabla \times \mathbf {\vec {C}} )=\mathbf {\vec {\nabla }} (\nabla \cdot \mathbf {\vec {C}} )-\mathbf {\vec {\nabla }} ^{2}\mathbf {\vec {C}} \!\,$, vendar se iz $(\mathbf {\vec {A}} \psi )\cdot (\mathbf {\vec {A}} \phi )=(\mathbf {\vec {A}} \cdot \mathbf {\vec {A}} )(\psi \phi )\!\,$ morda ne bo dalo izpeljati $(\mathbf {\vec {\nabla }} \psi )\cdot (\mathbf {\vec {\nabla }} \phi )=\mathbf {\vec {\nabla }} ^{2}(\psi \phi )\!\,$.

Podpisani indeks $c\!\,$ na količini pomeni, da se začasno šteje za konstanto. Ker konstanta ni spremenljivka, se jo lahko, ko se uporabi pravilo zamenjave (glej prejšnji odstavek), za razliko od spremenljivke premakne v obseg operatorja nabla ali zunaj njega, kot v naslednjem primeru:

${\begin{aligned}\nabla \cdot (\mathbf {\vec {A}} \times \mathbf {\vec {B}} )&=\nabla \cdot (\mathbf {\vec {A}} \times \mathbf {\vec {B}} _{\mathrm {c} })+\nabla \cdot (\mathbf {\vec {A}} _{\mathrm {c} }\times \mathbf {\vec {B}} )\\[2pt]&=\nabla \cdot (\mathbf {\vec {A}} \times \mathbf {\vec {B}} _{\mathrm {c} })-\nabla \cdot (\mathbf {\vec {B}} \times \mathbf {\vec {A}} _{\mathrm {c} })\\[2pt]&=(\nabla \times \mathbf {\vec {A}} )\cdot \mathbf {\vec {B}} _{\mathrm {c} }-(\nabla \times \mathbf {\vec {B}} )\cdot \mathbf {A} _{\mathrm {c} }\\[2pt]&=(\nabla \times \mathbf {\vec {A}} )\cdot \mathbf {\vec {B}} -(\nabla \times \mathbf {\vec {B}} )\cdot \mathbf {\vec {A}} \!\,.\end{aligned}}$

Drug način, da se označi, da je količina konstanta, je, da se jo kot podpisani indeks doda obsegu operatorja nabla, kot sledi:

$\mathbf {\vec {\nabla }} \left(\mathbf {\vec {A}} \cdot \mathbf {\vec {B}} \right)_{\mathbf {\vec {A}} }=\mathbf {\vec {A}} {\times }\!\left(\nabla {\times }\mathbf {\vec {B}} \right)+\left(\mathbf {\vec {A}} \cdot \nabla \right)\mathbf {\vec {B}} \!\,.$

V preostalem delu tega članka bo po potrebi uporabljen Feynmanov podpisni zapis.

Identitete prvih odvodov

Za skalarni polji $\psi \!\,$, $\phi \!\,$ in vektorski polji $\mathbf {\vec {A}} \!\,$, $\mathbf {\vec {B}} \!\,$ obstajajo naslednje identitete odvodov:

Distributivnost

• $\mathbf {\vec {\nabla }} (\psi +\phi )=\mathbf {\vec {\nabla }} \psi +\mathbf {\vec {\nabla }} \phi \!\,,$
• $\mathbf {\vec {\nabla }} (\mathbf {\vec {A}} +\mathbf {\vec {B}} )=\mathbf {\vec {\nabla }} \mathbf {\vec {A}} +\mathbf {\vec {\nabla }} \mathbf {\vec {B}} \!\,,$
• $\nabla \cdot (\mathbf {\vec {A}} +\mathbf {\vec {B}} )=\nabla \cdot \mathbf {\vec {A}} +\nabla \cdot \mathbf {\vec {B}} \!\,,$
• $\nabla \times (\mathbf {\vec {A}} +\mathbf {\vec {B}} )=\nabla \times \mathbf {\vec {A}} +\nabla \times \mathbf {\vec {B}} \!\,.$

Asociativnost prvih odvodov

• $(\mathbf {\vec {A}} \cdot \mathbf {\vec {\nabla }} )\psi =\mathbf {\vec {A}} \cdot (\mathbf {\vec {\nabla }} \psi )\!\,,$
• $(\mathbf {\vec {A}} \cdot \mathbf {\vec {\nabla }} )\mathbf {\vec {B}} =\mathbf {\vec {A}} \cdot (\mathbf {\vec {\nabla }} \mathbf {\vec {B}} )\!\,,$
• $(\mathbf {\vec {A}} \times \mathbf {\vec {\nabla }} )\psi =\mathbf {\vec {A}} \times (\mathbf {\vec {\nabla }} \psi )\!\,,$
• $(\mathbf {\vec {A}} \times \mathbf {\vec {\nabla }} )\mathbf {\vec {B}} =\mathbf {\vec {A}} \times (\mathbf {\vec {\nabla }} \mathbf {\vec {B}} )\!\,.$

Pravilo produkta za množenje s skalarjem

Za odvod produkta v infinitezimalnem računu ene spremenljivke obstajajo naslednje posplošitve:

• $\mathbf {\vec {\nabla }} (\psi \phi )=\phi \,\mathbf {\vec {\nabla }} \psi +\psi \,\mathbf {\vec {\nabla }} \phi \!\,,$
• ${\begin{aligned}\mathbf {\vec {\nabla }} (\psi \mathbf {\vec {A}} )&=(\mathbf {\vec {\nabla }} \psi )\mathbf {\vec {A}} ^{\top }+\psi \mathbf {\vec {\nabla }} \mathbf {\vec {A}} \\&=\mathbf {\vec {\nabla }} \psi \otimes \mathbf {\vec {A}} +\psi \,\mathbf {\vec {\nabla }} \mathbf {\vec {A}} \!\,,\end{aligned}}$
• $\nabla \cdot (\psi \mathbf {\vec {A}} )=\psi \,\nabla \cdot \mathbf {\vec {A}} +(\mathbf {\vec {\nabla }} \psi )\,\cdot \mathbf {\vec {A}} \!\,,$
• $\nabla \times (\psi \mathbf {\vec {A}} )=\psi \,\nabla \times \mathbf {\vec {A}} +(\mathbf {\vec {\nabla }} \psi )\times \mathbf {\vec {A}} \!\,,$
• $\nabla ^{2}(\psi \phi )=\psi \,\mathbf {\vec {\nabla }} ^{2\!}\phi +2\,\mathbf {\vec {\nabla }} \!\psi \cdot \!\mathbf {\vec {\nabla }} \phi +\phi \,\mathbf {\vec {\nabla }} ^{2\!}\psi \!\,.$

Pravilo količnika za deljenje s skalarjem

• $\mathbf {\vec {\nabla }} \left({\frac {\psi }{\phi }}\right)={\frac {\phi \,\mathbf {\vec {\nabla }} \psi -\psi \,\mathbf {\vec {\nabla }} \phi }{\phi ^{2}}}\!\,$
• $\mathbf {\vec {\nabla }} \left({\frac {\mathbf {\vec {A}} }{\phi }}\right)={\frac {\phi \,\mathbf {\vec {\nabla }} \mathbf {\vec {A}} -\mathbf {\vec {\nabla }} \phi \otimes \mathbf {\vec {A}} }{\phi ^{2}}}\!\,,$
• $\nabla \cdot \left({\frac {\mathbf {\vec {A}} }{\phi }}\right)={\frac {\phi \,\nabla \cdot \mathbf {\vec {A}} -\mathbf {\vec {\nabla }} \!\phi \cdot \mathbf {\vec {A}} }{\phi ^{2}}}\!\,,$
• $\nabla \times \left({\frac {\mathbf {\vec {A}} }{\phi }}\right)={\frac {\phi \,\nabla \times \mathbf {\vec {A}} -\mathbf {\vec {\nabla }} \!\phi \,\times \,\mathbf {\vec {A}} }{\phi ^{2}}}\!\,,$
• $\mathbf {\vec {\nabla }} ^{2}\left({\frac {\psi }{\phi }}\right)={\frac {\phi \,\mathbf {\vec {\nabla }} ^{2\!}\psi -2\,\phi \,\mathbf {\vec {\nabla }} \!\left({\frac {\psi }{\phi }}\right)\cdot \!\mathbf {\vec {\nabla }} \phi -\psi \,\mathbf {\vec {\nabla }} ^{2\!}\phi }{\phi ^{2}}}\!\,.$

Odvod kompozituma

Naj je $f(x)\!\,$ funkcija ene spremenljvike od skalarjev do skalarjev $\mathbf {\vec {r}} (t)=(x_{1}(t),\ldots ,x_{n}(t))\!\,$ parametrizirana krivulja $\phi \colon \mathbb {R} ^{n}\to \mathbb {R} \!\,$ funkcija od vektorjev do skalarjev in $\mathbf {\vec {A}} \colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}\!\,$ vektorsko polje. Obstajajo naslednji posebni primeri odvoda kompozituma za več spremenljivk:

• $\mathbf {\vec {\nabla }} (f\circ \phi )=\left(f'\circ \phi \right)\mathbf {\vec {\nabla }} \phi \!\,,$
• $(\mathbf {\vec {r}} \circ f)'=(\mathbf {\vec {r}} '\circ f)f'\!\,,$
• $(\phi \circ \mathbf {\vec {r}} )'=(\mathbf {\vec {\nabla }} \phi \circ \mathbf {\vec {r}} )\cdot \mathbf {\vec {r}} '\!\,,$
• $(\mathbf {\vec {A}} \circ \mathbf {\vec {r}} )'=\mathbf {\vec {r}} '\cdot (\mathbf {\vec {\nabla }} \mathbf {\vec {A}} \circ \mathbf {\vec {r}} )\!\,,$
• $\mathbf {\vec {\nabla }} (\phi \circ \mathbf {\vec {A}} )=(\mathbf {\vec {\nabla }} \mathbf {\vec {A}} )\cdot (\mathbf {\vec {\nabla }} \phi \circ \mathbf {\vec {A}} )\!\,,$
• $\nabla \cdot (\mathbf {\vec {r}} \circ \phi )=\mathbf {\vec {\nabla }} \phi \cdot (\mathbf {\vec {r}} '\circ \phi )\!\,,$
• $\nabla \times (\mathbf {\vec {r}} \circ \phi )=\mathbf {\vec {\nabla }} \phi \times (\mathbf {\vec {r}} '\circ \phi )\!\,,$
• $\mathbf {\vec {\nabla }} (\mathbf {\vec {r}} \circ \phi )=\mathbf {\vec {\nabla }} \phi \otimes (\mathbf {\vec {r}} '\circ \phi )\!\,.$

Za vektorsko transformacijo $\mathbf {\vec {u}} \colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}\!\,$ je:

$\nabla \cdot (\mathbf {\vec {A}} \circ \mathbf {\vec {u}} )=\operatorname {tr} \left((\mathbf {\vec {\nabla }} \mathbf {\vec {u}} )\cdot (\mathbf {\vec {\nabla }} \mathbf {\vec {A}} \circ \mathbf {\vec {u}} )\right)\!\,.$

Tukaj se vzame sled skalarnega produkta dveh tenzorjev drugega reda, ki ustreza produktu njunih matrik.

Pravilo skalarnega produkta

${\begin{aligned}\mathbf {\vec {\nabla }} (\mathbf {\vec {A}} \cdot \mathbf {\vec {B}} )&\ =\ (\mathbf {\vec {A}} \cdot \nabla )\mathbf {\vec {B}} \,+\,(\mathbf {\vec {B}} \cdot \nabla )\mathbf {\vec {A}} \,+\,\mathbf {\vec {A}} \times (\nabla {\times }\mathbf {\vec {B}} )\,+\,\mathbf {\vec {B}} \times (\nabla \times \mathbf {\vec {A}} )\\&\ =\ \mathbf {\vec {A}} \cdot \mathbf {\vec {J}} _{\mathbf {\vec {B}} }+\mathbf {\vec {B}} \cdot \mathbf {\vec {J}} _{\mathbf {\vec {A}} }\ =\ (\mathbf {\vec {\nabla }} \mathbf {\vec {B}} )\cdot \mathbf {\vec {A}} \,+\,(\mathbf {\vec {\nabla }} \mathbf {\vec {A}} )\cdot \mathbf {\vec {B}} \!\,,\end{aligned}}$

kjer je $\mathbf {\vec {J}} _{\mathbf {\vec {A}} }=(\mathbf {\vec {\nabla }} \!\mathbf {A} )^{\top }=(\partial A_{i}/\partial x_{j})_{ij}\!\,$ označuje Jacobijevo matriko vektorskega polja $\mathbf {\vec {A}} =(A_{1},\ldots ,A_{n})\!\,$.

Druga možnost je raba Feynmanovega podpisnega zapisa:

$\nabla (\mathbf {\vec {A}} \cdot \mathbf {\vec {B}} )=\nabla _{\mathbf {\vec {A}} }(\mathbf {\vec {A}} \cdot \mathbf {\vec {B}} )+\nabla _{\mathbf {\vec {B}} }(\mathbf {\vec {A}} \cdot \mathbf {\vec {B}} )\!\,.$

Glej te opombe.

Kot posebni primer, ko je $\mathbf {\vec {A}} =\mathbf {\vec {B}} \!\,$:

${\tfrac {1}{2}}\nabla \left(\mathbf {\vec {A}} \cdot \mathbf {\vec {A}} \right)\ =\ \mathbf {\vec {A}} \cdot \mathbf {\vec {J}} _{\mathbf {\vec {A}} }\ =\ (\mathbf {\vec {\nabla }} \mathbf {\vec {A}} )\cdot \mathbf {\vec {A}} \ =\ (\mathbf {\vec {A}} {\cdot }\nabla )\mathbf {\vec {A}} \,+\,\mathbf {\vec {A}} \times (\nabla \times \mathbf {\vec {A}} )\ =\ A\mathbf {\vec {\nabla }} A\!\,.$

Posplošitev formule za skalarni produkt na Riemannove mnogoterosti je definirajoča lastnost Riemannove povezave, ki odvaja vektorsko polje in tako da vektorsko 1-formo.

Pravilo vektorskega produkta

• $\nabla (\mathbf {\vec {A}} \times \mathbf {\vec {B}} )\ =\ (\mathbf {\vec {\nabla }} \mathbf {\vec {A}} )\times \mathbf {\vec {B}} \,-\,(\mathbf {\vec {\nabla }} \mathbf {\vec {B}} )\times \mathbf {\vec {A}} \!\,,$
• $\nabla \cdot (\mathbf {\vec {A}} \times \mathbf {\vec {B}} )\ =\ (\nabla \times \mathbf {\vec {A}} )\cdot \mathbf {\vec {B}} \,-\,\mathbf {\vec {A}} \cdot (\nabla \times \mathbf {\vec {B}} )\!\,,$
• ${\begin{aligned}\nabla \times (\mathbf {\vec {A}} \times \mathbf {\vec {B}} )&\ =\ \mathbf {\vec {A}} (\nabla \cdot \mathbf {\vec {B}} )\,-\,\mathbf {\vec {B}} (\nabla \cdot \mathbf {\vec {A}} )\,+\,(\mathbf {\vec {B}} \cdot \nabla )\mathbf {\vec {A}} \,-\,(\mathbf {\vec {A}} \cdot \nabla )\mathbf {\vec {B}} \\[2pt]&\ =\ \mathbf {\vec {A}} (\nabla \cdot \mathbf {\vec {B}} )\,+\,(\mathbf {\vec {B}} \cdot \nabla )\mathbf {\vec {A}} \,-\,(\mathbf {\vec {B}} (\nabla \cdot \mathbf {\vec {A}} )\,+\,(\mathbf {\vec {A}} \cdot \nabla )\mathbf {\vec {B}} )\\[2pt]&\ =\ \nabla \cdot \left(\mathbf {\vec {B}} \mathbf {\vec {A}} ^{\top }\right)\,-\,\nabla \cdot \left(\mathbf {\vec {A}} \mathbf {\vec {B}} ^{\top }\right)\\[2pt]&\ =\ \nabla \cdot \left(\mathbf {\vec {B}} \mathbf {\vec {A}} ^{\top }\,-\,\mathbf {\vec {A}} \mathbf {\vec {B}} ^{\top }\right)\!\,,\end{aligned}}$
• ${\begin{aligned}\mathbf {\vec {A}} \times (\nabla \times \mathbf {\vec {B}} )&\ =\ \nabla _{\mathbf {\vec {B}} }(\mathbf {\vec {A}} \cdot \mathbf {\vec {B}} )\,-\,(\mathbf {\vec {A}} {\cdot }\nabla )\mathbf {\vec {B}} \\[2pt]&\ =\ \mathbf {\vec {A}} \cdot \mathbf {\vec {J}} _{\mathbf {\vec {B}} }\,-\,(\mathbf {\vec {A}} \cdot \nabla )\mathbf {\vec {B}} \\[2pt]&\ =\ (\mathbf {\vec {\nabla }} \mathbf {\vec {B}} )\cdot \mathbf {\vec {A}} \,-\,\mathbf {\vec {A}} \cdot (\mathbf {\vec {\nabla }} \mathbf {\vec {B}} )\\[2pt]&\ =\ \mathbf {\vec {A}} \cdot (\mathbf {\vec {J}} _{\mathbf {\vec {B}} }\,-\,\mathbf {\vec {J}} _{\mathbf {\vec {B}} }^{\top })\!\,,\end{aligned}}$
• ${\begin{aligned}(\mathbf {\vec {A}} \times \nabla )\times \mathbf {\vec {B}} &\ =\ (\mathbf {\vec {\nabla }} \mathbf {\vec {B}} )\cdot \mathbf {\vec {A}} \,-\,\mathbf {\vec {A}} (\nabla \cdot \mathbf {\vec {B}} )\\[2pt]&\ =\ \mathbf {\vec {A}} \times (\nabla \times \mathbf {\vec {B}} )\,+\,(\mathbf {\vec {A}} \cdot \nabla )\mathbf {\vec {B}} \,-\,\mathbf {\vec {A}} (\nabla \cdot \mathbf {\vec {B}} )\!\,,\end{aligned}}$
• $(\mathbf {\vec {A}} \times \nabla )\cdot \mathbf {\vec {B}} \ =\ \mathbf {\vec {A}} \cdot (\nabla {\times }\mathbf {\vec {B}} )\!\,.$

Upoštevati je treba, da je matrika $\mathbf {\vec {J}} _{\mathbf {\vec {B}} }\,-\,\mathbf {\vec {J}} _{\mathbf {\vec {B}} }^{\top }\!\,$ poševnosimetrična.

Identitete drugih odvodov

Divergenca rotorja je enaka nič

Divergenca rotorja poljubnega zvezno dvakrat odvedljivega vektorskega polja $\mathbf {\vec {A}} \!\,$ je vedno enaka nič:

$\nabla \cdot (\nabla \times \mathbf {\vec {A}} )=0\!\,.$

To je posebni primer izničenja kvadrata zunanjega odvoda v De Rhamovem verižnem kompleksu.

Divergenca gradienta je Laplaceov operator

Laplaceov operator skalarnega polja je divergenca njegovega gradienta:

$\Delta \psi \equiv \nabla ^{2}\psi =\nabla \cdot (\mathbf {\vec {\nabla }} \psi )\!\,.$

Rezultat je skalarna količina.

Divergenca divergence ni definirana

Divergenca vektorskega polja $\mathbf {\vec {A}} \!\,$ je skalar in divergenca skalarne količine je nedefinirana. Zato je:

$\nabla \cdot (\nabla \cdot \mathbf {\vec {A}} ){\text{ je nedefinirano.}}\!\,$

Rotor gradienta je enak nič

Rotor gradienta poljubnega zvezdno dvakrat odvedljivega skalarnega polja $\phi \!\,$ (to je razreda odvedljivosti $C^{2}\!\,$) je vedno ničelni vektor:

$\nabla \times (\mathbf {\vec {\nabla }} \phi )=\mathbf {\vec {0}} \!\,.$

To se lahko preprosto dokaže, če se izrazi $\nabla \times (\mathbf {\vec {\nabla }} \phi )\!\,$ v kartezičnem koordinatnem sistemu s Schwarzevim izrekom (imenovanim tudi Clairautov izrek o enakosti mešanih parcialov). Ta rezultat je posebni primer izničenja kvadrata zunanjega odvoda v De Rhamovem verižnem kompleksu.

Rotor rotorja

$\nabla \times \left(\nabla \times \mathbf {\vec {A}} \right)\ =\ \nabla (\nabla \cdot \mathbf {\vec {A}} )\,-\,\mathbf {\vec {\nabla }} ^{2\!}\mathbf {\vec {A}} \!\,.$

Tukaj je $\mathbf {\vec {\nabla }} ^{2}\!\,$ vektorski Laplaceov operator, ki deluje na vektorsko polje $\mathbf {\vec {A}} \!\,$.

Rotor divergence ni definiran

Divergenca vektorskega polja $\mathbf {\vec {A}} \!\,$ je skalar in rotor skalarne količine je nedefiniran. Zato je:

$\nabla \times (\nabla \cdot \mathbf {\vec {A}} ){\text{ je nedefinirano.}}\!\,$

Asociativnost drugih odvodov

• $(\nabla \cdot \nabla )\psi =\nabla \cdot (\mathbf {\vec {\nabla }} \psi )=\nabla ^{2}\psi \!\,,$
• $(\nabla \cdot \nabla )\mathbf {\vec {A}} =\nabla \cdot (\mathbf {\vec {\nabla }} \mathbf {\vec {A}} )=\mathbf {\vec {\nabla }} ^{2}\mathbf {\vec {A}} \!\,,$
• $(\nabla \times \nabla )\psi =\nabla \times (\mathbf {\vec {\nabla }} \psi )=\mathbf {\vec {0}} \!\,,$
• $(\nabla \times \nabla )\mathbf {\vec {A}} =\nabla \times (\mathbf {\vec {\nabla }} \mathbf {\vec {A}} )=\mathbf {\vec {0}} \!\,.$

Diagram DCG: Nekatera pravila za druge odvode.

Mnemonik

Slika na desni je mnemonik za nekatere od teh identitet. Rabljene okrajšave so:

• D: divergenca,
• C: rotor (angleško curl),
• G: gradient,
• L: Laplaceov operator,
• CC: rotor rotorja.

Vsaka puščica je označena z rezultatom identitete, natančneje z rezultatom rabe operatorja na repu puščice na operator na njenem vrhu. Modra krožnica na sredini pomeni, da operator rotor rotorja obstaja, drugi dve rdeči krožnici (črtkani) pa pomenita, da DD (divergenca divergence) in GG (gradient gradienta) ne obstajata.

Povzetek pomembnih identitet

Odvajanje

Gradient

• $\mathbf {\vec {\nabla }} (\psi +\phi )=\mathbf {\vec {\nabla }} \psi +\mathbf {\vec {\nabla }} \phi \!\,,$
• $\mathbf {\vec {\nabla }} (\psi \phi )=\phi \mathbf {\vec {\nabla }} \psi +\psi \mathbf {\vec {\nabla }} \phi \!\,,$
• $\mathbf {\vec {\nabla }} (\psi \mathbf {\vec {A}} )=\mathbf {\vec {\nabla }} \psi \otimes \mathbf {\vec {A}} +\psi \mathbf {\vec {\nabla }} \mathbf {\vec {A}} \!\,,$
• $\mathbf {\vec {\nabla }} (\mathbf {\vec {A}} \cdot \mathbf {\vec {B}} )=(\mathbf {\vec {A}} \cdot \nabla )\mathbf {\vec {B}} +(\mathbf {\vec {B}} \cdot \nabla )\mathbf {\vec {A}} +\mathbf {\vec {A}} \times (\nabla \times \mathbf {\vec {B}} )+\mathbf {\vec {B}} \times (\nabla \times \mathbf {\vec {A}} )\!\,.$

Divergenca

• $\nabla \cdot (\mathbf {\vec {A}} +\mathbf {\vec {B}} )=\nabla \cdot \mathbf {\vec {A}} +\nabla \cdot \mathbf {\vec {B}} \!\,,$
• $\nabla \cdot \left(\psi \mathbf {\vec {A}} \right)=\psi \nabla \cdot \mathbf {\vec {A}} +\mathbf {\vec {A}} \cdot \mathbf {\vec {\nabla }} \psi \!\,,$
• $\nabla \cdot \left(\mathbf {\vec {A}} \times \mathbf {\vec {B}} \right)=(\nabla \times \mathbf {\vec {A}} )\cdot \mathbf {\vec {B}} -(\nabla \times \mathbf {\vec {B}} )\cdot \mathbf {\vec {A}} \!\,.$

Rotor

• $\nabla \times (\mathbf {\vec {A}} +\mathbf {\vec {B}} )=\nabla \times \mathbf {\vec {A}} +\nabla \times \mathbf {\vec {B}} \!\,,$
• $\nabla \times \left(\psi \mathbf {\vec {A}} \right)=\psi \,(\nabla \times \mathbf {\vec {A}} )-(\mathbf {\vec {A}} \times \nabla )\psi =\psi \,(\nabla \times \mathbf {\vec {A}} )+(\mathbf {\vec {\nabla }} \psi )\times \mathbf {\vec {A}} \!\,,$
• $\nabla \times \left(\psi \mathbf {\vec {\nabla }} \phi \right)=\mathbf {\vec {\nabla }} \psi \times \mathbf {\vec {\nabla }} \phi \!\,,$
• $\nabla \times \left(\mathbf {\vec {A}} \times \mathbf {\vec {B}} \right)=\mathbf {\vec {A}} \left(\nabla \cdot \mathbf {\vec {B}} \right)-\mathbf {\vec {B}} \left(\nabla \cdot \mathbf {\vec {A}} \right)+\left(\mathbf {\vec {B}} \cdot \nabla \right)\mathbf {\vec {A}} -\left(\mathbf {\vec {A}} \cdot \nabla \right)\mathbf {\vec {B}} \!\,.$

Operator nabla vektorskega-skalarnega produkta

• ${\begin{aligned}\left(\mathbf {\vec {A}} \cdot \nabla \right)\mathbf {\vec {B}} ={}&{\hphantom {{}-{}}}{\tfrac {1}{2}}{\Bigl [}\nabla (\mathbf {\vec {A}} \cdot \mathbf {\vec {B}} )-\mathbf {\vec {B}} (\nabla \cdot \mathbf {\vec {A}} )+\mathbf {\vec {A}} (\nabla \cdot \mathbf {\vec {B}} ){\Bigr ]}\\&-{\tfrac {1}{2}}{\Bigl [}\nabla \times (\mathbf {\vec {A}} \times \mathbf {\vec {B}} )+\mathbf {\vec {B}} \times (\nabla \times \mathbf {\vec {A}} )+\mathbf {\vec {A}} \times (\nabla \times \mathbf {\vec {B}} ){\Big ]}\!\,,\end{aligned}}$
• ${\begin{aligned}\left(\mathbf {\vec {A}} \cdot \nabla \right)\mathbf {\vec {A}} &={\tfrac {1}{2}}\nabla \left|\mathbf {\vec {A}} \right|^{2}-\mathbf {\vec {A}} \times \left(\nabla \times \mathbf {\vec {A}} \right)\\&={\tfrac {1}{2}}\nabla \left|\mathbf {\vec {A}} \right|^{2}+(\nabla \times \mathbf {\vec {A}} )\times \mathbf {\vec {A}} \!\,,\end{aligned}}$
• $\mathbf {A} \cdot \nabla (\mathbf {\vec {B}} \cdot \mathbf {\vec {C}} )=\mathbf {\vec {B}} \cdot (\mathbf {\vec {A}} \cdot \nabla )\mathbf {\vec {C}} +\mathbf {\vec {C}} \cdot (\mathbf {\vec {A}} \cdot \nabla )\mathbf {\vec {B}} \!\,.$

Drugi odvodi

• $\nabla \cdot (\nabla \times \mathbf {\vec {A}} )=0\!\,,$
• $\nabla \times (\mathbf {\vec {\nabla }} \psi )=\mathbf {\vec {0}} \!\,$
• $\nabla \cdot (\mathbf {\vec {\nabla }} \psi )=\mathbf {\vec {\nabla }} ^{2}\psi \!\,$ (skalarni Laplaceov operator),
• $\nabla \left(\nabla \cdot \mathbf {\vec {A}} \right)-\nabla \times \left(\nabla \times \mathbf {\vec {A}} \right)=\mathbf {\vec {\nabla }} ^{2}\mathbf {\vec {A}} \!\,$ (vektorski Laplaceov operator),
• $\nabla \cdot {\big [}\mathbf {\vec {\nabla }} \mathbf {\vec {A}} +(\mathbf {\vec {\nabla }} \mathbf {A} )^{\top }{\big ]}=\mathbf {\vec {\nabla }} ^{2}\mathbf {\vec {A}} +\nabla (\nabla \cdot \mathbf {\vec {A}} )\!\,,$
• $\nabla \cdot (\phi \mathbf {\vec {\nabla }} \psi )=\phi \mathbf {\vec {\nabla }} ^{2}\psi +\mathbf {\vec {\nabla }} \phi \cdot \mathbf {\vec {\nabla }} \psi \!\,,$
• $\psi \mathbf {\vec {\nabla }} ^{2}\phi -\phi \mathbf {\vec {\nabla }} ^{2}\psi =\nabla \cdot \left(\psi \mathbf {\vec {\nabla }} \phi -\phi \mathbf {\vec {\nabla }} \psi \right)\!\,,$
• $\mathbf {\vec {\nabla }} ^{2}(\phi \psi )=\phi \mathbf {\vec {\nabla }} ^{2}\psi +2(\mathbf {\vec {\nabla }} \phi )\cdot (\mathbf {\vec {\nabla }} \psi )+\left(\mathbf {\vec {\nabla }} ^{2}\phi \right)\psi \!\,,$
• $\mathbf {\vec {\nabla }} ^{2}(\psi \mathbf {\vec {A}} )=\mathbf {\vec {A}} \mathbf {\vec {\nabla }} ^{2}\psi +2(\mathbf {\vec {\nabla }} \psi \cdot \nabla )\mathbf {\vec {A}} +\psi \mathbf {\vec {\nabla }} ^{2}\mathbf {\vec {A}} \!\,,$
• $\nabla \cdot {\big [}(\mathbf {\vec {A}} \cdot \nabla )\mathbf {\vec {B}} {\big ]}=(\mathbf {\vec {A}} \cdot \nabla )(\nabla \cdot \mathbf {\vec {B}} )+(\mathbf {\vec {\nabla }} \mathbf {\vec {A}} )\,{\underline {:}}\,(\mathbf {\vec {\nabla }} \mathbf {\vec {B}} )\!\,,$
• $\nabla \times \left[\left(\mathbf {\vec {A}} \cdot \nabla \right)\mathbf {\vec {A}} \right]+\left[\left(\nabla \times \mathbf {\vec {A}} \right)\cdot \nabla \right]\mathbf {\vec {A}} =\left[\left(\mathbf {\vec {A}} \cdot \nabla \right)+\left(\nabla \cdot \mathbf {\vec {A}} \right)\right]\left(\nabla \times \mathbf {\vec {A}} \right)\!\,,$
• $\mathbf {\vec {\nabla }} ^{2}(\mathbf {\vec {A}} \cdot \mathbf {\vec {B}} )=\mathbf {A} \cdot \mathbf {\vec {\nabla }} ^{2}\mathbf {\vec {B}} -\mathbf {B} \cdot \mathbf {\vec {\nabla }} ^{2}\!\mathbf {\vec {A}} +2\nabla \cdot \left((\mathbf {\vec {B}} \cdot \nabla )\mathbf {\vec {A}} +\mathbf {\vec {B}} \times (\nabla \times \mathbf {\vec {A}} )\right)\!\,$ (Greenova vektorska identiteta).

Tretji odvodi

• $\mathbf {\vec {\nabla }} ^{2}(\mathbf {\vec {\nabla }} \psi )=\nabla \left(\nabla \cdot (\mathbf {\vec {\nabla }} \psi )\right)=\nabla \left(\mathbf {\vec {\nabla }} ^{2}\psi \right)\!\,,$
• $\mathbf {\vec {\nabla }} ^{2}(\nabla \cdot \mathbf {\vec {A}} )=\nabla \cdot \left(\nabla (\nabla \cdot \mathbf {\vec {A}} )\right)=\nabla \cdot \left(\mathbf {\vec {\nabla }} ^{2}\mathbf {A} \right)\!\,,$
• $\mathbf {\vec {\nabla }} ^{2}(\nabla \times \mathbf {\vec {A}} )=-\nabla \times \left(\nabla \times (\nabla \times \mathbf {\vec {A}} )\right)=\nabla \times \left(\mathbf {\vec {\nabla }} ^{2}\mathbf {\vec {A}} \right)\!\,.$

Integriranje

Spodaj simbol $\partial \!\,$ označuje »mejo« ploskve ali trdnega telesa.

Ploskovnoprostorninski integrali

V naslednjih izrekih o ploskovnoprostorninskih integralih $V\!\,$ označuje trirazsežno prostornino z ustrezno dvorazsežno mejo $S=\partial V\!\,$ (zaprta ploskev):

• $\iint _{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\subset \!\supset \;\psi \,\operatorname {d} \!\mathbf {\vec {S}} =\iiint _{V}\mathbf {\vec {\nabla }} \psi \,\operatorname {d} \!V\!\,,$

• $\iint _{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\subset \!\supset \;\mathbf {\vec {A}} \cdot \operatorname {d} \!\mathbf {\vec {S}} =\iiint _{V}\nabla \cdot \mathbf {\vec {A}} \,\operatorname {d} \!V\!\,$ (izrek Gaussa in Ostrogradskega),

• $\iint _{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\subset \!\supset \;\mathbf {\vec {A}} \times \operatorname {d} \!\mathbf {\vec {S}} =-\iiint _{V}\nabla \times \mathbf {\vec {A}} \,\operatorname {d} \!V\!\,,$

• $\iint _{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\subset \!\supset \;\psi \mathbf {\vec {\nabla }} \phi \cdot \operatorname {d} \!\mathbf {\vec {S}} =\iiint _{V}\left(\psi \mathbf {\vec {\nabla }} ^{2}\!\phi +\mathbf {\vec {\nabla }} \phi \cdot \mathbf {\vec {\nabla }} \psi \right)\,\operatorname {d} \!V\!\,$ (Greenova prva identiteta),

• $\iint _{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\subset \!\supset \;\left(\psi \mathbf {\vec {\nabla }} \phi -\phi \mathbf {\vec {\nabla }} \psi \right)\cdot \operatorname {d} \!\mathbf {\vec {S}} =\iint _{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\subset \!\supset \;\left(\psi {\frac {\partial \phi }{\partial n}}-\phi {\frac {\partial \psi }{\partial n}}\right)\,\operatorname {d} \!\mathbf {\vec {S}} =\iiint _{V}\left(\psi \mathbf {\vec {\nabla }} ^{2}\phi -\phi \mathbf {\vec {\nabla }} ^{2}\psi \right)\,\operatorname {d} \!V\!\,$ (Greenova druga identiteta),

• $\iiint _{V}\mathbf {\vec {A}} \cdot \mathbf {\vec {\nabla }} \psi \,\operatorname {d} \!V=\iint _{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\subset \!\supset \;\psi \mathbf {\vec {A}} \cdot \operatorname {d} \!\mathbf {\vec {S}} -\iiint _{V}\psi \nabla \cdot \mathbf {\vec {A}} \,\operatorname {d} \!V\!\,$ (integracija po delih),

• $\iiint _{V}\psi \nabla \cdot \mathbf {\vec {A}} \,\operatorname {d} \!V=\iint _{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\subset \!\supset \;\psi \mathbf {\vec {A}} \cdot \operatorname {d} \!\mathbf {\vec {S}} -\iiint _{V}\mathbf {\vec {A}} \cdot \mathbf {\vec {\nabla }} \psi \,\operatorname {d} \!V\!\,$ (integracija po delih),

• $\iiint _{V}\mathbf {\vec {A}} \cdot \left(\nabla \times \mathbf {\vec {B}} \right)\,\operatorname {d} \!V=\iint _{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\subset \!\supset \;\left(\mathbf {\vec {A}} \times \mathbf {\vec {B}} \right)\cdot \operatorname {d} \!\mathbf {\vec {S}} +\iiint _{V}\left(\nabla \times \mathbf {\vec {A}} \right)\cdot \mathbf {\vec {B}} \,\operatorname {d} \!V\!\,$ (integracija po delih),

• $\iint _{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\subset \!\supset \;\mathbf {\vec {A}} \times \left(\operatorname {d} \!\mathbf {\vec {S}} \cdot \left(\mathbf {\vec {B}} \mathbf {\vec {C}} ^{\top }\right)\right)=\iiint _{V}\mathbf {\vec {A}} \times \left(\nabla \cdot \left(\mathbf {\vec {B}} \mathbf {\vec {C}} ^{\top }\right)\right)\,\operatorname {d} \!V+\iiint _{V}\mathbf {\vec {B}} \cdot \left(\mathbf {\vec {\nabla }} \mathbf {\vec {A}} \right)\times \mathbf {\vec {C}} \,\operatorname {d} \!V\!\,,$

• $\iiint _{V}\left(\nabla \cdot \mathbf {\vec {B}} +\mathbf {\vec {B}} \cdot \nabla \right)\mathbf {\vec {A}} \,\operatorname {d} \!V=\iint _{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\subset \!\supset \;\left(\mathbf {\vec {B}} \cdot \operatorname {d} \!S\right)\mathbf {\vec {A}} \!\,.$

Krivuljnoploskovni integrali

V naslednjih izrekih o krivuljnoploskovnih integralih $S\!\,$ označuje dvorazsežno odprto ploskev z ustrezno enorazsežno mejo $C=\partial S\!\,$ (zaprta krivulja):

• $\oint _{\partial S}\mathbf {\vec {A}} \cdot \operatorname {d} \!{\boldsymbol {\vec {\ell }}}\ =\ \iint _{S}\left(\nabla \times \mathbf {\vec {A}} \right)\cdot \operatorname {d} \!\mathbf {\vec {S}} \!\,$ (Stokesov izrek),
• $\oint _{\partial S}\psi \,\operatorname {d} \!{\boldsymbol {\vec {\ell }}}\ =\ -\iint _{S}\mathbf {\vec {\nabla }} \psi \times \operatorname {d} \!\mathbf {\vec {S}} \!\,,$
• $\oint _{\partial S}\mathbf {\vec {A}} \times \operatorname {d} \!{\boldsymbol {\vec {\ell }}}\ =\ -\iint _{S}\left(\mathbf {\vec {\nabla }} \mathbf {\vec {A}} -(\nabla \cdot \mathbf {\vec {A}} )\mathbf {\vec {1}} \right)\cdot \operatorname {d} \!\mathbf {\vec {S}} \ =\ -\iint _{S}\left(\operatorname {d} \!\mathbf {\vec {S}} \times \nabla \right)\times \mathbf {\vec {A}} \!\,,$
• $\oint _{\partial S}\mathbf {\vec {A}} \times (\mathbf {\vec {B}} \times \operatorname {d} \!{\boldsymbol {\vec {\ell }}})\ =\ \iint _{S}\left(\nabla \times \left(\mathbf {\vec {A}} \mathbf {\vec {B}} ^{\top }\right)\right)\cdot \operatorname {d} \!\mathbf {\vec {S}} +\iint _{S}\left(\nabla \cdot \left(\mathbf {\vec {B}} \mathbf {\vec {A}} ^{\top }\right)\right)\times \operatorname {d} \!\mathbf {\vec {S}} \!\,,$
• $\oint _{\partial S}(\mathbf {\vec {B}} \cdot \operatorname {d} \!{\boldsymbol {\vec {\ell }}})\mathbf {\vec {A}} =\iint _{S}\left(\operatorname {d} \!\mathbf {\vec {S}} \cdot \left[\nabla \times \mathbf {\vec {B}} -\mathbf {\vec {B}} \times \nabla \right]\right)\mathbf {\vec {A}} \!\,.$

Integracija okrog zaprte krivulje v smeri urinega kazalca je negativ istega krivuljnega integrala v nasprotni smeri urinega kazalca (analogno zamenjavi limit v določenem integralu):

${}_{\partial S}\!\,$ $\mathbf {\vec {A}} \cdot \operatorname {d} \!{\boldsymbol {\vec {\ell }}}=-\!\,$ ${}_{\partial S}\!\,$ $\mathbf {\vec {A}} \cdot \operatorname {d} \!{\boldsymbol {\vec {\ell }}}\!\,.$

Krivuljni integrali v končnih točkah

V naslednjih izrekih o krivuljnih integralih v končnih točkah $P\!\,$ označuje enorazsežno odprto pot s predznačenima ničrazsežnima mejnima točkama $\mathbf {\vec {q}} -\mathbf {\vec {p}} =\partial P\!\,$ in integracija vzdolž $P\!\,$ poteka od $\mathbf {\vec {p}} \!\,$ do $\mathbf {\vec {q}} \!\,$:

• $\psi |_{\partial P}=\psi (\mathbf {\vec {q}} )-\psi (\mathbf {\vec {p}} )=\int _{P}\mathbf {\vec {\nabla }} \psi \cdot \operatorname {d} \!{\boldsymbol {\vec {\ell }}}\!\,$ (izrek o gradientu),
• $\mathbf {\vec {A}} |_{\partial P}=\mathbf {\vec {A}} (\mathbf {\vec {q}} )-\mathbf {\vec {A}} (\mathbf {\vec {p}} )=\int _{P}\left(\operatorname {d} \!{\boldsymbol {\vec {\ell }}}\cdot \nabla \right)\mathbf {\vec {A}} \!\,,$
• $\mathbf {\vec {A}} |_{\partial P}=\mathbf {\vec {A}} (\mathbf {\vec {q}} )-\mathbf {\vec {A}} (\mathbf {\vec {p}} )=\int _{P}\left(\mathbf {\vec {\nabla }} \mathbf {\vec {A}} \right)\cdot \operatorname {d} \!{\boldsymbol {\vec {\ell }}}+\int _{P}\left(\nabla \times \mathbf {\vec {A}} \right)\times \operatorname {d} \!{\boldsymbol {\vec {\ell }}}\!\,.$

Tenzorski integrali

Tenzorsko obliko izreka o vektorskem integralu se lahko dobi tako, da se vektor (ali enega od njiju) zamenja s tenzorjem, pod pogojem, da se vektor najprej pojavi le kot skrajni desni vektor vsakega integranda. Stokesov izrek na primer postane:

$\oint _{\partial S}\operatorname {d} \!{\boldsymbol {\vec {\ell }}}\cdot \mathbf {T} \ =\ \iint _{S}\operatorname {d} \!\mathbf {\vec {S}} \cdot \left(\nabla \times \mathbf {T} \right)\!\,.$

Skalarno polje se lahko obravnava tudi kot vektor in se ga nadomesti z vektorjem ali tenzorjem. Greenova prva identiteta na primer postane:

$\iint _{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\subset \!\supset \;\psi \,\operatorname {d} \!\mathbf {\vec {S}} \cdot \mathbf {\vec {\nabla }} \mathbf {\vec {A}} =\iiint _{V}\left(\psi \mathbf {\vec {\nabla }} ^{2}\mathbf {\vec {A}} +\mathbf {\vec {\nabla }} \psi \cdot \mathbf {\vec {\nabla }} \mathbf {\vec {A}} \right)\,\operatorname {d} \!V\!\,.$

Podobna pravila veljajo za algebrske in diferencialne formule. Za algebrske formule se lahko alternativno rabi skrajno levo vektorsko lego.

Glej tudi

• primerjava vektorske in geometrijske algebre
• operator nabla v valjnih in krogelnih koordinatah
• pravila odvajanja
• identitete zunanjega infinitezimalnega računa
• zunanji odvod
• seznam limit
• relacije vektorske algebre

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• Wangsness, Roald K.; Cloud, Michael J. (1986), Electromagnetic Fields (2. izd.), New York [etc.]: John Wiley & Sons, str. 587, COBISS 62564, ISBN 978-0-47-181186-2, LCCN 85024642, OCLC 12808819, OL 2542944M
• Wilson, Edwin Bidwell (1901), Vector Analysis, New York: Charles Scribner's Sons

Nadaljnje branje

• Balanis, Constantine A. (23. maj 1989), Advanced Engineering Electromagnetics, New York: John Wiley & Sons, Incorporated, ISBN 978-0-47-162194-2, LCCN 88033846, OCLC 18908383, OL 2055547M
• Griffiths, David Jeffrey (1999), Introduction to Electrodynamics (3. izd.), Upper Saddle River, New Jersey: Prentice Hall, COBISS 44519169, ISBN 978-0-13-805326-0, LCCN 98050525, OCLC 40251748, OL 387513M – prek Internet Archive
• Schey, Harry M. (1997), Div Grad Curl and all that: An informal text on vector calculus (3. izd.), New York: W. W. Norton & Company, Incorporated, ISBN 978-0-39-396997-9, LCCN 96004942, OL 968439M