Operator Nabla

În matematică și în special în calculul vectorial și analiza matematică , simbolul nabla ( $\mathbf {\nabla }$ ${\ displaystyle \ mathbf {\ nabla}}$ $\ mathbf {\ nabla}$ ) este utilizat pentru un anumit operator diferențial de tip vector .

Termenul derivă din numele unui instrument muzical cu coarde din tradiția popoarelor antice din Palestina, nebelul sau nabla. Este un instrument tradițional asemănător cu o lira și o harpă , cu o cutie acustică, totuși, cu un profil triunghiular, care amintește de cel al unei delte inversate. ^[1] ^[2] Simbolul atled, într-un triunghi inversat, seamănă cu vechile harpe și lire din Ur .

În contextul operatorilor diferențiali, simbolul nebelului a fost folosit pentru prima dată de matematicianul și fizicianul irlandez William Rowan Hamilton , sub forma delta nebel înclinată. ⊲

În greacă, simbolul ανάδελτα, anádelta sau o deltă inversată amintește de harpele și lirele din Ur. Acest simbol este numit, foarte rar și numai în contextul american, de asemenea atled ("delta" citit înapoi ) datorită formei sale de delta ( $\Delta$ ${\ displaystyle \ Delta}$ $\ Delta$ ) inversat. Numele cel mai frecvent utilizat în literatura anglo-saxonă este însă „ del ”, adică prima parte a cuvântului „delta”: de fapt, delta (în mod corespunzător, cu numărul „2” în supercript) este adesea folosit pentru a indica laplacianul.

Notația diferențială bazată pe nabla permite indicarea, cu o notație foarte sintetică, a operatorilor diferențiali iacobieni , gradient , divergență și rotație .

Dacă spațiul vectorial în care acționează nabla este unidimensional, definiția nabla coincide cu derivata obișnuită .

Simbolul „nabla” este disponibil în cod HTML ca ∇ și în cod LaTeX ca \nabla . În codificarea Unicode este reprezentată în celula U + 2207 sau, în notație zecimală, 8711.

Definiție

Într-un spațiu tridimensional $\mathbb {R} ^{3}$ ${\ displaystyle \ mathbb {R} ^ {3}}$ $\ mathbb {R} ^ {3}$ generat de un sistem de coordonate cartezian $X, y, z$ ${\ displaystyle x, y, z}$ $x, y, z$ cu versorii indicați ${\hat {\mathbf {i} }}$ ${\ displaystyle {\ hat {\ mathbf {i}}}}$ $\ hat {\ mathbf {i}}$ , ${\hat {\mathbf {j} }}$ ${\ displaystyle {\ hat {\ mathbf {j}}}}$ $\ hat {\ mathbf {j}}$ Și ${\hat {\mathbf {k} }}$ ${\ displaystyle {\ hat {\ mathbf {k}}}}$ $\ hat {\ mathbf {k}}$ , nabla este definit ca:

\nabla \,\,{\stackrel {\Delta }{=}}\,\,{\hat {\mathbf {i} }}{\partial  \over \partial x}+{\hat {\mathbf {j} }}{\partial  \over \partial y}+{\hat {\mathbf {k} }}{\partial  \over \partial z}

{\ displaystyle \ nabla \, \, {\ stackrel {\ Delta} {=}} \, \, {\ hat {\ mathbf {i}}} {\ partial \ over \ partial x} + {\ hat {\ mathbf {j}}} {\ partial \ over \ partial y} + {\ hat {\ mathbf {k}}} {\ partial \ over \ partial z}}

{\ displaystyle \ nabla \, \, {\ stackrel {\ Delta} {=}} \, \, {\ hat {\ mathbf {i}}} {\ partial \ over \ partial x} + {\ hat {\ mathbf {j}}} {\ partial \ over \ partial y} + {\ hat {\ mathbf {k}}} {\ partial \ over \ partial z}}

Generalizarea pentru un spațiu $\mathbb {R} ^{n,m}$ ${\ displaystyle \ mathbb {R} ^ {n, m}}$ $\ mathbb {R} ^ {n, m}$ cu funcții de $n$ ${\ displaystyle n}$ $n$ variabile a $m$ ${\ displaystyle m}$ $m$ valori, este scris:

\nabla =\sum _{i=1}^{n}{\hat {\mathbf {x} }}_{i}{\frac {\partial }{\partial x_{i}}}

{\ displaystyle \ nabla = \ sum _ {i = 1} ^ {n} {\ hat {\ mathbf {x}}} _ {i} {\ frac {\ partial} {\ partial x_ {i}}}}

{\ displaystyle \ nabla = \ sum _ {i = 1} ^ {n} {\ hat {\ mathbf {x}}} _ {i} {\ frac {\ partial} {\ partial x_ {i}}}}

Utilizare

Acest operator vă permite să scrieți operatorii diferențiali ai gradientului , divergenței , rotorului , derivatei direcționale , Laplacianului folosind o notație compactă:

Gradient :

\operatorname {grad} f=\nabla f

{\ displaystyle \ operatorname {grad} f = \ nabla f}

{\ displaystyle \ operatorname {grad} f = \ nabla f}

Divergență :

\operatorname {div} {\vec {v}}=\nabla \cdot {\vec {v}}

{\ displaystyle \ operatorname {div} {\ vec {v}} = \ nabla \ cdot {\ vec {v}}}

{\ displaystyle \ operatorname {div} {\ vec {v}} = \ nabla \ cdot {\ vec {v}}}

Rotor :

\operatorname {rot} {\vec {v}}=\nabla \times {\vec {v}}

{\ displaystyle \ operatorname {rot} {\ vec {v}} = \ nabla \ times {\ vec {v}}}

{\ displaystyle \ operatorname {rot} {\ vec {v}} = \ nabla \ times {\ vec {v}}}

Laplacian :

\nabla \cdot \nabla f=\nabla ^{2}f

{\ displaystyle \ nabla \ cdot \ nabla f = \ nabla ^ {2} f}

\ nabla \ cdot \ nabla f = \ nabla ^ 2 f

unde este $f$ ${\ displaystyle f}$ $f$ este o funcție reală a uneia sau mai multor variabile reale, în timp ce ${\vec {v}}$ ${\ displaystyle {\ vec {v}}}$ $\ vec v$ este un câmp , adică o funcție vectorială a uneia sau mai multor variabile reale. Simbolul $\cdot$ ${\ displaystyle \ cdot}$ $\ cdot$ reprezintă produsul punct , în timp ce $\times$ ${\ displaystyle \ times}$ $\ ori$ produsul vector .

Acest lucru face mai ușor să scrieți ecuații diferențiale chiar complicate.

Definiție intrinsecă

Definiția dată mai sus este de fapt o definiție informală care depinde de sistemul de coordonate ales. Cu toate acestea, nabla poate fi definit cu o definiție intrinsecă mai generală, independentă de sistemul de coordonate:

\nabla \star f=\lim _{V\to 0}{\frac {1}{V}}\oint _{\partial V}\operatorname {d} \mathbf {S} \star f

{\ displaystyle \ nabla \ star f = \ lim _ {V \ to 0} {\ frac {1} {V}} \ oint _ {\ partial V} \ operatorname {d} \ mathbf {S} \ star f}

\ nabla \ star f = \ lim_ {V \ to 0} \ frac {1} {V} \ oint _ {\ partial V} \ operatorname d \ mathbf S \ star f

in care $\star$ ${\ displaystyle \ star}$ $\ stea$ reprezintă un produs arbitrar (scalar, vector, tensor sau pentru un scalar), în timp ce $f$ ${\ displaystyle f}$ $f$ este un câmp scalar, vector sau tensor. $\partial V$ ${\ displaystyle \ partial V}$ $\ parțial V$ este suprafața limită a volumului $V.$ ${\ displaystyle V}$ $V.$ care în limită se reduce la un punct. În acest fel putem defini intrinsec gradientul, divergența, rotorul și ceilalți operatori diferențiali.

Coordonate sferice

Ecuațiile care transformă coordonatele polare în coordonate carteziene sunt:

{\tilde {X}}(r,\theta ,\phi )\,=\,{\begin{cases}x=r\sin \theta \cos \phi \\y=r\sin \theta \sin \phi \\z=r\cos \theta \\\end{cases}}

{\ displaystyle {\ tilde {X}} (r, \ theta, \ phi) \, = \, {\ begin {cases} x = r \ sin \ theta \ cos \ phi \\ y = r \ sin \ theta \ sin \ phi \\ z = r \ cos \ theta \\\ end {cases}}}

{\ displaystyle {\ tilde {X}} (r, \ theta, \ phi) \, = \, {\ begin {cases} x = r \ sin \ theta \ cos \ phi \\ y = r \ sin \ theta \ sin \ phi \\ z = r \ cos \ theta \\\ end {cases}}}

Folosind regula de derivare a lanțului putem scrie:

{\frac {\partial }{\partial r}}={\frac {\partial x}{\partial r}}{\frac {\partial }{\partial x}}+{\frac {\partial y}{\partial r}}{\frac {\partial }{\partial y}}+{\frac {\partial z}{\partial r}}{\frac {\partial }{\partial z}}

{\ displaystyle {\ frac {\ partial} {\ partial r}} = {\ frac {\ partial x} {\ partial r}} {\ frac {\ partial} {\ partial x}} + {\ frac {\ partial y} {\ partial r}} {\ frac {\ partial} {\ partial y}} + {\ frac {\ partial z} {\ partial r}} {\ frac {\ partial} {\ partial z}} }

\ frac {\ partial} {\ partial r} = \ frac {\ partial x} {\ partial r} \ frac {\ partial} {\ partial x} + \ frac {\ partial y} {\ partial r} \ frac {\ partial} {\ partial y} + \ frac {\ partial z} {\ partial r} \ frac {\ partial} {\ partial z}

{\frac {\partial }{\partial \theta }}={\frac {\partial x}{\partial \theta }}{\frac {\partial }{\partial x}}+{\frac {\partial y}{\partial \theta }}{\frac {\partial }{\partial y}}+{\frac {\partial z}{\partial \theta }}{\frac {\partial }{\partial z}}

{\ displaystyle {\ frac {\ partial} {\ partial \ theta}} = {\ frac {\ partial x} {\ partial \ theta}} {\ frac {\ partial} {\ partial x}} + {\ frac {\ partial y} {\ partial \ theta}} {\ frac {\ partial} {\ partial y}} + {\ frac {\ partial z} {\ partial \ theta}} {\ frac {\ partial} {\ parțial z}}}

\ frac {\ partial} {\ partial \ theta} = \ frac {\ partial x} {\ partial \ theta} \ frac {\ partial} {\ partial x} + \ frac {\ partial y} {\ partial \ theta } \ frac {\ partial} {\ partial y} + \ frac {\ partial z} {\ partial \ theta} \ frac {\ partial} {\ partial z}

{\frac {\partial }{\partial \phi }}={\frac {\partial x}{\partial \phi }}{\frac {\partial }{\partial x}}+{\frac {\partial y}{\partial \phi }}{\frac {\partial }{\partial y}}+{\frac {\partial z}{\partial \phi }}{\frac {\partial }{\partial z}}

{\ displaystyle {\ frac {\ partial} {\ partial \ phi}} = {\ frac {\ partial x} {\ partial \ phi}} {\ frac {\ partial} {\ partial x}} + {\ frac {\ partial y} {\ partial \ phi}} {\ frac {\ partial} {\ partial y}} + {\ frac {\ partial z} {\ partial \ phi}} {\ frac {\ partial} {\ parțial z}}}

\ frac {\ partial} {\ partial \ phi} = \ frac {\ partial x} {\ partial \ phi} \ frac {\ partial} {\ partial x} + \ frac {\ partial y} {\ partial \ phi } \ frac {\ partial} {\ partial y} + \ frac {\ partial z} {\ partial \ phi} \ frac {\ partial} {\ partial z}

același lucru, folosind notația cu matrici și vectori, scriem:

{\begin{pmatrix}{\frac {\partial }{\partial r}}\\{\frac {\partial }{\partial \theta }}\\{\frac {\partial }{\partial \phi }}\end{pmatrix}}={\begin{pmatrix}\sin \theta \cos \phi &\sin \theta \sin \phi &\cos \theta \\r\cos \theta \cos \phi &r\cos \theta \sin \phi &-r\sin \theta \\-r\sin \theta \sin \phi &r\sin \theta \cos \phi &0\end{pmatrix}}{\begin{pmatrix}{\frac {\partial }{\partial x}}\\{\frac {\partial }{\partial y}}\\{\frac {\partial }{\partial z}}\end{pmatrix}}

{\ displaystyle {\ begin {pmatrix} {\ frac {\ partial} {\ partial r}} \\ {\ frac {\ partial} {\ partial \ theta}} \\ {\ frac {\ partial} {\ partial \ phi}} \ end {pmatrix}} = {\ begin {pmatrix} \ sin \ theta \ cos \ phi & \ sin \ theta \ sin \ phi & \ cos \ theta \\ r \ cos \ theta \ cos \ phi & r \ cos \ theta \ sin \ phi & -r \ sin \ theta \\ - r \ sin \ theta \ sin \ phi & r \ sin \ theta \ cos \ phi & 0 \ end {pmatrix}} {\ begin {pmatrix} {\ frac {\ partial} {\ partial x}} \\ {\ frac {\ partial} {\ partial y}} \\ {\ frac {\ partial} {\ partial z}} \ end {pmatrix }}}

{\ displaystyle {\ begin {pmatrix} {\ frac {\ partial} {\ partial r}} \\ {\ frac {\ partial} {\ partial \ theta}} \\ {\ frac {\ partial} {\ partial \ phi}} \ end {pmatrix}} = {\ begin {pmatrix} \ sin \ theta \ cos \ phi & \ sin \ theta \ sin \ phi & \ cos \ theta \\ r \ cos \ theta \ cos \ phi & r \ cos \ theta \ sin \ phi & -r \ sin \ theta \\ - r \ sin \ theta \ sin \ phi & r \ sin \ theta \ cos \ phi & 0 \ end {pmatrix}} {\ begin {pmatrix} {\ frac {\ partial} {\ partial x}} \\ {\ frac {\ partial} {\ partial y}} \\ {\ frac {\ partial} {\ partial z}} \ end {pmatrix }}}

sau chiar într-o formă mai compactă:

\nabla _{r}\,=\,\left({\frac {\partial {\tilde {X}}}{\partial (r,\theta ,\phi )}}\right)^{T}\nabla \,=\,AB\nabla

{\ displaystyle \ nabla _ {r} \, = \, \ left ({\ frac {\ partial {\ tilde {X}}} {\ partial (r, \ theta, \ phi)}} \ right) ^ { T} \ nabla \, = \, AB \ nabla}

{\ displaystyle \ nabla _ {r} \, = \, \ left ({\ frac {\ partial {\ tilde {X}}} {\ partial (r, \ theta, \ phi)}} \ right) ^ { T} \ nabla \, = \, AB \ nabla}

după ce am definit:

\nabla _{r}={\begin{pmatrix}{\frac {\partial }{\partial r}}\\{\frac {\partial }{\partial \theta }}\\{\frac {\partial }{\partial \phi }}\end{pmatrix}}

{\ displaystyle \ nabla _ {r} = {\ begin {pmatrix} {\ frac {\ partial} {\ partial r}} \\ {\ frac {\ partial} {\ partial \ theta}} \\ {\ frac {\ partial} {\ partial \ phi}} \ end {pmatrix}}}

\ nabla_ {r} = \ begin {pmatrix} \ frac {\ partial} {\ partial r} \\ \ frac {\ partial} {\ partial \ theta} \\ \ frac {\ partial} {\ partial \ phi} \ end {pmatrix}

\nabla ={\begin{pmatrix}{\frac {\partial }{\partial x}}\\{\frac {\partial }{\partial y}}\\{\frac {\partial }{\partial z}}\end{pmatrix}}

{\ displaystyle \ nabla = {\ begin {pmatrix} {\ frac {\ partial} {\ partial x}} \\ {\ frac {\ partial} {\ partial y}} \\ {\ frac {\ partial} { \ partial z}} \ end {pmatrix}}}

\ nabla = \ begin {pmatrix} \ frac {\ partial} {\ partial x} \\ \ frac {\ partial} {\ partial y} \\ \ frac {\ partial} {\ partial z} \ end {pmatrix}

A={\begin{pmatrix}1&0&0\\0&r&0\\0&0&r\sin \theta \end{pmatrix}}

{\ displaystyle A = {\ begin {pmatrix} 1 & 0 & 0 \\ 0 & r & 0 \\ 0 & 0 & r \ sin \ theta \ end {pmatrix}}}

{\ displaystyle A = {\ begin {pmatrix} 1 & 0 & 0 \\ 0 & r & 0 \\ 0 & 0 & r \ sin \ theta \ end {pmatrix}}}

B={\begin{pmatrix}\sin \theta \,\cos \phi &\sin \theta \,\sin \phi &\cos \theta \\\cos \theta \,\cos \phi &\cos \theta \,\sin \phi &-\sin \theta \\-\sin \phi &\cos \phi &0\end{pmatrix}}

{\ displaystyle B = {\ begin {pmatrix} \ sin \ theta \, \ cos \ phi & \ sin \ theta \, \ sin \ phi & \ cos \ theta \\\ cos \ theta \, \ cos \ phi & \ cos \ theta \, \ sin \ phi & - \ sin \ theta \\ - \ sin \ phi & \ cos \ phi & 0 \ end {pmatrix}}}

B = \ begin {pmatrix} \ sin \ theta \, \ cos \ phi & \ sin \ theta \, \ sin \ phi & \ cos \ theta \\ \ cos \ theta \, \ cos \ phi & \ cos \ theta \, \ sin \ phi & - \ sin \ theta \\ - \ sin \ phi & \ cos \ phi & 0 \ end {pmatrix}

Rețineți că:

A^{-1}={\begin{pmatrix}1&0&0\\0&{\frac {1}{r}}&0\\0&0&{\frac {1}{r\sin \theta }}\end{pmatrix}}

{\ displaystyle A ^ {- 1} = {\ begin {pmatrix} 1 & 0 & 0 \\ 0 & {\ frac {1} {r}} & 0 \\ 0 & 0 & {\ frac {1} { r \ sin \ theta}} \ end {pmatrix}}}

{\ displaystyle A ^ {- 1} = {\ begin {pmatrix} 1 & 0 & 0 \\ 0 & {\ frac {1} {r}} & 0 \\ 0 & 0 & {\ frac {1} { r \ sin \ theta}} \ end {pmatrix}}}

B^{-1}=B^{T}=\left(b_{1},b_{2},b_{3}\right)

{\ displaystyle B ^ {- 1} = B ^ {T} = \ left (b_ {1}, b_ {2}, b_ {3} \ right)}

{\ displaystyle B ^ {- 1} = B ^ {T} = \ left (b_ {1}, b_ {2}, b_ {3} \ right)}

unde sunt indicate cu $b_{i}$ ${\ displaystyle b_ {i}}$ $bi$ versorii (ortonormali) ai bazei spațiului tangent la sferă

b_{1}={\hat {r}}={\begin{pmatrix}\sin \theta \cos \phi \\\sin \theta \sin \phi \\\cos \theta \end{pmatrix}}

{\ displaystyle b_ {1} = {\ hat {r}} = {\ begin {pmatrix} \ sin \ theta \ cos \ phi \\\ sin \ theta \ sin \ phi \\\ cos \ theta \ end {pmatrix }}}

{\ displaystyle b_ {1} = {\ hat {r}} = {\ begin {pmatrix} \ sin \ theta \ cos \ phi \\\ sin \ theta \ sin \ phi \\\ cos \ theta \ end {pmatrix }}}

b_{2}={\hat {\theta }}={\begin{pmatrix}\cos \theta \cos \phi \\\cos \theta \sin \phi \\-\sin \theta \end{pmatrix}}

{\ displaystyle b_ {2} = {\ hat {\ theta}} = {\ begin {pmatrix} \ cos \ theta \ cos \ phi \\\ cos \ theta \ sin \ phi \\ - \ sin \ theta \ end {pmatrix}}}

{\ displaystyle b_ {2} = {\ hat {\ theta}} = {\ begin {pmatrix} \ cos \ theta \ cos \ phi \\\ cos \ theta \ sin \ phi \\ - \ sin \ theta \ end {pmatrix}}}

b_{3}={\hat {\phi }}={\begin{pmatrix}-\sin \phi \\\cos \phi \\0\end{pmatrix}}

{\ displaystyle b_ {3} = {\ hat {\ phi}} = {\ begin {pmatrix} - \ sin \ phi \\\ cos \ phi \\ 0 \ end {pmatrix}}}

{\ displaystyle b_ {3} = {\ hat {\ phi}} = {\ begin {pmatrix} - \ sin \ phi \\\ cos \ phi \\ 0 \ end {pmatrix}}}

b_{2}={\frac {\partial b_{1}}{\partial \theta }}

{\ displaystyle b_ {2} = {\ frac {\ partial b_ {1}} {\ partial \ theta}}}

b_ {2} = \ frac {\ partial b_ {1}} {\ partial \ theta}

b_{3}={\frac {1}{\sin \theta }}{\frac {\partial b_{1}}{\partial \phi }}={\frac {1}{\cos \theta }}{\frac {\partial b_{2}}{\partial \phi }}

{\ displaystyle b_ {3} = {\ frac {1} {\ sin \ theta}} {\ frac {\ partial b_ {1}} {\ partial \ phi}} = {\ frac {1} {\ cos \ theta}} {\ frac {\ partial b_ {2}} {\ partial \ phi}}}

b_ {3} = \ frac {1} {\ sin \ theta} \ frac {\ partial b_ {1}} {\ partial \ phi} = \ frac {1} {\ cos \ theta} \ frac {\ partial b_ {2}} {\ partial \ phi}

b_{i}\cdot b_{j}=\delta _{ij}

{\ displaystyle b_ {i} \ cdot b_ {j} = \ delta _ {ij}}

b_ {i} \ cdot b_ {j} = \ delta_ {ij}

b_{i}\times b_{j}=\epsilon _{ijk}b_{k}

{\ displaystyle b_ {i} \ times b_ {j} = \ epsilon _ {ijk} b_ {k}}

b_ {i} \ times b_ {j} = \ epsilon_ {ijk} b_ {k}

Cu cele de mai sus, operatorul de gradient în coordonate polare este exprimat:

\nabla \,=\,\left({\frac {\partial {\tilde {X}}}{\partial {(r,\theta ,\phi )}}}\right)^{-T}\nabla _{r}\,=\,B^{-1}A^{-1}\nabla _{r}=b_{1}{\frac {\partial }{\partial r}}+b_{2}{\frac {1}{r}}{\frac {\partial }{\partial \theta }}+b_{3}{\frac {1}{r\,\sin \theta }}{\frac {\partial }{\partial \phi }}\ ={\hat {r}}{\frac {\partial }{\partial r}}+{\frac {\hat {\theta }}{r}}{\frac {\partial }{\partial \theta }}+{\frac {\hat {\phi }}{r\,\sin \theta }}{\frac {\partial }{\partial \phi }}\

{\ displaystyle \ nabla \, = \, \ left ({\ frac {\ partial {\ tilde {X}}} {\ partial {(r, \ theta, \ phi)}}} \ right) ^ {- T } \ nabla _ {r} \, = \, B ^ {- 1} A ^ {- 1} \ nabla _ {r} = b_ {1} {\ frac {\ partial} {\ partial r}} + b_ {2} {\ frac {1} {r}} {\ frac {\ partial} {\ partial \ theta}} + b_ {3} {\ frac {1} {r \, \ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} \ = {\ hat {r}} {\ frac {\ partial} {\ partial r}} + {\ frac {\ hat {\ theta}} {r}} {\ frac {\ partial} {\ partial \ theta}} + {\ frac {\ hat {\ phi}} {r \, \ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} \}

{\ displaystyle \ nabla \, = \, \ left ({\ frac {\ partial {\ tilde {X}}} {\ partial {(r, \ theta, \ phi)}}} \ right) ^ {- T } \ nabla _ {r} \, = \, B ^ {- 1} A ^ {- 1} \ nabla _ {r} = b_ {1} {\ frac {\ partial} {\ partial r}} + b_ {2} {\ frac {1} {r}} {\ frac {\ partial} {\ partial \ theta}} + b_ {3} {\ frac {1} {r \, \ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} \ = {\ hat {r}} {\ frac {\ partial} {\ partial r}} + {\ frac {\ hat {\ theta}} {r}} {\ frac {\ partial} {\ partial \ theta}} + {\ frac {\ hat {\ phi}} {r \, \ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} \}

Avem:

b_{1}{\frac {\partial }{\partial r}}\cdot b_{1}{\frac {\partial }{\partial r}}=b_{1}\cdot \left({\frac {\partial b_{1}}{\partial r}}\right){\frac {\partial }{\partial r}}+b_{1}\cdot b_{1}{\frac {\partial }{\partial r}}{\frac {\partial }{\partial r}}={\frac {\partial ^{2}}{\partial r^{2}}}

{\ displaystyle b_ {1} {\ frac {\ partial} {\ partial r}} \ cdot b_ {1} {\ frac {\ partial} {\ partial r}} = b_ {1} \ cdot \ left ({ \ frac {\ partial b_ {1}} {\ partial r}} \ right) {\ frac {\ partial} {\ partial r}} + b_ {1} \ cdot b_ {1} {\ frac {\ partial} {\ partial r}} {\ frac {\ partial} {\ partial r}} = {\ frac {\ partial ^ {2}} {\ partial r ^ {2}}}}

b_ {1} \ frac {\ partial} {\ partial r} \ cdot b_ {1} \ frac {\ partial} {\ partial r} = b_ {1} \ cdot \ left (\ frac {\ partial b_ {1 }} {\ partial r} \ right) \ frac {\ partial} {\ partial r} + b_ {1} \ cdot b_ {1} \ frac {\ partial} {\ partial r} \ frac {\ partial} { \ partial r} = \ frac {\ partial ^ {2}} {\ partial r ^ {2}}

b_{1}{\frac {\partial }{\partial r}}\cdot b_{2}{\frac {1}{r}}{\frac {\partial }{\partial \theta }}=b_{1}\cdot b_{2}{\frac {\partial }{\partial r}}{\frac {1}{r}}{\frac {\partial }{\partial \theta }}=0

{\ displaystyle b_ {1} {\ frac {\ partial} {\ partial r}} \ cdot b_ {2} {\ frac {1} {r}} {\ frac {\ partial} {\ partial \ theta}} = b_ {1} \ cdot b_ {2} {\ frac {\ partial} {\ partial r}} {\ frac {1} {r}} {\ frac {\ partial} {\ partial \ theta}} = 0 }

b_ {1} \ frac {\ partial} {\ partial r} \ cdot b_ {2} \ frac {1} {r} \ frac {\ partial} {\ partial \ theta} = b_ {1} \ cdot b_ { 2} \ frac {\ partial} {\ partial r} \ frac {1} {r} \ frac {\ partial} {\ partial \ theta} = 0

b_{1}{\frac {\partial }{\partial r}}\cdot b_{3}{\frac {1}{r\,\sin \theta }}{\frac {\partial }{\partial \phi }}=b_{1}\cdot b_{3}{\frac {\partial }{\partial r}}{\frac {1}{r\,\sin \theta }}{\frac {\partial }{\partial \phi }}=0

{\ displaystyle b_ {1} {\ frac {\ partial} {\ partial r}} \ cdot b_ {3} {\ frac {1} {r \, \ sin \ theta}} {\ frac {\ partial} { \ partial \ phi}} = b_ {1} \ cdot b_ {3} {\ frac {\ partial} {\ partial r}} {\ frac {1} {r \, \ sin \ theta}} {\ frac { \ partial} {\ partial \ phi}} = 0}

b_ {1} \ frac {\ partial} {\ partial r} \ cdot b_ {3} \ frac {1} {r \, \ sin \ theta} \ frac {\ partial} {\ partial \ phi} = b_ { 1} \ cdot b_ {3} \ frac {\ partial} {\ partial r} \ frac {1} {r \, \ sin \ theta} \ frac {\ partial} {\ partial \ phi} = 0

b_{2}{\frac {1}{r}}{\frac {\partial }{\partial \theta }}\cdot b_{1}{\frac {\partial }{\partial r}}=b_{2}\cdot \left({\frac {\partial b_{1}}{\partial \theta }}\right){\frac {1}{r}}{\frac {\partial }{\partial r}}+b_{2}\cdot b_{1}{\frac {1}{r}}{\frac {\partial }{\partial \theta }}{\frac {\partial }{\partial r}}={\frac {1}{r}}{\frac {\partial }{\partial r}}

{\ displaystyle b_ {2} {\ frac {1} {r}} {\ frac {\ partial} {\ partial \ theta}} \ cdot b_ {1} {\ frac {\ partial} {\ partial r}} = b_ {2} \ cdot \ left ({\ frac {\ partial b_ {1}} {\ partial \ theta}} \ right) {\ frac {1} {r}} {\ frac {\ partial} {\ partial r}} + b_ {2} \ cdot b_ {1} {\ frac {1} {r}} {\ frac {\ partial} {\ partial \ theta}} {\ frac {\ partial} {\ partial r }} = {\ frac {1} {r}} {\ frac {\ partial} {\ partial r}}}

b_ {2} \ frac {1} {r} \ frac {\ partial} {\ partial \ theta} \ cdot b_ {1} \ frac {\ partial} {\ partial r} = b_ {2} \ cdot \ left (\ frac {\ partial b_ {1}} {\ partial \ theta} \ right) \ frac {1} {r} \ frac {\ partial} {\ partial r} + b_ {2} \ cdot b_ {1} \ frac {1} {r} \ frac {\ partial} {\ partial \ theta} \ frac {\ partial} {\ partial r} = \ frac {1} {r} \ frac {\ partial} {\ partial r }

b_{2}{\frac {1}{r}}{\frac {\partial }{\partial \theta }}\cdot b_{2}{\frac {1}{r}}{\frac {\partial }{\partial \theta }}=b_{2}\cdot \left({\frac {\partial b_{2}}{\partial \theta }}\right){\frac {1}{r}}{\frac {1}{r}}{\frac {\partial }{\partial \theta }}+b_{2}\cdot b_{2}{\frac {1}{r}}{\frac {\partial }{\partial \theta }}{\frac {1}{r}}{\frac {\partial }{\partial \theta }}={\frac {1}{r^{2}}}{\frac {\partial ^{2}}{\partial \theta ^{2}}}

{\ displaystyle b_ {2} {\ frac {1} {r}} {\ frac {\ partial} {\ partial \ theta}} \ cdot b_ {2} {\ frac {1} {r}} {\ frac {\ partial} {\ partial \ theta}} = b_ {2} \ cdot \ left ({\ frac {\ partial b_ {2}} {\ partial \ theta}} \ right) {\ frac {1} {r }} {\ frac {1} {r}} {\ frac {\ partial} {\ partial \ theta}} + b_ {2} \ cdot b_ {2} {\ frac {1} {r}} {\ frac {\ partial} {\ partial \ theta}} {\ frac {1} {r}} {\ frac {\ partial} {\ partial \ theta}} = {\ frac {1} {r ^ {2}}} {\ frac {\ partial ^ {2}} {\ partial \ theta ^ {2}}}}

b_ {2} \ frac {1} {r} \ frac {\ partial} {\ partial \ theta} \ cdot b_ {2} \ frac {1} {r} \ frac {\ partial} {\ partial \ theta} = b_ {2} \ cdot \ left (\ frac {\ partial b_ {2}} {\ partial \ theta} \ right) \ frac {1} {r} \ frac {1} {r} \ frac {\ partial } {\ partial \ theta} + b_ {2} \ cdot b_ {2} \ frac {1} {r} \ frac {\ partial} {\ partial \ theta} \ frac {1} {r} \ frac {\ partial} {\ partial \ theta} = \ frac {1} {r ^ {2}} \ frac {\ partial ^ {2}} {\ partial \ theta ^ {2}}

b_{2}{\frac {1}{r}}{\frac {\partial }{\partial \theta }}\cdot b_{3}{\frac {1}{r\,\sin \theta }}{\frac {\partial }{\partial \phi }}=b_{2}\cdot b_{3}{\frac {1}{r}}{\frac {\partial }{\partial \theta }}{\frac {1}{r\,\sin \theta }}{\frac {\partial }{\partial \phi }}=0

{\ displaystyle b_ {2} {\ frac {1} {r}} {\ frac {\ partial} {\ partial \ theta}} \ cdot b_ {3} {\ frac {1} {r \, \ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} = b_ {2} \ cdot b_ {3} {\ frac {1} {r}} {\ frac {\ partial} {\ partial \ theta }} {\ frac {1} {r \, \ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} = 0}

b_ {2} \ frac {1} {r} \ frac {\ partial} {\ partial \ theta} \ cdot b_ {3} \ frac {1} {r \, \ sin \ theta} \ frac {\ partial} {\ partial \ phi} = b_ {2} \ cdot b_ {3} \ frac {1} {r} \ frac {\ partial} {\ partial \ theta} \ frac {1} {r \, \ sin \ theta } \ frac {\ partial} {\ partial \ phi} = 0

b_{3}{\frac {1}{r\,\sin \theta }}{\frac {\partial }{\partial \phi }}\cdot b_{1}{\frac {\partial }{\partial r}}=b_{3}\cdot \left({\frac {1}{\sin \theta }}{\frac {\partial b_{1}}{\partial \phi }}\right){\frac {1}{r}}{\frac {\partial }{\partial r}}+b_{3}\cdot b_{1}{\frac {1}{r\,\sin \theta }}{\frac {\partial }{\partial \phi }}{\frac {\partial }{\partial r}}={\frac {1}{r}}{\frac {\partial }{\partial r}}

{\ displaystyle b_ {3} {\ frac {1} {r \, \ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} \ cdot b_ {1} {\ frac {\ partial} {\ partial r}} = b_ {3} \ cdot \ left ({\ frac {1} {\ sin \ theta}} {\ frac {\ partial b_ {1}} {\ partial \ phi}} \ right) {\ frac {1} {r}} {\ frac {\ partial} {\ partial r}} + b_ {3} \ cdot b_ {1} {\ frac {1} {r \, \ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} {\ frac {\ partial} {\ partial r}} = {\ frac {1} {r}} {\ frac {\ partial} {\ partial r} }}

b_ {3} \ frac {1} {r \, \ sin \ theta} \ frac {\ partial} {\ partial \ phi} \ cdot b_ {1} \ frac {\ partial} {\ partial r} = b_ { 3} \ cdot \ left (\ frac {1} {\ sin \ theta} \ frac {\ partial b_ {1}} {\ partial \ phi} \ right) \ frac {1} {r} \ frac {\ partial } {\ partial r} + b_ {3} \ cdot b_ {1} \ frac {1} {r \, \ sin \ theta} \ frac {\ partial} {\ partial \ phi} \ frac {\ partial} { \ partial r} = \ frac {1} {r} \ frac {\ partial} {\ partial r}

b_{3}{\frac {1}{r\,\sin \theta }}{\frac {\partial }{\partial \phi }}\cdot b_{2}{\frac {1}{r}}{\frac {\partial }{\partial \theta }}=b_{3}\cdot \left({\frac {\cos \theta }{\sin \theta }}{\frac {1}{\cos \theta }}{\frac {\partial b_{2}}{\partial \phi }}\right){\frac {1}{r^{2}}}{\frac {\partial }{\partial \theta }}+b_{3}\cdot b_{2}{\frac {1}{r\,\sin \theta }}{\frac {\partial }{\partial \phi }}{\frac {1}{r}}{\frac {\partial }{\partial \theta }}=

{\ displaystyle b_ {3} {\ frac {1} {r \, \ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} \ cdot b_ {2} {\ frac {1} { r}} {\ frac {\ partial} {\ partial \ theta}} = b_ {3} \ cdot \ left ({\ frac {\ cos \ theta} {\ sin \ theta}} {\ frac {1} { \ cos \ theta}} {\ frac {\ partial b_ {2}} {\ partial \ phi}} \ right) {\ frac {1} {r ^ {2}}} {\ frac {\ partial} {\ partial \ theta}} + b_ {3} \ cdot b_ {2} {\ frac {1} {r \, \ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} {\ frac { 1} {r}} {\ frac {\ partial} {\ partial \ theta}} =}

b_ {3} \ frac {1} {r \, \ sin \ theta} \ frac {\ partial} {\ partial \ phi} \ cdot b_ {2} \ frac {1} {r} \ frac {\ partial} {\ partial \ theta} = b_ {3} \ cdot \ left (\ frac {\ cos \ theta} {\ sin \ theta} \ frac {1} {\ cos \ theta} \ frac {\ partial b_ {2} } {\ partial \ phi} \ right) \ frac {1} {r ^ {2}} \ frac {\ partial} {\ partial \ theta} + b_ {3} \ cdot b_ {2} \ frac {1} {r \, \ sin \ theta} \ frac {\ partial} {\ partial \ phi} \ frac {1} {r} \ frac {\ partial} {\ partial \ theta} =

={\frac {\cot \theta }{r^{2}}}{\frac {\partial }{\partial \theta }}

{\ displaystyle = {\ frac {\ cot \ theta} {r ^ {2}}} {\ frac {\ partial} {\ partial \ theta}}}

= \ frac {\ cot \ theta} {r ^ {2}} \ frac {\ partial} {\ partial \ theta}

b_{3}{\frac {1}{r\,\sin \theta }}{\frac {\partial }{\partial \phi }}\cdot b_{3}{\frac {1}{r\,\sin \theta }}{\frac {\partial }{\partial \phi }}=b_{3}\cdot \left({\frac {\partial b_{3}}{\partial \phi }}\right){\frac {1}{r\,\sin \theta }}{\frac {1}{r\,\sin \theta }}{\frac {\partial }{\partial \phi }}+b_{3}\cdot b_{3}{\frac {1}{r\,\sin \theta }}{\frac {\partial }{\partial \phi }}{\frac {1}{r\,\sin \theta }}{\frac {\partial }{\partial \phi }}=

{\ displaystyle b_ {3} {\ frac {1} {r \, \ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} \ cdot b_ {3} {\ frac {1} { r \, \ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} = b_ {3} \ cdot \ left ({\ frac {\ partial b_ {3}} {\ partial \ phi} } \ right) {\ frac {1} {r \, \ sin \ theta}} {\ frac {1} {r \, \ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} + b_ {3} \ cdot b_ {3} {\ frac {1} {r \, \ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} {\ frac {1} {r \ , \ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} =}

b_ {3} \ frac {1} {r \, \ sin \ theta} \ frac {\ partial} {\ partial \ phi} \ cdot b_ {3} \ frac {1} {r \, \ sin \ theta} \ frac {\ partial} {\ partial \ phi} = b_ {3} \ cdot \ left (\ frac {\ partial b_ {3}} {\ partial \ phi} \ right) \ frac {1} {r \, \ sin \ theta} \ frac {1} {r \, \ sin \ theta} \ frac {\ partial} {\ partial \ phi} + b_ {3} \ cdot b_ {3} \ frac {1} {r \ , \ sin \ theta} \ frac {\ partial} {\ partial \ phi} \ frac {1} {r \, \ sin \ theta} \ frac {\ partial} {\ partial \ phi} =

={\frac {1}{r^{2}\,\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \phi ^{2}}}

{\ displaystyle = {\ frac {1} {r ^ {2} \, \ sin ^ {2} \ theta}} {\ frac {\ partial ^ {2}} {\ partial \ phi ^ {2}}} }

= \ frac {1} {r ^ {2} \, \ sin ^ {2} \ theta} \ frac {\ partial ^ {2}} {\ partial \ phi ^ {2}}

din care obținem expresia laplacianului în coordonate polare:

\nabla \cdot \nabla =\nabla ^{2}={\frac {\partial ^{2}}{\partial r^{2}}}+{\frac {1}{r^{2}}}{\frac {\partial ^{2}}{\partial \theta ^{2}}}+{\frac {1}{r^{2}\,\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \phi ^{2}}}+{\frac {2}{r}}{\frac {\partial }{\partial r}}+{\frac {\cot \theta }{r^{2}}}{\frac {\partial }{\partial \theta }}=

{\ displaystyle \ nabla \ cdot \ nabla = \ nabla ^ {2} = {\ frac {\ partial ^ {2}} {\ partial r ^ {2}}} + {\ frac {1} {r ^ {2 }}} {\ frac {\ partial ^ {2}} {\ partial \ theta ^ {2}}} + {\ frac {1} {r ^ {2} \, \ sin ^ {2} \ theta}} {\ frac {\ partial ^ {2}} {\ partial \ phi ^ {2}}} + {\ frac {2} {r}} {\ frac {\ partial} {\ partial r}} + {\ frac {\ cot \ theta} {r ^ {2}}} {\ frac {\ partial} {\ partial \ theta}} =}

{\ displaystyle \ nabla \ cdot \ nabla = \ nabla ^ {2} = {\ frac {\ partial ^ {2}} {\ partial r ^ {2}}} + {\ frac {1} {r ^ {2 }}} {\ frac {\ partial ^ {2}} {\ partial \ theta ^ {2}}} + {\ frac {1} {r ^ {2} \, \ sin ^ {2} \ theta}} {\ frac {\ partial ^ {2}} {\ partial \ phi ^ {2}}} + {\ frac {2} {r}} {\ frac {\ partial} {\ partial r}} + {\ frac {\ cot \ theta} {r ^ {2}}} {\ frac {\ partial} {\ partial \ theta}} =}

={\frac {1}{r^{2}}}\left({\frac {\partial }{\partial r}}r^{2}{\frac {\partial }{\partial r}}+{\frac {\partial ^{2}}{\partial \theta ^{2}}}+{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \phi ^{2}}}+\cot \theta {\frac {\partial }{\partial \theta }}\right)=

{\ displaystyle = {\ frac {1} {r ^ {2}}} \ left ({\ frac {\ partial} {\ partial r}} r ^ {2} {\ frac {\ partial} {\ partial r }} + {\ frac {\ partial ^ {2}} {\ partial \ theta ^ {2}}} + {\ frac {1} {\ sin ^ {2} \ theta}} {\ frac {\ partial ^ {2}} {\ partial \ phi ^ {2}}} + \ cot \ theta {\ frac {\ partial} {\ partial \ theta}} \ right) =}

{\ displaystyle = {\ frac {1} {r ^ {2}}} \ left ({\ frac {\ partial} {\ partial r}} r ^ {2} {\ frac {\ partial} {\ partial r }} + {\ frac {\ partial ^ {2}} {\ partial \ theta ^ {2}}} + {\ frac {1} {\ sin ^ {2} \ theta}} {\ frac {\ partial ^ {2}} {\ partial \ phi ^ {2}}} + \ cot \ theta {\ frac {\ partial} {\ partial \ theta}} \ right) =}

={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}r^{2}{\frac {\partial }{\partial r}}+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial }{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \phi ^{2}}}

{\ displaystyle = {\ frac {1} {r ^ {2}}} {\ frac {\ partial} {\ partial r}} r ^ {2} {\ frac {\ partial} {\ partial r}} + {\ frac {1} {r ^ {2} \ sin \ theta}} {\ frac {\ partial} {\ partial \ theta}} \ left (\ sin \ theta {\ frac {\ partial} {\ partial \ theta}} \ right) + {\ frac {1} {r ^ {2} \ sin ^ {2} \ theta}} {\ frac {\ partial ^ {2}} {\ partial \ phi ^ {2}} }}

{\ displaystyle = {\ frac {1} {r ^ {2}}} {\ frac {\ partial} {\ partial r}} r ^ {2} {\ frac {\ partial} {\ partial r}} + {\ frac {1} {r ^ {2} \ sin \ theta}} {\ frac {\ partial} {\ partial \ theta}} \ left (\ sin \ theta {\ frac {\ partial} {\ partial \ theta}} \ right) + {\ frac {1} {r ^ {2} \ sin ^ {2} \ theta}} {\ frac {\ partial ^ {2}} {\ partial \ phi ^ {2}} }}

O altă modalitate mai convenabilă de a obține laplacianul folosește noțiuni de calcul tensorial ( notația Einstein pentru indicii însumați):

$\nabla ^{2}={\frac {1}{\sqrt {g}}}\partial _{i}{\sqrt {g}}\,g^{ij}\partial _{j}$ ${\ displaystyle \ nabla ^ {2} = {\ frac {1} {\ sqrt {g}}} \ partial _ {i} {\ sqrt {g}} \, g ^ {ij} \ partial _ {j} }$ ${\ displaystyle \ nabla ^ {2} = {\ frac {1} {\ sqrt {g}}} \ partial _ {i} {\ sqrt {g}} \, g ^ {ij} \ partial _ {j} }$ cu $g^{ij}={\frac {\partial {\tilde {X}}}{\partial {q_{i}}}}\cdot {\frac {\partial {\tilde {X}}}{\partial {q_{j}}}}$ ${\ displaystyle g ^ {ij} = {\ frac {\ partial {\ tilde {X}}} {\ partial {q_ {i}}}} \ cdot {\ frac {\ partial {\ tilde {X}}} {\ partial {q_ {j}}}}}$ ${\ displaystyle g ^ {ij} = {\ frac {\ partial {\ tilde {X}}} {\ partial {q_ {i}}}} \ cdot {\ frac {\ partial {\ tilde {X}}} {\ partial {q_ {j}}}}}$ , ${\sqrt {g}}={\sqrt {\mathrm {det} \left(g_{ij}\right)}}\,,\,g_{ij}=\left(g^{ij}\right)^{-1}$ ${\ displaystyle {\ sqrt {g}} = {\ sqrt {\ mathrm {det} \ left (g_ {ij} \ right)}} \ ,, \, g_ {ij} = \ left (g ^ {ij} \ dreapta) ^ {- 1}}$ ${\ displaystyle {\ sqrt {g}} = {\ sqrt {\ mathrm {det} \ left (g_ {ij} \ right)}} \ ,, \, g_ {ij} = \ left (g ^ {ij} \ dreapta) ^ {- 1}}$

Operatorii sunt, de asemenea, ușor de găsit $x\times \nabla$ ${\ displaystyle x \ times \ nabla}$ $x \ times \ nabla$ (legendriano) e $\left(x\times \nabla \right)^{2}$ ${\ displaystyle \ left (x \ times \ nabla \ right) ^ {2}}$ $\ left (x \ times \ nabla \ right) ^ {2}$ , care sunt strâns legate de $L$ ${\ displaystyle L}$ $L$ Și $L^{2}$ ${\ displaystyle L ^ {2}}$ $L ^ {2}$ în teoria momentelor unghiulare ale mecanicii cuantice, de fapt:

x\times \nabla =r\,b_{1}\times \left(b_{1}{\frac {\partial }{\partial r}}+b_{2}{\frac {1}{r}}{\frac {\partial }{\partial \theta }}+b_{3}{\frac {1}{r\,\sin \theta }}{\frac {\partial }{\partial \phi }}\right)=b_{3}{\frac {\partial }{\partial \theta }}-b_{2}{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \phi }}

{\ displaystyle x \ times \ nabla = r \, b_ {1} \ times \ left (b_ {1} {\ frac {\ partial} {\ partial r}} + b_ {2} {\ frac {1} { r}} {\ frac {\ partial} {\ partial \ theta}} + b_ {3} {\ frac {1} {r \, \ sin \ theta}} {\ frac {\ partial} {\ partial \ phi }} \ right) = b_ {3} {\ frac {\ partial} {\ partial \ theta}} - b_ {2} {\ frac {1} {\ sin \ theta}} {\ frac {\ partial} { \ partial \ phi}}}

x \ times \ nabla = r \, b_ {1} \ times \ left (b_ {1} \ frac {\ partial} {\ partial r} + b_ {2} \ frac {1} {r} \ frac {\ partial} {\ partial \ theta} + b_ {3} \ frac {1} {r \, \ sin \ theta} \ frac {\ partial} {\ partial \ phi} \ right) = b_ {3} \ frac { \ partial} {\ partial \ theta} -b_ {2} \ frac {1} {\ sin \ theta} \ frac {\ partial} {\ partial \ phi}

și calculând:

b_{2}{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \phi }}\cdot b_{2}{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \phi }}=b_{2}\cdot \left({\frac {\cos \theta }{\sin \theta }}{\frac {1}{\cos \theta }}{\frac {\partial b_{2}}{\partial \phi }}\right){\frac {1}{\sin \theta }}{\frac {\partial }{\partial \phi }}+b_{2}\cdot b_{2}{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \phi }}{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \phi }}={\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \phi ^{2}}}

{\ displaystyle b_ {2} {\ frac {1} {\ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} \ cdot b_ {2} {\ frac {1} {\ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} = b_ {2} \ cdot \ left ({\ frac {\ cos \ theta} {\ sin \ theta}} {\ frac {1} { \ cos \ theta}} {\ frac {\ partial b_ {2}} {\ partial \ phi}} \ right) {\ frac {1} {\ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} + b_ {2} \ cdot b_ {2} {\ frac {1} {\ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} {\ frac {1} {\ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} = {\ frac {1} {\ sin ^ {2} \ theta}} {\ frac {\ partial ^ {2}} {\ parțial \ phi ^ {2}}}}

b_ {2} \ frac {1} {\ sin \ theta} \ frac {\ partial} {\ partial \ phi} \ cdot b_ {2} \ frac {1} {\ sin \ theta} \ frac {\ partial} {\ partial \ phi} = b_ {2} \ cdot \ left (\ frac {\ cos \ theta} {\ sin \ theta} \ frac {1} {\ cos \ theta} \ frac {\ partial b_ {2} } {\ partial \ phi} \ right) \ frac {1} {\ sin \ theta} \ frac {\ partial} {\ partial \ phi} + b_ {2} \ cdot b_ {2} \ frac {1} { \ sin \ theta} \ frac {\ partial} {\ partial \ phi} \ frac {1} {\ sin \ theta} \ frac {\ partial} {\ partial \ phi} = \ frac {1} {\ sin ^ {2} \ theta} \ frac {\ partial ^ {2}} {\ partial \ phi ^ {2}}

b_{2}{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \phi }}\cdot b_{3}{\frac {\partial }{\partial \theta }}=b_{2}\cdot \left({\frac {1}{\sin \theta }}{\frac {\partial b_{3}}{\partial \phi }}\right){\frac {\partial }{\partial \theta }}+b_{2}\cdot b_{3}{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \phi }}{\frac {\partial }{\partial \theta }}=-\cot \theta {\frac {\partial }{\partial \theta }}

{\ displaystyle b_ {2} {\ frac {1} {\ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} \ cdot b_ {3} {\ frac {\ partial} {\ partial \ theta}} = b_ {2} \ cdot \ left ({\ frac {1} {\ sin \ theta}} {\ frac {\ partial b_ {3}} {\ partial \ phi}} \ right) {\ frac {\ partial} {\ partial \ theta}} + b_ {2} \ cdot b_ {3} {\ frac {1} {\ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} {\ frac {\ partial} {\ partial \ theta}} = - \ cot \ theta {\ frac {\ partial} {\ partial \ theta}}}

b_ {2} \ frac {1} {\ sin \ theta} \ frac {\ partial} {\ partial \ phi} \ cdot b_ {3} \ frac {\ partial} {\ partial \ theta} = b_ {2} \ cdot \ left (\ frac {1} {\ sin \ theta} \ frac {\ partial b_ {3}} {\ partial \ phi} \ right) \ frac {\ partial} {\ partial \ theta} + b_ { 2} \ cdot b_ {3} \ frac {1} {\ sin \ theta} \ frac {\ partial} {\ partial \ phi} \ frac {\ partial} {\ partial \ theta} = - \ cot \ theta \ frac {\ partial} {\ partial \ theta}

b_{3}{\frac {\partial }{\partial \theta }}\cdot b_{2}{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \phi }}=b_{3}\cdot \left({\frac {\partial b_{2}}{\partial \theta }}\right){\frac {1}{\sin \theta }}{\frac {\partial }{\partial \phi }}+b_{3}\cdot b_{2}{\frac {\partial }{\partial \theta }}{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \phi }}=0

{\ displaystyle b_ {3} {\ frac {\ partial} {\ partial \ theta}} \ cdot b_ {2} {\ frac {1} {\ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} = b_ {3} \ cdot \ left ({\ frac {\ partial b_ {2}} {\ partial \ theta}} \ right) {\ frac {1} {\ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} + b_ {3} \ cdot b_ {2} {\ frac {\ partial} {\ partial \ theta}} {\ frac {1} {\ sin \ theta}} {\ frac {\ partial} {\ partial \ phi}} = 0}

b_ {3} \ frac {\ partial} {\ partial \ theta} \ cdot b_ {2} \ frac {1} {\ sin \ theta} \ frac {\ partial} {\ partial \ phi} = b_ {3} \ cdot \ left (\ frac {\ partial b_ {2}} {\ partial \ theta} \ right) \ frac {1} {\ sin \ theta} \ frac {\ partial} {\ partial \ phi} + b_ { 3} \ cdot b_ {2} \ frac {\ partial} {\ partial \ theta} \ frac {1} {\ sin \ theta} \ frac {\ partial} {\ partial \ phi} = 0

b_{3}{\frac {\partial }{\partial \theta }}\cdot b_{3}{\frac {\partial }{\partial \theta }}=b_{3}\cdot \left({\frac {\partial b_{3}}{\partial \theta }}\right){\frac {\partial }{\partial \theta }}+b_{3}\cdot b_{3}{\frac {\partial }{\partial \theta }}{\frac {\partial }{\partial \theta }}={\frac {\partial ^{2}}{\partial \theta ^{2}}}

{\ displaystyle b_ {3} {\ frac {\ partial} {\ partial \ theta}} \ cdot b_ {3} {\ frac {\ partial} {\ partial \ theta}} = b_ {3} \ cdot \ left ({\ frac {\ partial b_ {3}} {\ partial \ theta}} \ right) {\ frac {\ partial} {\ partial \ theta}} + b_ {3} \ cdot b_ {3} {\ frac {\ partial} {\ partial \ theta}} {\ frac {\ partial} {\ partial \ theta}} = {\ frac {\ partial ^ {2}} {\ partial \ theta ^ {2}}}}

b_ {3} \ frac {\ partial} {\ partial \ theta} \ cdot b_ {3} \ frac {\ partial} {\ partial \ theta} = b_ {3} \ cdot \ left (\ frac {\ partial b_ {3}} {\ partial \ theta} \ right) \ frac {\ partial} {\ partial \ theta} + b_ {3} \ cdot b_ {3} \ frac {\ partial} {\ partial \ theta} \ frac {\ partial} {\ partial \ theta} = \ frac {\ partial ^ {2}} {\ partial \ theta ^ {2}}

primesti:

\left(x\times \nabla \right)^{2}={\frac {\partial ^{2}}{\partial \theta ^{2}}}+{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \phi ^{2}}}+\cot \theta {\frac {\partial }{\partial \theta }}

{\ displaystyle \ left (x \ times \ nabla \ right) ^ {2} = {\ frac {\ partial ^ {2}} {\ partial \ theta ^ {2}}} + {\ frac {1} {\ sin ^ {2} \ theta}} {\ frac {\ partial ^ {2}} {\ partial \ phi ^ {2}}} + \ cot \ theta {\ frac {\ partial} {\ partial \ theta}} }

\ left (x \ times \ nabla \ right) ^ {2} = \ frac {\ partial ^ {2}} {\ partial \ theta ^ {2}} + \ frac {1} {\ sin ^ {2} \ theta} \ frac {\ partial ^ {2}} {\ partial \ phi ^ {2}} + \ cot \ theta \ frac {\ partial} {\ partial \ theta}

Operatorul $\left(x\times \nabla \right)^{2}$ ${\ displaystyle \ left (x \ times \ nabla \ right) ^ {2}}$ $\ left (x \ times \ nabla \ right) ^ {2}$ reprezintă partea unghiulară a $\nabla ^{2}$ ${\ displaystyle \ nabla ^ {2}}$ $\ nabla ^ {2}$ și se poate scrie o altă expresie importantă pentru laplacian:

\nabla ^{2}={\frac {1}{r^{2}}}\left({\frac {\partial }{\partial r}}r^{2}{\frac {\partial }{\partial r}}+\left(x\times \nabla \right)^{2}\right)={\frac {\partial ^{2}}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial }{\partial r}}+\left({\frac {x\times \nabla }{r}}\right)^{2}

{\ displaystyle \ nabla ^ {2} = {\ frac {1} {r ^ {2}}} \ left ({\ frac {\ partial} {\ partial r}} r ^ {2} {\ frac {\ partial} {\ partial r}} + \ left (x \ times \ nabla \ right) ^ {2} \ right) = {\ frac {\ partial ^ {2}} {\ partial r ^ {2}}} + {\ frac {2} {r}} {\ frac {\ partial} {\ partial r}} + \ left ({\ frac {x \ times \ nabla} {r}} \ right) ^ {2}}

\ nabla ^ {2} = \ frac {1} {r ^ {2}} \ left (\ frac {\ partial} {\ partial r} r ^ {2} \ frac {\ partial} {\ partial r} + \ left (x \ times \ nabla \ right) ^ {2} \ right) = \ frac {\ partial ^ {2}} {\ partial r ^ {2}} + \ frac {2} {r} \ frac { \ partial} {\ partial r} + \ left (\ frac {x \ times \ nabla} {r} \ right) ^ {2}

Notă

Elemente conexe

Portale Matematica : accedi alle voci di Wikipedia che trattano di matematica

[1] Nebel în "Enciclopedia italiană" - Treccani

[2] Nebel in Vocabulary - Treccani

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[2]