Rotor (matematika)

U vektorskoj analizi i teoriji polja, rotor ili rotacija (rot, eng. curl) je veličina koja odražava svojstva vektorskoga polja u prostoru. Najviše se primjenjuje u fizici, pogotovo u elektromagnetizmu i hidrodinamici.

Pogledajmo linijski integral vektorskog polja ${\displaystyle {\overrightarrow {W}}}$ duž zatvorne krivulje ${\displaystyle C}$ koja ograničava površinu ${\displaystyle S}$. Premostimo krivulju nekim lukom, tako da je vanjska krivulja razdvojena na dvije (${\displaystyle C_{1}+C_{2}=C}$). Pri integriranju sada udio imaju samo vanjski dijelovi početne linije, jer se po luku integrira jednom u jedom, a drugi put u suprotom smijeru pa se taj integral poništava (v. sl.). Naravno, isto se događa i za velik broj razdioba početne površine ${\displaystyle S}$:

${\displaystyle \oint {\overrightarrow {W}}d{\vec {S}}=\int \limits _{C}{\overrightarrow {W}}d{\vec {S}}=\sum _{i=1}^{N}\int \limits _{C_{i}}{\overrightarrow {W}}d{\vec {S}}_{i}.}$

Uzmimo sada omjer te vrijednosti i infinitezimalno malog dijela površine ${\displaystyle A_{i}}$ koji okružuje krivulja ${\displaystyle C_{i}}$. Pustimo li da ${\displaystyle N\mapsto \infty }$, odnosno ${\displaystyle A_{i}\mapsto 0}$, dobivamo graničnu vrijednost koja predstavlja skalarnu veličinu pridruženu određenoj točki prostora, pa je stoga možemo smatrati komponentom vektora. Pomnožimo li dati izraz s vektorom normale ${\displaystyle {\hat {n}}}$, dolazimo upravo do definicije rotacje ili rotora vektorskog polja:

${\displaystyle {\hat {n}}\cdot {\mbox{rot}}{\overrightarrow {W}}{\stackrel {def.}{=}}\lim _{A_{i}\rightarrow 0}{\frac {\int \limits _{C_{i}}{\overrightarrow {W}}d{\vec {S}}}{A_{i}}}=\lim _{\Delta S\rightarrow 0}{\frac {\oint {\overrightarrow {W}}d{\vec {S}}}{\Delta S}}.}$

Nije nužno da ploha omeđena krvuljom koju promatramo leži u ravnini, traži se jedino da ta ploha nema singularnosti.

Nadalje, pretpostavlja se da se vektor normale ${\displaystyle {\hat {n}}}$ ne mijenja dok se element plohe smanjuje k nuli. Rotor je, kao i Divergencija, također invarijanta vektorskog polja.

Kako bismo izveli izraz za rotor u kartezijevu sustavu, napravimo integraciju po rubu pravokutnika paralelnog s ${\displaystyle xOy}$ - ravinom (${\displaystyle {\hat {n}}={\hat {z}}}$), kao na sl.

${\displaystyle \oint {\overrightarrow {W}}d{\vec {S}}=\int \limits _{C_{1}}{\overrightarrow {W}}d{\vec {S}}+\int \limits _{C_{2}}{\overrightarrow {W}}d{\vec {S}}+\int \limits _{C_{3}}{\overrightarrow {W}}d{\vec {S}}+\int \limits _{C_{4}}{\overrightarrow {W}}d{\vec {S}}=}$

${\displaystyle =\int \limits _{C_{1}}W_{x}(x,y_{0},z_{0})dx+\int \limits _{C_{2}}W_{y}(x_{0}+\Delta x,y,z_{0})dy-}$

${\displaystyle -\int \limits _{C_{3}}W_{x}(x,y_{0}+\Delta y,z_{0})dx-\int \limits _{C_{4}}W_{y}(x_{0},y,z_{0})dy=}$

${\displaystyle =\int {\Bigl [}W_{x}(x,y_{0},z_{0})-W_{x}(x,y_{0}+\Delta y,z_{0}){\Bigr ]}dx+}$

${\displaystyle +\int {\Bigl [}W_{y}(x_{0}+\Delta x,y,z_{0})-W_{y}(x_{0},y,z_{0}){\Bigr ]}dy=}$

${\displaystyle ={\frac {\partial W_{y}}{\partial x}}\cdot \Delta x\Delta y-{\frac {\partial W_{x}}{\partial y}}\cdot \Delta x\Delta y=}$

${\displaystyle =\Delta S\cdot {\Bigl (}{\frac {\partial W_{y}}{\partial x}}-{\frac {\partial W_{x}}{\partial y}}{\Bigr )}.}$

Uvršatavanjem u definiciju rotacije, te potpunom analogijom, imamo:

${\displaystyle {\hat {z}}\cdot {\mbox{rot}}{\overrightarrow {W}}=\lim _{\Delta S\rightarrow 0}{\frac {\oint {\overrightarrow {W}}d{\vec {S}}}{\Delta S}}=\lim _{\Delta S\rightarrow 0}{\frac {{\Bigl (}{\frac {\partial W_{y}}{\partial x}}-{\frac {\partial W_{x}}{\partial y}}{\Bigr )}\Delta S}{\Delta S}}={\Bigl (}{\frac {\partial W_{y}}{\partial x}}-{\frac {\partial W_{x}}{\partial y}}{\Bigr )}=({\mbox{rot}}{\overrightarrow {W}})_{z}.}$

${\displaystyle ({\mbox{rot}}{\overrightarrow {W}})_{x}={\Bigl (}{\frac {\partial W_{z}}{\partial y}}-{\frac {\partial W_{y}}{\partial z}}{\Bigr )}}$

${\displaystyle ({\mbox{rot}}{\overrightarrow {W}})_{y}={\Bigl (}{\frac {\partial W_{x}}{\partial z}}-{\frac {\partial W_{z}}{\partial x}}{\Bigr )}}$

${\displaystyle ({\mbox{rot}}{\overrightarrow {W}})_{z}={\Bigl (}{\frac {\partial W_{y}}{\partial x}}-{\frac {\partial W_{x}}{\partial y}}{\Bigr )}}$

${\displaystyle {\mbox{rot}}{\overrightarrow {W}}={\hat {x}}{\Bigl (}{\frac {\partial W_{z}}{\partial y}}-{\frac {\partial W_{y}}{\partial z}}{\Bigr )}+{\hat {y}}{\Bigl (}{\frac {\partial W_{x}}{\partial z}}-{\frac {\partial W_{z}}{\partial x}}{\Bigr )}+{\hat {z}}{\Bigl (}{\frac {\partial W_{y}}{\partial x}}-{\frac {\partial W_{x}}{\partial y}}{\Bigr )}.}$

Očito u danoj fomuli možemo prepoznati simbolički zapisanu determinantu:

${\displaystyle {\mbox{rot}}{\overrightarrow {W}}=\left|{\begin{array}{ccc}\displaystyle {\hat {x}}&\displaystyle {\hat {y}}&\displaystyle {\hat {z}}\\\displaystyle {\frac {\partial }{\partial x}}&\displaystyle {\frac {\partial }{\partial y}}&\displaystyle {\frac {\partial }{\partial z}}\\\displaystyle {W_{x}}&\displaystyle {W_{y}}&\displaystyle {W_{z}}\end{array}}\right|.}$

Nadalje, očito je

${\displaystyle {\mbox{rot}}{\overrightarrow {W}}={\Bigl (}{\hat {x}}{\frac {\partial }{\partial x}}+{\hat {y}}{\frac {\partial }{\partial y}}+{\hat {z}}{\frac {\partial }{\partial z}}{\Bigr )}\times ({\hat {x}}W_{x}+{\hat {y}}W_{y}+{\hat {z}}W_{z})={\vec {\nabla }}\times {\overrightarrow {W}},}$

pa ${\displaystyle {\mbox{rot}}{\overrightarrow {W}}}$ često označavamo s ${\displaystyle {\vec {\nabla }}\times {\overrightarrow {W}}}$, gdje je ${\displaystyle {\vec {\nabla }}}$ Hamiltonov operator.

Za rotaciju vrijedi Stokesov teorem

${\displaystyle \int \limits _{S}{\mbox{rot}}{\overrightarrow {W}}\cdot d{\vec {A}}=\int \limits _{C}{\overrightarrow {W}}\cdot d{\vec {S}}.}$

• u cilindričnom:

${\displaystyle |({\mbox{rot}}{\overrightarrow {W}})_{\rho }|={\frac {1}{\rho }}{\frac {\partial W_{z}}{\partial \varphi }}-{\frac {\partial W_{\varphi }}{\partial z}}}$

${\displaystyle |({\mbox{rot}}{\overrightarrow {W}})_{\varphi }|={\frac {\partial W_{\rho }}{\partial z}}-{\frac {\partial W_{z}}{\partial \rho }}}$

${\displaystyle |({\mbox{rot}}{\overrightarrow {W}})_{z}|={\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}(\rho W_{\varphi })-{\frac {1}{\rho }}{\frac {\partial W_{\rho }}{\partial \varphi }}}$

${\displaystyle {\mbox{rot}}{\overrightarrow {W}}={\biggl [}{\frac {1}{\rho }}{\frac {\partial W_{z}}{\partial \varphi }}-{\frac {\partial W_{\varphi }}{\partial z}}{\biggr ]}{\hat {\rho }}+{\biggl [}{\frac {\partial W_{\rho }}{\partial z}}-{\frac {\partial W_{z}}{\partial \rho }}{\biggr ]}{\hat {\varphi }}+{\biggl [}{\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}(\rho W_{\varphi })-{\frac {1}{\rho }}{\frac {\partial W_{\rho }}{\partial \varphi }}{\biggr ]}{\hat {z}}}$

• u sfernom:

${\displaystyle |({\mbox{rot}}{\overrightarrow {W}})_{r}|={\frac {1}{r\sin \vartheta }}{\frac {\partial }{\partial \vartheta }}(W_{\varphi }\sin \vartheta )-{\frac {1}{r\sin \vartheta }}{\frac {\partial W_{\vartheta }}{\partial \varphi }}}$

${\displaystyle |({\mbox{rot}}{\overrightarrow {W}})_{\vartheta }|={\frac {1}{r\sin \vartheta }}{\frac {\partial W_{r}}{\partial \varphi }}-{\frac {1}{r}}{\frac {\partial }{\partial r}}(rW_{\varphi })}$

${\displaystyle |({\mbox{rot}}{\overrightarrow {W}})_{\varphi }|={\frac {1}{r}}{\frac {\partial }{\partial r}}(rW_{\vartheta })-{\frac {1}{r}}{\frac {\partial W_{r}}{\partial \vartheta }}}$

${\displaystyle {\mbox{rot}}{\overrightarrow {W}}={\biggl [}{\frac {1}{r\sin \vartheta }}{\frac {\partial }{\partial \vartheta }}(W_{\varphi }\sin \vartheta )-{\frac {1}{r\sin \vartheta }}{\frac {\partial W_{\vartheta }}{\partial \varphi }}{\biggr ]}{\hat {r}}+{\biggl [}{\frac {1}{r\sin \vartheta }}{\frac {\partial W_{r}}{\partial \varphi }}-{\frac {1}{r}}{\frac {\partial }{\partial r}}(rW_{\varphi }){\biggr ]}{\hat {\vartheta }}+{\biggl [}{\frac {1}{r}}{\frac {\partial }{\partial r}}(rW_{\vartheta })-{\frac {1}{r}}{\frac {\partial W_{r}}{\partial \vartheta }}{\biggr ]}{\hat {\varphi }}.}$

Neka su dana vektorska polja ${\displaystyle {\vec {u}}}$ i ${\displaystyle {\vec {v}}}$, skalar ${\displaystyle U}$, skalarna funkcija ${\displaystyle f(U)}$ i radij-vektor ${\displaystyle {\vec {r}}}$. Tada vrijedi:

• ${\displaystyle {\textrm {rot}}({\vec {u}}+{\vec {v}})={\textrm {rot}}{\vec {u}}+{\textrm {rot}}{\vec {v}}}$
• ${\displaystyle {\textrm {rot}}(U\cdot {\vec {v}})=U\cdot {\textrm {rot}}{\vec {v}}-{\vec {v}}\times {\mbox{grad}}U}$
• ${\displaystyle {\textrm {rot}}[f(U)\cdot {\vec {v}}]=f(U)\cdot {\textrm {rot}}{\vec {v}}-{\vec {v}}\times f_{U}^{'}(U){\textrm {grad}}U}$
• ${\displaystyle {\textrm {rot}}{\vec {r}}=0}$
• ${\displaystyle {\mbox{rot}}({\vec {u}}\times {\vec {v}})={\vec {u}}{\mbox{div}}{\vec {v}}-{\vec {v}}{\mbox{div}}{\vec {u}}+({\vec {v}}\cdot {\vec {\nabla }}){\vec {u}}-({\vec {u}}\cdot {\vec {\nabla }}){\vec {v}}}$
• ${\displaystyle {\mbox{grad}}({\vec {u}}\cdot {\vec {v}})={\vec {v}}\times {\mbox{rot}}{\vec {u}}+{\vec {u}}\times {\mbox{rot}}{\vec {v}}+({\vec {v}}\cdot {\vec {\nabla }}){\vec {u}}+({\vec {u}}\cdot {\vec {\nabla }}){\vec {v}}}$
• ${\displaystyle {\mbox{div}}({\vec {u}}\times {\vec {v}})={\vec {v}}{\mbox{rot}}{\vec {u}}-{\vec {u}}{\mbox{rot}}{\vec {v}}.}$

• Rotor elektrostaskog polja točkastog naboja, ${\displaystyle {\overrightarrow {E}}={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q}{r^{3}}}{\vec {r}}}$:

${\displaystyle {\mbox{rot}}{\overrightarrow {E}}={\mbox{rot}}{\Bigl (}{\frac {1}{4\pi \varepsilon _{0}}}{\frac {q}{r^{3}}}{\vec {r}}{\Bigr )}{\stackrel {(2.)}{=}}{\frac {q}{4\pi \varepsilon _{0}r^{3}}}{\mbox{rot}}{\vec {r}}-{\vec {r}}\times {\mbox{grad}}{\frac {q}{4\pi \varepsilon _{0}r^{3}}}=-{\frac {3q}{4\pi \varepsilon _{0}r^{4}}}{\frac {\vec {r}}{r}}\times {\vec {r}}=[{\vec {r}}\times {\vec {r}}=0]=0.}$

• Rotor vektorskog polja obodne kružne brzine, ${\displaystyle {\vec {v}}={\vec {\omega }}\times {\vec {r}}}$ (v. sl.).

${\displaystyle {\vec {v}}={\vec {\omega }}\times {\vec {r}}=\left|{\begin{array}{ccc}{\hat {x}}&{\hat {y}}&{\hat {z}}\\\omega _{x}&\omega _{y}&\omega _{z}\\x&y&z\end{array}}\right|={\hat {x}}(z\omega _{y}-y\omega _{z})+{\hat {y}}(x\omega _{z}-z\omega _{x})+{\hat {z}}(y\omega _{x}-x\omega _{y});}$

${\displaystyle {\mbox{rot}}{\vec {v}}=\left|{\begin{array}{ccc}{\hat {x}}&{\hat {y}}&{\hat {z}}\\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\(z\omega _{y}-y\omega _{z})&(x\omega _{z}-z\omega _{x})&(y\omega _{x}-x\omega _{y})\end{array}}\right|=2\omega _{x}{\hat {x}}+2\omega _{y}{\hat {y}}+2\omega _{z}{\hat {z}}=2{\vec {\omega }}.}$

Odatle se lako mogu iščitati komponente kutne brzine:

${\displaystyle \omega _{x}={\frac {1}{2}}{\Bigl (}{\frac {\partial v_{z}}{\partial y}}-{\frac {\partial v_{y}}{\partial z}}{\Bigr )}}$

${\displaystyle \omega _{y}={\frac {1}{2}}{\Bigl (}{\frac {\partial v_{x}}{\partial z}}-{\frac {\partial v_{z}}{\partial x}}{\Bigr )}}$

${\displaystyle \omega _{z}={\frac {1}{2}}{\Bigl (}{\frac {\partial v_{y}}{\partial x}}-{\frac {\partial v_{x}}{\partial y}}{\Bigr )}.}$

Na ovom primjeru primijetimo: vektor brzine ${\displaystyle {\vec {v}}}$ je polarni vektor, a vektor ${\displaystyle {\mbox{rot}}{\vec {v}}}$ je aksijalni vektor. Međutim, to vrijedi i općenito: rotor polarnog vektora je aksijalni vektor, a rotor aksijalnog vektora je polarni vektor.

• Tok polja
• Gradijent
• Divergencija
• Vektorsko polje
• Elektromagnetizam
• Električno polje
• Magnetsko polje
• Hidrodinamika
• Vektorske operacije u zakrivljenim koordinatama

• Gradijent, divergencija i rotacija
• Divergencija i rotacija. Specijalna polja
• Wolfram: Curl