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.
Definicija Shematski prikaz uz definiciju rotacije vektorskoga polja Pogledajmo linijski integral vektorskog polja W → {\displaystyle {\overrightarrow {W}}} duž zatvorne krivulje C {\displaystyle C} koja ograničava površinu S {\displaystyle S} . Premostimo krivulju nekim lukom, tako da je vanjska krivulja razdvojena na dvije ( C 1 + C 2 = C {\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 S {\displaystyle S} :
∮ W → d S → = ∫ C W → d S → = ∑ i = 1 N ∫ C i W → d S → i . {\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 A i {\displaystyle A_{i}} koji okružuje krivulja C i {\displaystyle C_{i}} . Pustimo li da N ↦ ∞ {\displaystyle N\mapsto \infty } , odnosno A i ↦ 0 {\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 n ^ {\displaystyle {\hat {n}}} , dolazimo upravo do definicije rotacje ili rotora vektorskog polja:
n ^ ⋅ rot W → = d e f . lim A i → 0 ∫ C i W → d S → A i = lim Δ S → 0 ∮ W → d S → Δ S . {\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}}.}
Svojstva i pretpostavke 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 n ^ {\displaystyle {\hat {n}}} ne mijenja dok se element plohe smanjuje k nuli. Rotor je, kao i Divergencija, također invarijanta vektorskog polja.
Rotor u kartezijevu sustavu Shematski prikaz uz definiciju rotacije vektorskoga polja Kako bismo izveli izraz za rotor u kartezijevu sustavu, napravimo integraciju po rubu pravokutnika paralelnog s x O y {\displaystyle xOy} - ravinom ( n ^ = z ^ {\displaystyle {\hat {n}}={\hat {z}}} ), kao na sl.
∮ W → d S → = ∫ C 1 W → d S → + ∫ C 2 W → d S → + ∫ C 3 W → d S → + ∫ C 4 W → d S → = {\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}}=} = ∫ C 1 W x ( x , y 0 , z 0 ) d x + ∫ C 2 W y ( x 0 + Δ x , y , z 0 ) d y − {\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-} − ∫ C 3 W x ( x , y 0 + Δ y , z 0 ) d x − ∫ C 4 W y ( x 0 , y , z 0 ) d y = {\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=} = ∫ [ W x ( x , y 0 , z 0 ) − W x ( x , y 0 + Δ y , z 0 ) ] d x + {\displaystyle =\int {\Bigl [}W_{x}(x,y_{0},z_{0})-W_{x}(x,y_{0}+\Delta y,z_{0}){\Bigr ]}dx+} + ∫ [ W y ( x 0 + Δ x , y , z 0 ) − W y ( x 0 , y , z 0 ) ] d y = {\displaystyle +\int {\Bigl [}W_{y}(x_{0}+\Delta x,y,z_{0})-W_{y}(x_{0},y,z_{0}){\Bigr ]}dy=} = ∂ W y ∂ x ⋅ Δ x Δ y − ∂ W x ∂ y ⋅ Δ x Δ y = {\displaystyle ={\frac {\partial W_{y}}{\partial x}}\cdot \Delta x\Delta y-{\frac {\partial W_{x}}{\partial y}}\cdot \Delta x\Delta y=} = Δ S ⋅ ( ∂ W y ∂ x − ∂ W x ∂ 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:
z ^ ⋅ rot W → = lim Δ S → 0 ∮ W → d S → Δ S = lim Δ S → 0 ( ∂ W y ∂ x − ∂ W x ∂ y ) Δ S Δ S = ( ∂ W y ∂ x − ∂ W x ∂ y ) = ( rot W → ) z . {\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}.} ( rot W → ) x = ( ∂ W z ∂ y − ∂ W y ∂ z ) {\displaystyle ({\mbox{rot}}{\overrightarrow {W}})_{x}={\Bigl (}{\frac {\partial W_{z}}{\partial y}}-{\frac {\partial W_{y}}{\partial z}}{\Bigr )}} ( rot W → ) y = ( ∂ W x ∂ z − ∂ W z ∂ x ) {\displaystyle ({\mbox{rot}}{\overrightarrow {W}})_{y}={\Bigl (}{\frac {\partial W_{x}}{\partial z}}-{\frac {\partial W_{z}}{\partial x}}{\Bigr )}} ( rot W → ) z = ( ∂ W y ∂ x − ∂ W x ∂ y ) {\displaystyle ({\mbox{rot}}{\overrightarrow {W}})_{z}={\Bigl (}{\frac {\partial W_{y}}{\partial x}}-{\frac {\partial W_{x}}{\partial y}}{\Bigr )}} rot W → = x ^ ( ∂ W z ∂ y − ∂ W y ∂ z ) + y ^ ( ∂ W x ∂ z − ∂ W z ∂ x ) + z ^ ( ∂ W y ∂ x − ∂ W x ∂ y ) . {\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:
rot W → = | x ^ y ^ z ^ ∂ ∂ x ∂ ∂ y ∂ ∂ z W x W y W z | . {\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
rot W → = ( x ^ ∂ ∂ x + y ^ ∂ ∂ y + z ^ ∂ ∂ z ) × ( x ^ W x + y ^ W y + z ^ W z ) = ∇ → × W → , {\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 rot W → {\displaystyle {\mbox{rot}}{\overrightarrow {W}}} često označavamo s ∇ → × W → {\displaystyle {\vec {\nabla }}\times {\overrightarrow {W}}} , gdje je ∇ → {\displaystyle {\vec {\nabla }}} Hamiltonov operator.
Rotacija i Stokesov teorem Za rotaciju vrijedi Stokesov teorem
∫ S rot W → ⋅ d A → = ∫ C W → ⋅ d S → . {\displaystyle \int \limits _{S}{\mbox{rot}}{\overrightarrow {W}}\cdot d{\vec {A}}=\int \limits _{C}{\overrightarrow {W}}\cdot d{\vec {S}}.}
Izrazi za rotaciju u drugim koordinatnim sustavima | ( rot W → ) ρ | = 1 ρ ∂ W z ∂ φ − ∂ W φ ∂ z {\displaystyle |({\mbox{rot}}{\overrightarrow {W}})_{\rho }|={\frac {1}{\rho }}{\frac {\partial W_{z}}{\partial \varphi }}-{\frac {\partial W_{\varphi }}{\partial z}}} | ( rot W → ) φ | = ∂ W ρ ∂ z − ∂ W z ∂ ρ {\displaystyle |({\mbox{rot}}{\overrightarrow {W}})_{\varphi }|={\frac {\partial W_{\rho }}{\partial z}}-{\frac {\partial W_{z}}{\partial \rho }}} | ( rot W → ) z | = 1 ρ ∂ ∂ ρ ( ρ W φ ) − 1 ρ ∂ W ρ ∂ φ {\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 }}} rot W → = [ 1 ρ ∂ W z ∂ φ − ∂ W φ ∂ z ] ρ ^ + [ ∂ W ρ ∂ z − ∂ W z ∂ ρ ] φ ^ + [ 1 ρ ∂ ∂ ρ ( ρ W φ ) − 1 ρ ∂ W ρ ∂ φ ] z ^ {\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}}} | ( rot W → ) r | = 1 r sin ϑ ∂ ∂ ϑ ( W φ sin ϑ ) − 1 r sin ϑ ∂ W ϑ ∂ φ {\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 }}} | ( rot W → ) ϑ | = 1 r sin ϑ ∂ W r ∂ φ − 1 r ∂ ∂ r ( r W φ ) {\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 })} | ( rot W → ) φ | = 1 r ∂ ∂ r ( r W ϑ ) − 1 r ∂ W r ∂ ϑ {\displaystyle |({\mbox{rot}}{\overrightarrow {W}})_{\varphi }|={\frac {1}{r}}{\frac {\partial }{\partial r}}(rW_{\vartheta })-{\frac {1}{r}}{\frac {\partial W_{r}}{\partial \vartheta }}} rot W → = [ 1 r sin ϑ ∂ ∂ ϑ ( W φ sin ϑ ) − 1 r sin ϑ ∂ W ϑ ∂ φ ] r ^ + [ 1 r sin ϑ ∂ W r ∂ φ − 1 r ∂ ∂ r ( r W φ ) ] ϑ ^ + [ 1 r ∂ ∂ r ( r W ϑ ) − 1 r ∂ W r ∂ ϑ ] φ ^ . {\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 }}.}
Rotacija i algebarske operacije Neka su dana vektorska polja u → {\displaystyle {\vec {u}}} i v → {\displaystyle {\vec {v}}} , skalar U {\displaystyle U} , skalarna funkcija f ( U ) {\displaystyle f(U)} i radij-vektor r → {\displaystyle {\vec {r}}} . Tada vrijedi:
rot ( u → + v → ) = rot u → + rot v → {\displaystyle {\textrm {rot}}({\vec {u}}+{\vec {v}})={\textrm {rot}}{\vec {u}}+{\textrm {rot}}{\vec {v}}} rot ( U ⋅ v → ) = U ⋅ rot v → − v → × grad U {\displaystyle {\textrm {rot}}(U\cdot {\vec {v}})=U\cdot {\textrm {rot}}{\vec {v}}-{\vec {v}}\times {\mbox{grad}}U} rot [ f ( U ) ⋅ v → ] = f ( U ) ⋅ rot v → − v → × f U ′ ( U ) 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} rot r → = 0 {\displaystyle {\textrm {rot}}{\vec {r}}=0} rot ( u → × v → ) = u → div v → − v → div u → + ( v → ⋅ ∇ → ) u → − ( u → ⋅ ∇ → ) v → {\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}}} grad ( u → ⋅ v → ) = v → × rot u → + u → × rot v → + ( v → ⋅ ∇ → ) u → + ( u → ⋅ ∇ → ) 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}}} div ( u → × v → ) = v → rot u → − u → rot v → . {\displaystyle {\mbox{div}}({\vec {u}}\times {\vec {v}})={\vec {v}}{\mbox{rot}}{\vec {u}}-{\vec {u}}{\mbox{rot}}{\vec {v}}.}
Primjeri Rotor elektrostaskog polja točkastog naboja , E → = 1 4 π ε 0 q r 3 r → {\displaystyle {\overrightarrow {E}}={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q}{r^{3}}}{\vec {r}}} : rot E → = rot ( 1 4 π ε 0 q r 3 r → ) = ( 2. ) q 4 π ε 0 r 3 rot r → − r → × grad q 4 π ε 0 r 3 = − 3 q 4 π ε 0 r 4 r → r × r → = [ r → × r → = 0 ] = 0. {\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 , v → = ω → × r → {\displaystyle {\vec {v}}={\vec {\omega }}\times {\vec {r}}} (v. sl.). Shematski prikaz uz rotaciju polja obodne brzine v → = ω → × r → = | x ^ y ^ z ^ ω x ω y ω z x y z | = x ^ ( z ω y − y ω z ) + y ^ ( x ω z − z ω x ) + z ^ ( y ω x − x ω y ) ; {\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});} rot v → = | x ^ y ^ z ^ ∂ ∂ x ∂ ∂ y ∂ ∂ z ( z ω y − y ω z ) ( x ω z − z ω x ) ( y ω x − x ω y ) | = 2 ω x x ^ + 2 ω y y ^ + 2 ω z z ^ = 2 ω → . {\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:
ω x = 1 2 ( ∂ v z ∂ y − ∂ v y ∂ z ) {\displaystyle \omega _{x}={\frac {1}{2}}{\Bigl (}{\frac {\partial v_{z}}{\partial y}}-{\frac {\partial v_{y}}{\partial z}}{\Bigr )}} ω y = 1 2 ( ∂ v x ∂ z − ∂ v z ∂ x ) {\displaystyle \omega _{y}={\frac {1}{2}}{\Bigl (}{\frac {\partial v_{x}}{\partial z}}-{\frac {\partial v_{z}}{\partial x}}{\Bigr )}} ω z = 1 2 ( ∂ v y ∂ x − ∂ v x ∂ y ) . {\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 v → {\displaystyle {\vec {v}}} je polarni vektor, a vektor rot v → {\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.
Vezani pojmovi
Vanjske poveznice Gradijent, divergencija i rotacija Divergencija i rotacija. Specijalna polja Wolfram: Curl