Definicija
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Svojstva i pretpostavke
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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
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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
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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
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| ( 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
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Vezani pojmovi
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Vanjske poveznice
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