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.

Definicija

Datoteka:Shema rotacija.png

Shematski prikaz uz definiciju rotacije vektorskoga polja

Pogledajmo linijski integral vektorskog polja $\overrightarrow{W}$ duž zatvorne krivulje $C$ koja ograničava površinu $S$. Premostimo krivulju nekim lukom, tako da je vanjska krivulja razdvojena na dvije ($C_1+C_2=C$). 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 $S$:

$\oint \overrightarrow{W}d\overrightarrow{S}=\underset{C}{\int }\overrightarrow{W}d\overrightarrow{S}=\sum _{i=1}^N\underset{C_i}{\int }\overrightarrow{W}d{\overrightarrow{S}}_i.$

Uzmimo sada omjer te vrijednosti i infinitezimalno malog dijela površine $A_i$ koji okružuje krivulja $C_i$. Pustimo li da $N\mapsto \mathrm{\infty }$, odnosno $A_i\mapsto 0$, 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 $\hat{n}$, dolazimo upravo do definicije rotacje ili rotora vektorskog polja:

$\hat{n}\cdot \text{rot}\overrightarrow{W}\stackrel{def.}{=}\underset{A_i\to 0}{\lim }\frac{\underset{C_i}{\int }\overrightarrow{W}d\overrightarrow{S}}{A_i}=\underset{\mathrm{\Delta }S\to 0}{\lim }\frac{\oint \overrightarrow{W}d\overrightarrow{S}}{\mathrm{\Delta }S}.$

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

Rotor u kartezijevu sustavu

Datoteka:Rotor pravokutni.png

Shematski prikaz uz definiciju rotacije vektorskoga polja

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

$\oint \overrightarrow{W}d\overrightarrow{S}=\underset{C_1}{\int }\overrightarrow{W}d\overrightarrow{S}+\underset{C_2}{\int }\overrightarrow{W}d\overrightarrow{S}+\underset{C_3}{\int }\overrightarrow{W}d\overrightarrow{S}+\underset{C_4}{\int }\overrightarrow{W}d\overrightarrow{S}=$

$=\underset{C_1}{\int }{W}_x(x,y_0,z_0)dx+\underset{C_2}{\int }{W}_y(x_0+\mathrm{\Delta }x,y,z_0)dy-$

$-\underset{C_3}{\int }{W}_x(x,y_0+\mathrm{\Delta }y,z_0)dx-\underset{C_4}{\int }{W}_y(x_0,y,z_0)dy=$

$=\int [W_x(x,y_0,z_0)-W_x(x,y_0+\mathrm{\Delta }y,z_0)]dx+$

$+\int [W_y(x_0+\mathrm{\Delta }x,y,z_0)-W_y(x_0,y,z_0)]dy=$

$=\frac{\partial W_y}{\partial x}\cdot \mathrm{\Delta }x\mathrm{\Delta }y-\frac{\partial W_x}{\partial y}\cdot \mathrm{\Delta }x\mathrm{\Delta }y=$

$=\mathrm{\Delta }S\cdot (\frac{\partial W_y}{\partial x}-\frac{\partial W_x}{\partial y}).$

Uvršatavanjem u definiciju rotacije, te potpunom analogijom, imamo:

$\hat{z}\cdot \text{rot}\overrightarrow{W}=\underset{\mathrm{\Delta }S\to 0}{\lim }\frac{\oint \overrightarrow{W}d\overrightarrow{S}}{\mathrm{\Delta }S}=\underset{\mathrm{\Delta }S\to 0}{\lim }\frac{(\frac{\partial W_y}{\partial x}-\frac{\partial W_x}{\partial y})\mathrm{\Delta }S}{\mathrm{\Delta }S}=(\frac{\partial W_y}{\partial x}-\frac{\partial W_x}{\partial y})=(\text{rot}\overrightarrow{W}{)}_z.$

$(\text{rot}\overrightarrow{W}{)}_x=(\frac{\partial W_z}{\partial y}-\frac{\partial W_y}{\partial z})$

$(\text{rot}\overrightarrow{W}{)}_y=(\frac{\partial W_x}{\partial z}-\frac{\partial W_z}{\partial x})$

$(\text{rot}\overrightarrow{W}{)}_z=(\frac{\partial W_y}{\partial x}-\frac{\partial W_x}{\partial y})$

$\text{rot}\overrightarrow{W}=\hat{x}(\frac{\partial W_z}{\partial y}-\frac{\partial W_y}{\partial z})+\hat{y}(\frac{\partial W_x}{\partial z}-\frac{\partial W_z}{\partial x})+\hat{z}(\frac{\partial W_y}{\partial x}-\frac{\partial W_x}{\partial y}).$

Očito u danoj fomuli možemo prepoznati simbolički zapisanu determinantu:

$\text{rot}\overrightarrow{W}=\left|\begin{array}{ccc}\hat{x}& \hat{y}& \hat{z}\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ W_x& W_y& W_z\end{array}\right|.$

Nadalje, očito je

$\text{rot}\overrightarrow{W}=(\hat{x}\frac{\partial }{\partial x}+\hat{y}\frac{\partial }{\partial y}+\hat{z}\frac{\partial }{\partial z})\times (\hat{x}{W}_x+\hat{y}{W}_y+\hat{z}{W}_z)=\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{W},$

pa $\text{rot}\overrightarrow{W}$ često označavamo s $\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{W}$, gdje je $\overrightarrow{\mathrm{\nabla }}$ Hamiltonov operator.

Rotacija i Stokesov teorem

Za rotaciju vrijedi Stokesov teorem

$\underset{S}{\int }\text{rot}\overrightarrow{W}\cdot d\overrightarrow{A}=\underset{C}{\int }\overrightarrow{W}\cdot d\overrightarrow{S}.$

Izrazi za rotaciju u drugim koordinatnim sustavima

• u cilindričnom:

$|(\text{rot}\overrightarrow{W}{)}_{\rho }|=\frac{1}{\rho }\frac{\partial W_z}{\partial \phi }-\frac{\partial W_{\phi }}{\partial z}$

$|(\text{rot}\overrightarrow{W}{)}_{\phi }|=\frac{\partial W_{\rho }}{\partial z}-\frac{\partial W_z}{\partial \rho }$

$|(\text{rot}\overrightarrow{W}{)}_z|=\frac{1}{\rho }\frac{\partial }{\partial \rho }(\rho W_{\phi })-\frac{1}{\rho }\frac{\partial W_{\rho }}{\partial \phi }$

$\text{rot}\overrightarrow{W}=[\frac{1}{\rho }\frac{\partial W_z}{\partial \phi }-\frac{\partial W_{\phi }}{\partial z}]\hat{\rho }+[\frac{\partial W_{\rho }}{\partial z}-\frac{\partial W_z}{\partial \rho }]\hat{\phi }+[\frac{1}{\rho }\frac{\partial }{\partial \rho }(\rho W_{\phi })-\frac{1}{\rho }\frac{\partial W_{\rho }}{\partial \phi }]\hat{z}$

• u sfernom:

$|(\text{rot}\overrightarrow{W}{)}_r|=\frac{1}{r\sin \vartheta }\frac{\partial }{\partial \vartheta }(W_{\phi }\sin \vartheta )-\frac{1}{r\sin \vartheta }\frac{\partial W_{\vartheta }}{\partial \phi }$

$|(\text{rot}\overrightarrow{W}{)}_{\vartheta }|=\frac{1}{r\sin \vartheta }\frac{\partial W_r}{\partial \phi }-\frac{1}{r}\frac{\partial }{\partial r}(r{W}_{\phi })$

$|(\text{rot}\overrightarrow{W}{)}_{\phi }|=\frac{1}{r}\frac{\partial }{\partial r}(r{W}_{\vartheta })-\frac{1}{r}\frac{\partial W_r}{\partial \vartheta }$

$\text{rot}\overrightarrow{W}=[\frac{1}{r\sin \vartheta }\frac{\partial }{\partial \vartheta }(W_{\phi }\sin \vartheta )-\frac{1}{r\sin \vartheta }\frac{\partial W_{\vartheta }}{\partial \phi }]\hat{r}+[\frac{1}{r\sin \vartheta }\frac{\partial W_r}{\partial \phi }-\frac{1}{r}\frac{\partial }{\partial r}(r{W}_{\phi })]\hat{\vartheta }+[\frac{1}{r}\frac{\partial }{\partial r}(r{W}_{\vartheta })-\frac{1}{r}\frac{\partial W_r}{\partial \vartheta }]\hat{\phi }.$

Rotacija i algebarske operacije

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

• $\text{rot}(\overrightarrow{u}+\overrightarrow{v})=\text{rot}\overrightarrow{u}+\text{rot}\overrightarrow{v}$
• $\text{rot}(U\cdot \overrightarrow{v})=U\cdot \text{rot}\overrightarrow{v}-\overrightarrow{v}\times \text{grad}U$
• $\text{rot}[f(U)\cdot \overrightarrow{v}]=f(U)\cdot \text{rot}\overrightarrow{v}-\overrightarrow{v}\times f_U^{\text{'}}(U)\text{grad}U$
• $\text{rot}\overrightarrow{r}=0$
• $\text{rot}(\overrightarrow{u}\times \overrightarrow{v})=\overrightarrow{u}\text{div}\overrightarrow{v}-\overrightarrow{v}\text{div}\overrightarrow{u}+(\overrightarrow{v}\cdot \overrightarrow{\mathrm{\nabla }})\overrightarrow{u}-(\overrightarrow{u}\cdot \overrightarrow{\mathrm{\nabla }})\overrightarrow{v}$
• $\text{grad}(\overrightarrow{u}\cdot \overrightarrow{v})=\overrightarrow{v}\times \text{rot}\overrightarrow{u}+\overrightarrow{u}\times \text{rot}\overrightarrow{v}+(\overrightarrow{v}\cdot \overrightarrow{\mathrm{\nabla }})\overrightarrow{u}+(\overrightarrow{u}\cdot \overrightarrow{\mathrm{\nabla }})\overrightarrow{v}$
• $\text{div}(\overrightarrow{u}\times \overrightarrow{v})=\overrightarrow{v}\text{rot}\overrightarrow{u}-\overrightarrow{u}\text{rot}\overrightarrow{v}.$

Primjeri

• Rotor elektrostaskog polja točkastog naboja, $\overrightarrow{E}=\frac{1}{4\pi {\epsilon }_0}\frac{q}{r^3}\overrightarrow{r}$:

$\text{rot}\overrightarrow{E}=\text{rot}\left(\frac{1}{4\pi {\epsilon }_0}\frac{q}{r^3}\overrightarrow{r}\right)\stackrel{(2.)}{=}\frac{q}{4\pi {\epsilon }_0{r}^3}\text{rot}\overrightarrow{r}-\overrightarrow{r}\times \text{grad}\frac{q}{4\pi {\epsilon }_0{r}^3}=-\frac{3q}{4\pi {\epsilon }_0{r}^4}\frac{\overrightarrow{r}}{r}\times \overrightarrow{r}=[\overrightarrow{r}\times \overrightarrow{r}=0]=0.$

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

Datoteka:Brzina rot.png

Shematski prikaz uz rotaciju polja obodne brzine

$\overrightarrow{v}=\overrightarrow{\omega }\times \overrightarrow{r}=\left|\begin{array}{ccc}\hat{x}& \hat{y}& \hat{z}\\ {\omega }_x& {\omega }_y& {\omega }_z\\ x& y& z\end{array}\right|=\hat{x}(z{\omega }_y-y{\omega }_z)+\hat{y}(x{\omega }_z-z{\omega }_x)+\hat{z}(y{\omega }_x-x{\omega }_y);$

$\text{rot}\overrightarrow{v}=\left|\begin{array}{ccc}\hat{x}& \hat{y}& \hat{z}\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ (z{\omega }_y-y{\omega }_z)& (x{\omega }_z-z{\omega }_x)& (y{\omega }_x-x{\omega }_y)\end{array}\right|=2{\omega }_x\hat{x}+2{\omega }_y\hat{y}+2{\omega }_z\hat{z}=2\overrightarrow{\omega }.$

Odatle se lako mogu iščitati komponente kutne brzine:

${\omega }_x=\frac{1}{2}(\frac{\partial v_z}{\partial y}-\frac{\partial v_y}{\partial z})$

${\omega }_y=\frac{1}{2}(\frac{\partial v_x}{\partial z}-\frac{\partial v_z}{\partial x})$

${\omega }_z=\frac{1}{2}(\frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y}).$

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

Vezani pojmovi

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

Vanjske veze

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