Vektorske operacije u zakrivljenim koordinatama

Ovo je popis najbitnijih relacija i operacija vektorskog računa u sfernom, cilindričnom, paraboličnom i kartezijevu sustavu.

Operacija	Kartezijeve koordinate (x,y,z)	Cilindrične koordinate (ρ,φ,z)	Sferne koordinate (r,ϑ,φ)	Parabolične koordinate (σ,τ,z)
Definicija koordinata	$\begin{matrix} ρ & = & \sqrt{x^{2} + y^{2}} \\ φ & = & \arctan (y / x) \\ z & = & z \end{matrix}$	$\begin{matrix} x & = & ρ \cos φ \\ y & = & ρ \sin φ \\ z & = & z \end{matrix}$	$\begin{matrix} x & = & r \sin ϑ \cos φ \\ y & = & r \sin ϑ \sin φ \\ z & = & r \cos ϑ \end{matrix}$	$\begin{matrix} x & = & σ τ \\ y & = & \frac{1}{2} (τ^{2} - σ^{2}) \\ z & = & z \end{matrix}$
Definicija koordinata	$\begin{matrix} r & = & \sqrt{x^{2} + y^{2} + z^{2}} \\ ϑ & = & \arccos (z / r) \\ φ & = & \arctan (y / x) \end{matrix}$	$\begin{matrix} r & = & \sqrt{ρ^{2} + z^{2}} \\ ϑ & = & \arctan (ρ / z) \\ φ & = & φ \end{matrix}$	$\begin{matrix} ρ & = & r \sin (ϑ) \\ φ & = & φ \\ z & = & r \cos (ϑ) \end{matrix}$	$\begin{matrix} ρ \cos φ & = & σ τ \\ ρ \sin φ & = & \frac{1}{2} (τ^{2} - σ^{2}) \\ z & = & z \end{matrix}$
Definicija jediničnih vektora	$\begin{matrix} \hat{𝝆} & = & \frac{x}{ρ} \hat{𝐱} + \frac{y}{ρ} \hat{𝐲} \\ \hat{𝝋} & = & - \frac{y}{ρ} \hat{𝐱} + \frac{x}{ρ} \hat{𝐲} \\ \hat{𝐳} & = & \hat{𝐳} \end{matrix}$	$\begin{matrix} \hat{𝐱} & = & \cos φ \hat{𝝆} - \sin φ \hat{𝝋} \\ \hat{𝐲} & = & \sin φ \hat{𝝆} + \cos φ \hat{𝝋} \\ \hat{𝐳} & = & \hat{𝐳} \end{matrix}$	$\begin{matrix} \hat{𝐱} & = & \sin ϑ \cos φ \hat{𝒓} + \cos ϑ \cos φ \hat{𝝑} - \sin φ \hat{𝝋} \\ \hat{𝐲} & = & \sin ϑ \sin φ \hat{𝒓} + \cos ϑ \sin φ \hat{𝝑} + \cos φ \hat{𝝋} \\ \hat{𝐳} & = & \cos ϑ \hat{𝒓} - \sin ϑ \hat{𝝑} \end{matrix}$	$\begin{matrix} \hat{𝝈} & = & \frac{τ}{\sqrt{τ^{2} + σ^{2}}} \hat{𝐱} - \frac{σ}{\sqrt{τ^{2} + σ^{2}}} \hat{𝐲} \\ \hat{𝝉} & = & \frac{σ}{\sqrt{τ^{2} + σ^{2}}} \hat{𝐱} + \frac{τ}{\sqrt{τ^{2} + σ^{2}}} \hat{𝐲} \\ \hat{𝐳} & = & \hat{𝐳} \end{matrix}$
Definicija jediničnih vektora	$\begin{matrix} \hat{𝐫} & = & \frac{x \hat{𝐱} + y \hat{𝐲} + z \hat{𝐳}}{r} \\ \hat{𝝑} & = & \frac{x z \hat{𝐱} + y z \hat{𝐲} - ρ^{2} \hat{𝐳}}{r ρ} \\ \hat{𝝋} & = & \frac{- y \hat{𝐱} + x \hat{𝐲}}{ρ} \end{matrix}$	$\begin{matrix} \hat{𝐫} & = & \frac{ρ}{r} \hat{𝝆} + \frac{z}{r} \hat{𝐳} \\ \hat{𝝑} & = & \frac{z}{r} \hat{𝝆} - \frac{ρ}{r} \hat{𝐳} \\ \hat{𝝋} & = & \hat{𝝋} \end{matrix}$	$\begin{matrix} \hat{𝝆} & = & \sin ϑ \hat{𝐫} + \cos ϑ \hat{𝝑} \\ \hat{𝝋} & = & \hat{𝝋} \\ \hat{𝐳} & = & \cos ϑ \hat{𝐫} - \sin ϑ \hat{𝝑} \end{matrix}$
A vektorsko polje $𝐀$	$A_{x} \hat{𝐱} + A_{y} \hat{𝐲} + A_{z} \hat{𝐳}$	$A_{ρ} \hat{𝝆} + A_{φ} \hat{𝝋} + A_{z} \hat{𝒛}$	$A_{r} \hat{𝒓} + A_{ϑ} \hat{𝝑} + A_{φ} \hat{𝝋}$	$A_{σ} \hat{𝝈} + A_{τ} \hat{𝝉} + A_{φ} \hat{𝒛}$
Gradijent $\nabla f$	$\frac{\partial f}{\partial x} \hat{𝐱} + \frac{\partial f}{\partial y} \hat{𝐲} + \frac{\partial f}{\partial z} \hat{𝐳}$	$\frac{\partial f}{\partial ρ} \hat{𝝆} + \frac{1}{ρ} \frac{\partial f}{\partial φ} \hat{𝝋} + \frac{\partial f}{\partial z} \hat{𝒛}$	$\frac{\partial f}{\partial r} \hat{𝒓} + \frac{1}{r} \frac{\partial f}{\partial ϑ} \hat{𝝑} + \frac{1}{r \sin ϑ} \frac{\partial f}{\partial φ} \hat{𝝋}$	$\frac{1}{\sqrt{σ^{2} + τ^{2}}} \frac{\partial f}{\partial σ} \hat{𝝈} + \frac{1}{\sqrt{σ^{2} + τ^{2}}} \frac{\partial f}{\partial τ} \hat{𝝉} + \frac{\partial f}{\partial z} \hat{𝒛}$
Divergencija $\nabla \cdot 𝐀$	$\frac{\partial A_{x}}{\partial x} + \frac{\partial A_{y}}{\partial y} + \frac{\partial A_{z}}{\partial z}$	$\frac{1}{ρ} \frac{\partial (ρ A_{ρ})}{\partial ρ} + \frac{1}{ρ} \frac{\partial A_{φ}}{\partial φ} + \frac{\partial A_{z}}{\partial z}$	$\frac{1}{r^{2}} \frac{\partial (r^{2} A_{r})}{\partial r} + \frac{1}{r \sin ϑ} \frac{\partial}{\partial ϑ} (A_{ϑ} \sin ϑ) + \frac{1}{r \sin ϑ} \frac{\partial A_{φ}}{\partial φ}$	$\frac{1}{σ^{2} + τ^{2}} \frac{\partial A_{σ}}{\partial σ} + \frac{1}{σ^{2} + τ^{2}} \frac{\partial A_{τ}}{\partial τ} + \frac{\partial A_{z}}{\partial z}$
Rotacija $\nabla \times 𝐀$	$\begin{matrix} (\frac{\partial A_{z}}{\partial y} - \frac{\partial A_{y}}{\partial z}) \hat{𝐱} & + \\ (\frac{\partial A_{x}}{\partial z} - \frac{\partial A_{z}}{\partial x}) \hat{𝐲} & + \\ (\frac{\partial A_{y}}{\partial x} - \frac{\partial A_{x}}{\partial y}) \hat{𝐳} \end{matrix}$	$\begin{matrix} (\frac{1}{ρ} \frac{\partial A_{z}}{\partial φ} - \frac{\partial A_{φ}}{\partial z}) \hat{𝝆} & + \\ (\frac{\partial A_{ρ}}{\partial z} - \frac{\partial A_{z}}{\partial ρ}) \hat{𝝋} & + \\ \frac{1}{ρ} (\frac{\partial (ρ A_{φ})}{\partial ρ} - \frac{\partial A_{ρ}}{\partial φ}) \hat{𝒛} \end{matrix}$	$\begin{matrix} \frac{1}{r \sin ϑ} (\frac{\partial}{\partial ϑ} (A_{φ} \sin ϑ) - \frac{\partial A_{ϑ}}{\partial φ}) \hat{𝒓} & + \\ \frac{1}{r} (\frac{1}{\sin ϑ} \frac{\partial A_{r}}{\partial φ} - \frac{\partial}{\partial r} (r A_{φ})) \hat{𝝑} & + \\ \frac{1}{r} (\frac{\partial}{\partial r} (r A_{ϑ}) - \frac{\partial A_{r}}{\partial ϑ}) \hat{𝝋} \end{matrix}$	$\begin{matrix} (\frac{1}{\sqrt{σ^{2} + τ^{2}}} \frac{\partial A_{z}}{\partial τ} - \frac{\partial A_{τ}}{\partial z}) \hat{𝝈} & - \\ (\frac{1}{\sqrt{σ^{2} + τ^{2}}} \frac{\partial A_{z}}{\partial σ} - \frac{\partial A_{σ}}{\partial z}) \hat{𝝉} & + \\ \frac{1}{\sqrt{σ^{2} + τ^{2}}} (\frac{\partial (ρ A_{φ})}{\partial ρ} - \frac{\partial A_{ρ}}{\partial φ}) \hat{𝒛} \end{matrix}$
Laplasijan $Δ f = \nabla^{2} f$	$\frac{\partial^{2} f}{\partial x^{2}} + \frac{\partial^{2} f}{\partial y^{2}} + \frac{\partial^{2} f}{\partial z^{2}}$	$\frac{1}{ρ} \frac{\partial}{\partial ρ} (ρ \frac{\partial f}{\partial ρ}) + \frac{1}{ρ^{2}} \frac{\partial^{2} f}{\partial φ^{2}} + \frac{\partial^{2} f}{\partial z^{2}}$	$\frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial f}{\partial r}) + \frac{1}{r^{2} \sin ϑ} \frac{\partial}{\partial ϑ} (\sin ϑ \frac{\partial f}{\partial ϑ}) + \frac{1}{r^{2} \sin^{2} ϑ} \frac{\partial^{2} f}{\partial φ^{2}}$	$\frac{1}{σ^{2} + τ^{2}} (\frac{\partial^{2} f}{\partial σ^{2}} + \frac{\partial^{2} f}{\partial τ^{2}}) + \frac{\partial^{2} f}{\partial z^{2}}$
Vekotrski laplasijan $Δ 𝐀 = \nabla^{2} 𝐀$	$Δ A_{x} \hat{𝐱} + Δ A_{y} \hat{𝐲} + Δ A_{z} \hat{𝐳}$	$\begin{matrix} (Δ A_{ρ} - \frac{A_{ρ}}{ρ^{2}} - \frac{2}{ρ^{2}} \frac{\partial A_{φ}}{\partial φ}) \hat{𝝆} & + \\ (Δ A_{φ} - \frac{A_{φ}}{ρ^{2}} + \frac{2}{ρ^{2}} \frac{\partial A_{ρ}}{\partial φ}) \hat{𝝋} & + \\ (Δ A_{z}) \hat{𝒛} \end{matrix}$	$\begin{matrix} (Δ A_{r} - \frac{2 A_{r}}{r^{2}} - \frac{2}{r^{2} \sin ϑ} \frac{\partial (A_{ϑ} \sin ϑ)}{\partial ϑ} - \frac{2}{r^{2} \sin ϑ} \frac{\partial A_{φ}}{\partial φ}) \hat{𝒓} & + \\ (Δ A_{ϑ} - \frac{A_{ϑ}}{r^{2} \sin^{2} ϑ} + \frac{2}{r^{2}} \frac{\partial A_{r}}{\partial ϑ} - \frac{2 \cos ϑ}{r^{2} \sin^{2} ϑ} \frac{\partial A_{φ}}{\partial φ}) \hat{𝝑} & + \\ (Δ A_{φ} - \frac{A_{φ}}{r^{2} \sin^{2} ϑ} + \frac{2}{r^{2} \sin ϑ} \frac{\partial A_{r}}{\partial φ} + \frac{2 \cos ϑ}{r^{2} \sin^{2} ϑ} \frac{\partial A_{ϑ}}{\partial φ}) \hat{𝝋} \end{matrix}$
Element duljine	$d 𝐥 = d x \hat{𝐱} + d y \hat{𝐲} + d z \hat{𝐳}$	$d 𝐥 = d ρ \hat{𝝆} + ρ d φ \hat{𝝋} + d z \hat{𝒛}$	$d 𝐥 = d r \hat{𝐫} + r d ϑ \hat{𝝑} + r \sin ϑ d φ \hat{𝝋}$	$d 𝐥 = \sqrt{σ^{2} + τ^{2}} d σ \hat{𝝈} + \sqrt{σ^{2} + τ^{2}} d τ \hat{𝝉} + d z \hat{𝒛}$
Element površine	$\begin{matrix} d 𝐒 = & d y d z \hat{𝐱} + \\ d x d z \hat{𝐲} + \\ d x d y \hat{𝐳} \end{matrix}$	$\begin{matrix} d 𝐒 = & ρ d φ d z \hat{𝝆} + \\ d ρ d z \hat{𝝋} + \\ ρ d ρ d φ \hat{𝐳} \end{matrix}$	$\begin{matrix} d 𝐒 = & r^{2} \sin ϑ d ϑ d φ \hat{𝐫} + \\ r \sin ϑ d r d φ \hat{𝝑} + \\ r d r d ϑ \hat{𝝋} \end{matrix}$	$\begin{matrix} d 𝐒 = & \sqrt{σ^{2} + τ^{2}}, d τ d z \hat{𝝈} + \\ \sqrt{σ^{2} + τ^{2}} d σ d z \hat{𝝉} + \\ σ^{2} + τ^{2} d σ, d τ \hat{𝐳} \end{matrix}$
Element obujma	$d V = d x d y d z$	$d V = ρ d ρ d φ d z$	$d V = r^{2} \sin ϑ d r d ϑ d φ$	$d V = (σ^{2} + τ^{2}) d σ d τ d z,$
Netrivijalne kombinacije vektorskih operacija: $d i v g r a d f = \nabla \cdot (\nabla f) = \nabla^{2} f = Δ f$ (laplasijan) $r o t g r a d f = \nabla \times (\nabla f) = 𝟎$ $d i v r o t 𝐀 = \nabla \cdot (\nabla \times 𝐀) = 0$ $r o t r o t 𝐀 = \nabla \times (\nabla \times 𝐀) = \nabla (\nabla \cdot 𝐀) - \nabla^{2} 𝐀$ $Δ f g = f Δ g + 2 \nabla f \cdot \nabla g + g Δ f$

Vezani pojmovi

Vektorske operacije u zakrivljenim koordinatama

Vezani pojmovi

Navigacijski izbornik

Traži