Silindirik ve küresel koordinatlarda del

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Bu liste eğrisel koordinat sistemleri ile çalışılırken genel olarak kullanılan vektör hesabı formüllerinin bir listesidir .

Not[değiştir | kaynağı değiştir]

  • Bu sayfada standart fizik gösterim kullanır. küresel koordinatlar için, θ açısı yarıçap vektörünün z ekseni ile olan açısıdır ve Söz konusu noktaya orijinden bağlanır. ϕ açısı yarıçap vektörünün x-y yüzeyine izdüşümü ile ve x ekseni ile olan açıdır.Bazı kaynaklar θ ve ϕ yi ters tanıtırlar, bu anlam bağlamında böyle bir bağlantı kurulmamalıdır.
  • atan2(y, x) fonsiyonu kendi etki ve görüntü nedeniyle matematiksel fonksiyon arctan(y/x) yerine kullanılabilir,klasik arctan(y/x) görüntüsü (-π/2, +π/2)aralığında idi,buradaki atan2(y, x) (-π, π] aralığındadır. (Küresel koordinatlarda Del için ifadelerin düzeltilmesi gerekebilir)
  • Dönüşümler kartezyen koordinatlardan silindirik ve küreseledir.
del operatörü ile Silindirik küresel ve parabolik silindirik koordinatlar tablosu
işlem Kartezyen koordinatlar (x,y,z) Silindirik koordinatlar (ρ,φ,z) Küresel koordinatlar (r,θ,φ) Parabolik silindrik koordinatlar (σ,τ,z)
Koordinat Tanımları \begin{align}
\rho & = \sqrt{x^2+y^2} \\
\phi & = \arctan(y/x) \\
z & = z \end{align} \begin{align}
x & = \rho\cos\phi \\
y & = \rho\sin\phi \\
z & = z \end{align} \begin{align}
x & = r\sin\theta\cos\phi \\
y & = r\sin\theta\sin\phi \\
z & = r\cos\theta \end{align} \begin{align}
x & = \sigma \tau\\
y & = \tfrac{1}{2} \left( \tau^{2} - \sigma^{2} \right) \\
z & = z \end{align}
\begin{align}
r      & = \sqrt{x^2+y^2+z^2} \\
\theta & = \arccos(z/r)\\
\phi   & = \arctan(y/x) \end{align} \begin{align}
r      & = \sqrt{\rho^2 + z^2} \\
\theta & = \arctan{(\rho/z)}\\
\phi   & = \phi \end{align} \begin{align}
\rho & = r\sin(\theta) \\
\phi & = \phi\\
z    & = r\cos(\theta) \end{align} \begin{align}
\rho\cos\phi & = \sigma \tau\\
\rho\sin\phi & = \frac{1}{2} \left( \tau^{2} - \sigma^{2} \right) \\
z & = z \end{align}
Birim Vektölerin Tanımları \begin{align}
\boldsymbol{\hat \rho} & = \frac{x\mathbf{\hat x} + y\mathbf{\hat y}}{\sqrt{x^2+y^2}} \\
\boldsymbol{\hat\phi} & =  \frac{-y\mathbf{\hat x} + x\mathbf{\hat y}}{\sqrt{x^2+y^2}}  \\
\mathbf{\hat z}       & =  \mathbf{\hat z}
\end{align} \begin{align}
\mathbf{\hat x} & = \cos\phi\boldsymbol{\hat \rho}-\sin\phi\boldsymbol{\hat\phi} \\
\mathbf{\hat y} & =\sin\phi\boldsymbol{\hat \rho}+\cos\phi\boldsymbol{\hat\phi} \\
\mathbf{\hat z} & = \mathbf{\hat z}
\end{align} \begin{align}
\mathbf{\hat x} & = \cos\phi\left( \sin\theta\boldsymbol{\hat r}+ \cos\theta\boldsymbol{\hat\theta}\right) -\sin\phi\boldsymbol{\hat\phi} \\
\mathbf{\hat y} & = \sin\phi\left( \sin\theta\boldsymbol{\hat r}+\ cos\theta\boldsymbol{\hat\theta}\right) +\cos\phi\boldsymbol{\hat\phi} \\
\mathbf{\hat z} & = \cos\theta        \boldsymbol{\hat r}-\sin\theta        \boldsymbol{\hat\theta}
\end{align} \begin{align}
\boldsymbol{\hat \sigma} & = \frac{\tau}{\sqrt{\tau^2+\sigma^2}}\mathbf{\hat x}-\frac{\sigma}{\sqrt{\tau^2+\sigma^2}}\mathbf{\hat y} \\
\boldsymbol{\hat\tau}    & = \frac{\sigma}{\sqrt{\tau^2+\sigma^2}}\mathbf{\hat x}+\frac{\tau}{\sqrt{\tau^2+\sigma^2}}\mathbf{\hat y} \\
\mathbf{\hat z}          & = \mathbf{\hat z}
\end{align}
\begin{align}
\mathbf{\hat r}    &= \frac{ x \mathbf{\hat x} \!+\! y \mathbf{\hat y} \!+\! z \mathbf{\hat z} }{\sqrt{x^2+y^2+z^2}}\\
\boldsymbol{\hat\theta} &= \frac{ z \left( x \mathbf{\hat x}\!+\!y \mathbf{\hat y} \right) \!-\! \left( x^2+y^2 \right) \mathbf{\hat z} }{ \sqrt{x^2+y^2+z^2} \sqrt{x^2+y^2} }\\
\boldsymbol{\hat\phi} &= \frac{-y \mathbf{\hat x}+ x \mathbf{\hat y}}{\sqrt{x^2+y^2}}
\end{align} \begin{align}
\mathbf{\hat r}         & = \frac{\rho}{\sqrt{\rho^2 +z^2}}\boldsymbol{\hat \rho}+\frac{   z}{\sqrt{\rho^2 +z^2}}\mathbf{\hat z} \\
\boldsymbol{\hat\theta} & = \frac{z}{\sqrt{\rho^2 +z^2}}\boldsymbol{\hat \rho}-\frac{\rho}{\sqrt{\rho^2 +z^2}}\mathbf{\hat z} \\
\boldsymbol{\hat\phi}   & = \boldsymbol{\hat\phi}
\end{align} \begin{align}
\boldsymbol{\hat \rho} & =  \sin\theta\mathbf{\hat r}+\cos\theta\boldsymbol{\hat\theta} \\
\boldsymbol{\hat\phi} & = \boldsymbol{\hat\phi} \\
\mathbf{\hat z}       & = \cos\theta\mathbf{\hat r}-\sin\theta\boldsymbol{\hat\theta}
\end{align} \begin{matrix}
\end{matrix}
Bir vektör alanı\mathbf{A} A_x\mathbf{\hat x} + A_y\mathbf{\hat y} + A_z\mathbf{\hat z} A_\rho\boldsymbol{\hat \rho} + A_\phi\boldsymbol{\hat \phi} + A_z\boldsymbol{\hat z} A_r\boldsymbol{\hat r} + A_\theta\boldsymbol{\hat \theta} + A_\phi\boldsymbol{\hat \phi} A_\sigma\boldsymbol{\hat \sigma} + A_\tau\boldsymbol{\hat \tau} + A_\phi\boldsymbol{\hat z}
Gradyan

\nabla f

{\partial f \over \partial x}\mathbf{\hat x} + {\partial f \over \partial y}\mathbf{\hat y}
+ {\partial f \over \partial z}\mathbf{\hat z} {\partial f \over \partial \rho}\boldsymbol{\hat \rho}
+ {1 \over \rho}{\partial f \over \partial \phi}\boldsymbol{\hat \phi}
+ {\partial f \over \partial z}\boldsymbol{\hat z} {\partial f \over \partial r}\boldsymbol{\hat r}
+ {1 \over r}{\partial f \over \partial \theta}\boldsymbol{\hat \theta}
+ {1 \over r\sin\theta}{\partial f \over \partial \phi}\boldsymbol{\hat \phi}  \frac{1}{\sqrt{\sigma^{2} + \tau^{2}}} {\partial f \over \partial \sigma}\boldsymbol{\hat \sigma} + \frac{1}{\sqrt{\sigma^{2} + \tau^{2}}} {\partial f \over \partial \tau}\boldsymbol{\hat \tau} + {\partial f \over \partial z}\boldsymbol{\hat z}
Diverjans

\nabla \cdot \mathbf{A}

{\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z} {1 \over \rho}{\partial \left( \rho A_\rho  \right) \over \partial \rho}
+ {1 \over \rho}{\partial A_\phi \over \partial \phi}
+ {\partial A_z \over \partial z} {1 \over r^2}{\partial \left( r^2 A_r \right) \over \partial r}
+ {1 \over r\sin\theta}{\partial \over \partial \theta} \left(  A_\theta\sin\theta \right)
+ {1 \over r\sin\theta}{\partial A_\phi \over \partial \phi}  \frac{1}{\sigma^{2} + \tau^{2}}\left({\partial (\sqrt{\sigma^2+\tau^2} A_\sigma) \over \partial \sigma} + {\partial (\sqrt{\sigma^2+\tau^2} A_\tau) \over \partial \tau}\right) + {\partial A_z \over \partial z}
Curl \nabla \times \mathbf{A} \begin{matrix}
\displaystyle\left({\partial A_z \over \partial y} - {\partial A_y \over \partial z}\right) \mathbf{\hat x} & + \\
\displaystyle\left({\partial A_x \over \partial z} - {\partial A_z \over \partial x}\right) \mathbf{\hat y} & + \\
\displaystyle\left({\partial A_y \over \partial x} - {\partial A_x \over \partial y}\right) \mathbf{\hat z} & \ \end{matrix} \begin{matrix}
\displaystyle\left({1 \over \rho}{\partial A_z \over \partial \phi}
- {\partial A_\phi \over \partial z}\right) \boldsymbol{\hat \rho} & + \\
\displaystyle\left({\partial A_\rho \over \partial z} - {\partial A_z \over \partial \rho}\right) \boldsymbol{\hat \phi} & + \\
\displaystyle{1 \over \rho}\left({\partial \left( \rho A_\phi \right) \over \partial \rho}
- {\partial A_\rho \over \partial \phi}\right) \boldsymbol{\hat z} & \ \end{matrix} \begin{matrix}
\displaystyle{1 \over r\sin\theta}\left({\partial \over \partial \theta} \left( A_\phi\sin\theta \right)
- {\partial A_\theta \over \partial \phi}\right) \boldsymbol{\hat r} & + \\
\displaystyle{1 \over r}\left({1 \over \sin\theta}{\partial A_r \over \partial \phi}
- {\partial \over \partial r} \left( r A_\phi \right) \right) \boldsymbol{\hat \theta} & + \\
\displaystyle{1 \over r}\left({\partial \over \partial r} \left( r A_\theta \right)
- {\partial A_r \over \partial \theta}\right) \boldsymbol{\hat \phi} & \ \end{matrix} \begin{matrix}
\displaystyle\left(\frac{1}{\sqrt{\sigma^{2} + \tau^{2}}}{\partial A_z \over \partial \tau}
- {\partial A_\tau \over \partial z}\right) \boldsymbol{\hat \sigma} & - \\
\displaystyle\left(\frac{1}{\sqrt{\sigma^{2} + \tau^{2}}}{\partial A_z \over \partial \sigma}- {\partial A_\sigma \over \partial z}\right) \boldsymbol{\hat \tau} & + \\
\displaystyle\frac{1}{\sqrt{\sigma^{2} + \tau^{2}}}\left({\partial \left( \sqrt{\sigma^{2} + \tau^{2}} A_\sigma \right) \over \partial \tau}
- {\partial \left( \sqrt{\sigma^{2} + \tau^{2}} A_\tau \right) \over \partial \sigma}\right) \boldsymbol{\hat z} & \ \end{matrix}
Laplace işlemcisi\Delta f = \nabla^2 f {\partial^2 f \over \partial x^2} + {\partial^2 f \over \partial y^2} + {\partial^2 f \over \partial z^2} {1 \over \rho}{\partial \over \partial \rho}\left(\rho {\partial f \over \partial \rho}\right)
+ {1 \over \rho^2}{\partial^2 f \over \partial \phi^2}
+ {\partial^2 f \over \partial z^2} {1 \over r^2}{\partial \over \partial r}\!\left(r^2 {\partial f \over \partial r}\right)
\!+\!{1 \over r^2\!\sin\theta}{\partial \over \partial \theta}\!\left(\sin\theta {\partial f \over \partial \theta}\right)
\!+\!{1 \over r^2\!\sin^2\theta}{\partial^2 f \over \partial \phi^2}  \frac{1}{\sigma^{2} + \tau^{2}}
\left(  \frac{\partial^{2} f}{\partial \sigma^{2}} +
\frac{\partial^{2} f}{\partial \tau^{2}} \right) +
\frac{\partial^{2} f}{\partial z^{2}}
Vektör Laplasyeni \Delta \mathbf{A} = \nabla^2 \mathbf{A} \Delta A_x \mathbf{\hat x} + \Delta A_y \mathbf{\hat y} + \Delta A_z \mathbf{\hat z} \begin{matrix}
\displaystyle\left(\Delta A_\rho - {A_\rho \over \rho^2}
- {2 \over \rho^2}{\partial A_\phi \over \partial \phi}\right) \boldsymbol{\hat \rho} & + \\
\displaystyle\left(\Delta A_\phi - {A_\phi \over \rho^2}
+ {2 \over \rho^2}{\partial A_\rho \over \partial \phi}\right) \boldsymbol{\hat\phi} & + \\
\displaystyle\left(\Delta A_z \right) \boldsymbol{\hat z}  & \ \end{matrix} \begin{matrix}
\left(\Delta A_r - {2 A_r \over r^2}
- {2 \over r^2\sin\theta}{\partial \left(A_\theta \sin\theta\right) \over \partial\theta}
- {2 \over r^2\sin\theta}{\partial A_\phi \over \partial \phi}\right) \boldsymbol{\hat r} & + \\
\left(\Delta A_\theta - {A_\theta \over r^2\sin^2\theta}
+ {2 \over r^2}{\partial A_r \over \partial \theta}
- {2 \cos\theta \over r^2\sin^2\theta}{\partial A_\phi \over \partial \phi}\right) \boldsymbol{\hat\theta} & + \\
\left(\Delta A_\phi - {A_\phi \over r^2\sin^2\theta}
+ {2 \over r^2\sin\theta}{\partial A_r \over \partial \phi}
+ {2 \cos\theta \over r^2\sin^2\theta}{\partial A_\theta \over \partial \phi}\right) \boldsymbol{\hat\phi} & \end{matrix}
Malzeme türevi[1]

(\mathbf{A} \cdot \nabla) \mathbf{B}

\begin{matrix}
\displaystyle\left(A_x \frac{\partial}{\partial x}+A_y \frac{\partial}{\partial y} + A_z \frac{\partial}{\partial z}\right) B_x \boldsymbol{\hat{x}} + \\
\displaystyle\left(A_x \frac{\partial}{\partial x}+A_y \frac{\partial}{\partial y} + A_z \frac{\partial}{\partial z}\right) B_y \boldsymbol{\hat{y}} + \\
\displaystyle\left(A_x \frac{\partial}{\partial x}+A_y \frac{\partial}{\partial y} + A_z \frac{\partial}{\partial z}\right) B_z \boldsymbol{\hat{z}}
\end{matrix} \begin{matrix}
\left(A_\rho \frac{\partial B_\rho}{\partial \rho}+\frac{A_\phi}{\rho}\frac{\partial B_\rho}{\partial \phi}+A_z\frac{\partial B_\rho}{\partial z}-\frac{A_\phi B_\phi}{\rho}\right) \boldsymbol{\hat\rho} \!+\!\\
\left(A_\rho \frac{\partial B_\phi}{\partial \rho}+\frac{A_\phi}{\rho}\frac{\partial B_\phi}{\partial \phi}+A_z\frac{\partial B_\phi}{\partial z}+\frac{A_\phi B_\rho}{\rho}\right) \boldsymbol{\hat\phi}\!+\!\\
\left(A_\rho \frac{\partial B_z   }{\partial \rho}+\frac{A_\phi}{\rho}\frac{\partial B_z   }{\partial \phi}+A_z\frac{\partial B_z   }{\partial z}                           \right) \boldsymbol{\hat z}
\end{matrix} \begin{matrix}
\left(A_r \frac{\partial B_r}{\partial r}\!+\!\frac{A_\theta}{r}\frac{\partial B_r}{\partial \theta}\!+\!\frac{A_\phi}{r\sin(\theta)}\frac{\partial B_r}{\partial \phi}\!-\!\frac{A_\theta B_\theta\!+\!A_\phi B_\phi}{r}\right)                  \boldsymbol{\hat r} \!+\!\\
\left(A_r \frac{\partial B_\theta}{\partial r}\!+\!\frac{A_\theta}{r}\frac{\partial B_\theta}{\partial \theta}\!+\!\frac{A_\phi}{r\sin(\theta)}\frac{\partial B_\theta}{\partial \phi}\!+\!\frac{A_\theta B_r}{r}-\frac{A_\phi B_\phi\cot(\theta)}{r}\right)  \boldsymbol{\hat\theta} \!+\!\\
\left(A_r \frac{\partial B_\phi}{\partial r}\!+\!\frac{A_\theta}{r}\frac{\partial B_\phi}{\partial \theta}\!+\!\frac{A_\phi}{r\sin(\theta)}\frac{\partial B_\phi}{\partial \phi}\!+\!\frac{A_\phi B_r}{r}\!+\!\frac{A_\phi B_\theta \cot(\theta)}{r}\right)\boldsymbol{\hat\phi}
\end{matrix}
Diferansiyel yer değiştirme  \begin{align} d\mathbf{l}&=d\mathbf{x} + d\mathbf{y} + d\mathbf{z} \\
        &= dx\mathbf{\hat x} + dy\mathbf{\hat y} + dz\mathbf{\hat z} \end{align} \begin{align} d\mathbf{l}&=d\boldsymbol{\rho} + d\boldsymbol{\phi} + d\mathbf{z} \\
        &= d\rho\boldsymbol{\hat \rho} + \rho d\phi\boldsymbol{\hat \phi} + dz\boldsymbol{\hat z} \end{align} \begin{align} d\mathbf{l}&=d\mathbf{r} + d\boldsymbol{\theta} + d\boldsymbol{\phi} \\
        &= dr\mathbf{\hat r} + rd\theta\boldsymbol{\hat \theta} + r\sin\theta d\phi\boldsymbol{\hat \phi} \end{align} d\mathbf{l} = \sqrt{\sigma^{2} + \tau^{2}} d\sigma\boldsymbol{\hat \sigma} + \sqrt{\sigma^{2} + \tau^{2}} d\tau\boldsymbol{\hat \tau} + dz\boldsymbol{\hat z}
Diferansiyel yüzey normali \begin{align}
d\mathbf{S}&= d\mathbf{y}\times d\mathbf{z} + d\mathbf{z}\times d\mathbf{x} + d\mathbf{x}\times d\mathbf{y} \\
&= dy\,dz\,\mathbf{\hat x} + dx\,dz\,\mathbf{\hat y} + dx\,dy\,\mathbf{\hat z} 
\end{align} \begin{align}
d\mathbf{S}&= d\boldsymbol{\phi} \times d\mathbf{z} + d\mathbf{z} \times d\boldsymbol{\rho} + d\boldsymbol{\rho} \times d\boldsymbol{\phi} \\
&= \rho\, d\phi\, dz\,\boldsymbol{\hat \rho} + 
  d\rho \,dz\,\boldsymbol{\hat \phi} + 
  \rho \,d\rho d\phi \,\mathbf{\hat z}
\end{align} \begin{align}
d\mathbf{S}&=
d\boldsymbol{\theta} \times d\boldsymbol{\phi} + d\boldsymbol{\phi} \times d\mathbf{r} + d\mathbf{r} \times d\boldsymbol{\theta}\\
&= r^2 \sin\theta \,d\theta \,d\phi \,\mathbf{\hat r} + 
 r\sin\theta \,d\phi \,dr\,\boldsymbol{\hat \theta} + 
 r\,dr\,d\theta\,\boldsymbol{\hat \phi}
\end{align} \begin{matrix}
d\mathbf{S} = & \sqrt{\sigma^{2} + \tau^{2}}, d\tau\, dz\,\boldsymbol{\hat \sigma} + \\
& \sqrt{\sigma^{2} + \tau^{2}} d\sigma\,dz\,\boldsymbol{\hat \tau} + \\
& \sigma^{2} + \tau^{2} d\sigma, d\tau \,\mathbf{\hat z}
\end{matrix}
Diferansiyel hacim dV = dx\,dy\,dz \, dV = \rho\, d\rho\, d\phi\, dz\, dV = r^2\sin\theta \,dr\,d\theta\, d\phi\, dV = \left( \sigma^{2} + \tau^{2} \right) d\sigma d\tau dz,
önemsiz olmayan hesaplama kuralları:
  1. \operatorname{div}\ \operatorname{grad} f = \nabla \cdot (\nabla f) = \nabla^2 f = \Delta f (Laplasyen)
  2. \operatorname{curl}\ \operatorname{grad} f = \nabla \times (\nabla f) = \mathbf{0}
  3. \operatorname{div}\ \operatorname{curl} \mathbf{A} = \nabla \cdot (\nabla \times \mathbf{A}) = 0
  4. \operatorname{curl}\ \operatorname{curl} \mathbf{A} = \nabla \times (\nabla \times \mathbf{A})
= \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A} (Vektör çarpımı için Lagrange formülünü kullanarak)
  5. \Delta (f g) = f \Delta g + 2 \nabla f \cdot \nabla g + g \Delta f

Ayrıca bakınız[değiştir | kaynağı değiştir]

Kaynakça[değiştir | kaynağı değiştir]

  1. ^ Weisstein, Eric W.. "Convective Operator". Mathworld. http://mathworld.wolfram.com/ConvectiveOperator.html. Erişim tarihi: 23 March 2011. 

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