Silindirik ve küresel koordinatlarda del sayfasının kaynağını görüntüle

Bu liste [[curvilinear coordinates|eğrisel]] [[koordinat sistem]]leri ile çalışılırken genel olarak kullanılan [[vektör hesabı]] formüllerinin bir listesidir.

== Not ==
* 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.
<div style="overflow:auto">

{| class="wikitable" style="background: white"
|+[[del]] operatörü ile Silindirik küresel ve parabolik silindirik koordinatlar tablosu
<!-- Header -->
|-
! style="background: white" | işlem
! style="background: white" | [[Kartezyen koordinatlar]] (x, y,z)
! style="background: white" | [[Silindirik koordinatlar]] (ρ,φ,z)
! style="background: white" | [[Küresel koordinatlar]] (r,θ,φ)
! style="background: white" | [[Parabolik silindrik koordinatlar]] (σ,τ,z)

<!-- Definition of coordinates -->
|- align="center"
! rowspan="2" style="background: white" | Koordinat Tanımları
| <math>\begin{align}
\rho & = \sqrt{x^2+y^2} \\
\phi & = \arctan(y/x) \\
z & = z \end{align}</math>
| <math>\begin{align}
x & = \rho\cos\phi \\
y & = \rho\sin\phi \\
z & = z \end{align}</math>
| <math>\begin{align}
x & = r\sin\theta\cos\phi \\
y & = r\sin\theta\sin\phi \\
z & = r\cos\theta \end{align}</math>
| <math>\begin{align}
x & = \sigma \tau\\
y & = \tfrac{1}{2} \left( \tau^{2} - \sigma^{2} \right) \\
z & = z \end{align}</math>
|- align="center"
| <math>\begin{align}
r      & = \sqrt{x^2+y^2+z^2} \\
\theta & = \arccos(z/r)\\
\phi   & = \arctan(y/x) \end{align}</math>
| <math>\begin{align}
r      & = \sqrt{\rho^2 + z^2} \\
\theta & = \arctan{(\rho/z)}\\
\phi   & = \phi \end{align}</math>
| <math>\begin{align}
\rho & = r\sin(\theta) \\
\phi & = \phi\\
z    & = r\cos(\theta) \end{align}</math>
| <math>\begin{align}
\rho\cos\phi & = \sigma \tau\\
\rho\sin\phi & = \frac{1}{2} \left( \tau^{2} - \sigma^{2} \right) \\
z & = z \end{align}</math>

<!-- Definition of unit vectors -->
|- align="center"
! rowspan="2" style="background: white" | Birim Vektölerin Tanımları
| <math>\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}</math>
| <math>\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}</math>
| <math>\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}</math>
| <math>\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}</math>
|- align="center"
| <math>\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}</math>
| <math>\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}</math>
| <math>\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}</math>
| <math>\begin{matrix}
\end{matrix}</math>

<!-- Definition of A -->
|- align="center"
! style="background: white" | Bir vektör alanı<math>\mathbf{A}</math>
| <math>A_x\mathbf{\hat x} + A_y\mathbf{\hat y} + A_z\mathbf{\hat z}</math>
| <math>A_\rho\boldsymbol{\hat \rho} + A_\phi\boldsymbol{\hat \phi} + A_z\boldsymbol{\hat z}</math>
| <math>A_r\boldsymbol{\hat r} + A_\theta\boldsymbol{\hat \theta} + A_\phi\boldsymbol{\hat \phi}</math>
| <math>A_\sigma\boldsymbol{\hat \sigma} + A_\tau\boldsymbol{\hat \tau} + A_\phi\boldsymbol{\hat z}</math>

<!-- grad f -->
|- align="center"
! style="background: white" | [[Gradient|Gradyan]]
<math>\nabla f</math>
| <math>{\partial f \over \partial x}\mathbf{\hat x} + {\partial f \over \partial y}\mathbf{\hat y}
+ {\partial f \over \partial z}\mathbf{\hat z}</math>
| <math>{\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}</math>
| <math>{\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}</math>
| <math> \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}</math>

<!-- div A -->
|- align="center"
! style="background: white" | [[Divergence|Diverjans]]
<math>\nabla \cdot \mathbf{A}</math>
| <math>{\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z}</math>
| <math>{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}</math>
| <math>{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}</math>
| <math> \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}</math>

<!-- curl A -->
|- align="center"
! style="background: white" | [[Curl (mathematics)|Curl]] <math>\nabla \times \mathbf{A}</math>
| <math>\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}</math>
| <math>\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}</math>
| <math>\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}</math>
| <math>\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}</math>

<!-- Laplacian f -->
|- align="center"
! style="background: white" | [[Laplace operator|Laplace işlemcisi]]<math>\Delta f = \nabla^2 f</math>
| <math>{\partial^2 f \over \partial x^2} + {\partial^2 f \over \partial y^2} + {\partial^2 f \over \partial z^2}</math>
| <math>{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}</math>
| <math>{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}</math>
| <math> \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}}
</math>

<!-- vector Laplacian A -->
|- align="center"
! style="background: white" | [[Vektör Laplasyeni]] <math>\Delta \mathbf{A} = \nabla^2 \mathbf{A}</math>
| <math>\Delta A_x \mathbf{\hat x} + \Delta A_y \mathbf{\hat y} + \Delta A_z \mathbf{\hat z} </math>
| <math>\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}</math>
| align="center" | <math>\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}</math>

<!-- Material derivative (A dot del)B -->
|- align="center"
! style="background: white" | [[Malzeme türevi]]<ref name="Mathworld">{{Web kaynağı | url = http://mathworld.wolfram.com/ConvectiveOperator.html | başlık = Convective Operator | yazar = Weisstein, Eric W. | tarih =  | çalışma = Mathworld | yayıncı =  | erişimtarihi = 23 Mart 2011 | arşivurl = https://web.archive.org/web/20160303221612/http://mathworld.wolfram.com/ConvectiveOperator.html | arşivtarihi = 3 Mart 2016 | ölüurl = hayır }}</ref>
<math>(\mathbf{A} \cdot \nabla) \mathbf{B}</math>
<!--         Cartesian -->
| <math>\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}</math>
<!--         Cylindrical \frac{\partial B_}{\partial } -->
| <math>\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}</math>
<!--         Sp -->
| align="center" | <math>\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}</math>

<!--Diferansiyel değiştirme-->
|- align="center"
! style="background: white" | Diferansiyel yer değiştirme
|<math> \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}</math>
| <math>\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}</math>
| <math>\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}</math>
| <math>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}</math>

<!-- Differentials normal area -->
|- align="center"
! style="background: white" | Diferansiyel yüzey normali
| <math>\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}</math>
| <math>\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}</math>
| <math>\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}</math>
| <math>\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}</math>

<!-- Differentials volume -->
|- align="center"
! style="background: white" | Diferansiyel hacim
| <math>dV = dx\,dy\,dz \,</math>
| <math>dV = \rho\, d\rho\, d\phi\, dz\,</math>
| <math>dV = r^2\sin\theta \,dr\,d\theta\, d\phi\,</math>
| <math>dV = \left( \sigma^{2} + \tau^{2} \right) d\sigma d\tau dz,</math>

<!-- nabla's on nabla's -->
|-
| colspan=5 | <strong>''önemli birtakım hesaplama kuralları:''</strong>
<ol>
<li>'''''<math>\operatorname{div}\ \operatorname{grad} f = \nabla \cdot (\nabla f) = \nabla^2 f = \Delta f</math> ([[Laplasyen]])'''''</li>
<li>'''''<math>\operatorname{curl}\ \operatorname{grad} f = \nabla \times (\nabla f) = \mathbf{0}</math>'''''</li>
<li>'''''<math>\operatorname{div}\ \operatorname{curl} \mathbf{A} = \nabla \cdot (\nabla \times \mathbf{A}) = 0</math>'''''</li>
<li>'''''<math>\operatorname{curl}\ \operatorname{curl} \mathbf{A} = \nabla \times (\nabla \times \mathbf{A})
= \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}</math> (Vektör çarpımı için Lagrange formülünü kullanarak)'''''</li>
<li>'''''<math>\Delta (f g) = f \Delta g + 2 \nabla f \cdot \nabla g + g \Delta f</math>'''''</li>
</ol>
|}
</div>

== Ayrıca bakınız ==
* [[Del işlemcisi]]
* [[Ortogonal koordinatlar]]
* [[Eğrisel koordinatlar]]
* [[Silindirik ve küresel koordinatlarda vektör alanı]]
* [[Birim vektör]]

== Kaynakça ==
{{Kaynakça}}

== Dış bağlantılar ==
* [http://www.csulb.edu/~woollett/ Maxima Computer Algebra system scripts]{{Webarşiv|url=https://web.archive.org/web/20140906133257/http://www.csulb.edu/~woollett/ |tarih=6 Eylül 2014 }} to generate some of these operators in cylindrical and spherical coordinates.

[[Kategori:Vektör hesabı]]
[[Kategori:Koordinat sistemleri]]