विभिन्न निर्देशांकों में डेल संक्रिया

इस पृष्ट पर विभिन्न निर्देशांक निकायों (coordinate systems) में कार्य करते समय प्रयोग में आने वाले सदिश कैलकुलस के प्रमुख सूत्र दिये गये हैं।


** कार्तीय एवं अन्य निर्देशांक निकायों में डेल संक्रिया (del operator) को प्रदर्शित करने वाली तालिका**

## टिप्पणी
* इस पृष्ट पर भौतिकी के मानक प्रतीकों (संकेतों) का प्रयोग हुआ है।

* गोलीय निर्देशांक में \theta *z* - अक्ष एवं रेडिअस वेक्टर (त्रिज्या सदिश) के बीच का कोण है।
\phi *x* - अक्ष एवं रेडिअस वेक्टर के *x-y* समतल पर प्रक्षेप (projection) के बीच का कोण है। कुछ (अमेरिकी गणित) के स्रोतों में यह संकेत परस्पर अदला-बदली करके लिये जाते हैं।

* arctan(y/x) फलन के स्थान पर atan2(y, x) फलन का प्रयोग किया गया है। क्योंकि arctan(y/x) का इमेज (परास) (-π/2, +π/2) होती है जबकि atan2(y, x) की (-π, π].

संक्रिया (Operation)
कार्तीय निर्देशांक (x,y,z)
बेलनाकार निर्देशांक (s,φ,z)
गोलीय निर्देशांक (r,θ,φ)
परबलीय बेलनाकार निर्देशांक (ο,τ,z)

निर्देशांकों कीपरिभाषा

\begin{matrix}
  s & = & \sqrt{x^2+y^2} \\
  \phi & = & \arctan(y/x) \\
    z & = & z \end{matrix}

\begin{matrix}
  x & = & s\cos\phi \\
  y & = & s\sin\phi \\
  z & = & z \end{matrix}

\begin{matrix}
  x & = & r\sin\theta\cos\phi \\
  y & = & r\sin\theta\sin\phi \\
  z & = & r\cos\theta \end{matrix}

\begin{matrix}
  x & = & \sigma \tau\\
  y & = & \frac{1}{2} \left(\tau^{2} - \sigma^{2} \right) \\
  z & = & z \end{matrix}



\begin{matrix}
  r   & = & \sqrt{x^2+y^2+z^2} \\
  \theta & = & \arctan{\left(\frac{\sqrt{x^2+y^2}}{z}\right)}\\
  \phi  & = & \arctan(y/x) \\ \end{matrix}

\begin{matrix}
  r   & = & \sqrt{s^2 + z^2} \\
  \theta & = & \arctan{(s/z)}\\
  \phi  & = & \phi \end{matrix}

\begin{matrix}
  s & = & r\sin(\theta) \\
  \phi & = & \phi\\
  z  & = & r\cos(\theta) \end{matrix}

\begin{matrix}
  s\cos\phi & = & \sigma \tau\\
  s\sin\phi & = & \frac{1}{2} \left(\tau^{2} - \sigma^{2} \right) \\
  z & = & z \end{matrix}

सदिशोंकीunitपरिभाषा

\begin{matrix}
  \boldsymbol{\hat s} & = & \frac{x}{s}\mathbf{\hat x}+\frac{y}{s}\mathbf{\hat y} \\
  \boldsymbol{\hat\phi} & = & -\frac{y}{s}\mathbf{\hat x}+\frac{x}{s}\mathbf{\hat y} \\
  \mathbf{\hat z}    & = & \mathbf{\hat z}
  \end{matrix}

\begin{matrix}
  \mathbf{\hat x} & = & \cos\phi\boldsymbol{\hat s}-\sin\phi\boldsymbol{\hat\phi} \\
  \mathbf{\hat y} & = & \sin\phi\boldsymbol{\hat s}+\cos\phi\boldsymbol{\hat\phi} \\
  \mathbf{\hat z} & = & \mathbf{\hat z}
  \end{matrix}

\begin{matrix}
  \mathbf{\hat x} & = & \sin\theta\cos\phi\boldsymbol{\hat r}+\cos\theta\cos\phi\boldsymbol{\hat\theta}-\sin\phi\boldsymbol{\hat\phi} \\
  \mathbf{\hat y} & = & \sin\theta\sin\phi\boldsymbol{\hat r}+\cos\theta\sin\phi\boldsymbol{\hat\theta}+\cos\phi\boldsymbol{\hat\phi} \\
  \mathbf{\hat z} & = & \cos\theta    \boldsymbol{\hat r}-\sin\theta    \boldsymbol{\hat\theta} \\
  \end{matrix}

\begin{matrix}
  \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{matrix}



\begin{matrix}
  \mathbf{\hat r}     & = & \frac{x\mathbf{\hat x}+y\mathbf{\hat y}+z\mathbf{\hat z}}{r} \\
  \boldsymbol{\hat\theta} & = & \frac{xz\mathbf{\hat x}+yz\mathbf{\hat y}-s^2\mathbf{\hat z}}{r s} \\
  \boldsymbol{\hat\phi}  & = & \frac{-y\mathbf{\hat x}+x\mathbf{\hat y}}{s}
  \end{matrix}

\begin{matrix}
  \mathbf{\hat r}     & = & \frac{s}{r}\boldsymbol{\hat s}+\frac{  z}{r}\mathbf{\hat z} \\
  \boldsymbol{\hat\theta} & = & \frac{z  }{r}\boldsymbol{\hat s}-\frac{s}{r}\mathbf{\hat z} \\
  \boldsymbol{\hat\phi}  & = & \boldsymbol{\hat\phi}
  \end{matrix}

\begin{matrix}
  \boldsymbol{\hat s} & = & \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{matrix}

\begin{matrix}
  \end{matrix}

A vector field \mathbf{A}
A_x\mathbf{\hat x} + A_y\mathbf{\hat y} + A_z\mathbf{\hat z}
A_s\boldsymbol{\hat s} + 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}

Gradient \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 s}\boldsymbol{\hat s} 
 + {1 \over s}{\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}

Divergence \nabla \cdot \mathbf{A}

{\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z}

{1 \over s}{\partial \left(s A_s \right) \over \partial s} 
 + {1 \over s}{\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}}{\partial A_\sigma \over \partial \sigma} + \frac{1}{\sigma^{2} + \tau^{2}}{\partial A_\tau \over \partial \tau} + {\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 s}{\partial A_z \over \partial \phi}
  - {\partial A_\phi \over \partial z}\right) \boldsymbol{\hat s} & + \\
 \displaystyle\left({\partial A_s \over \partial z} - {\partial A_z \over \partial s}\right) \boldsymbol{\hat \phi} & + \\
 \displaystyle{1 \over s}\left({\partial \left(s A_\phi \right) \over \partial s} 
  - {\partial A_s \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(s A_\phi \right) \over \partial s} 
  - {\partial A_s \over \partial \phi}\right) \boldsymbol{\hat z} & \ \end{matrix}

Laplace operator \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 s}{\partial \over \partial s}\left(s {\partial f \over \partial s}\right) 
 + {1 \over s^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}}



Vector Laplacian \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_s - {A_s \over s^2} 
  - {2 \over s^2}{\partial A_\phi \over \partial \phi}\right) \boldsymbol{\hat s} & + \\
 \displaystyle\left(\Delta A_\phi - {A_\phi \over s^2} 
  + {2 \over s^2}{\partial A_s \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}


Differential displacement

d\mathbf{l} = dx\mathbf{\hat x} + dy\mathbf{\hat y} + dz\mathbf{\hat z}

d\mathbf{l} = ds\boldsymbol{\hat s} + s d\phi\boldsymbol{\hat \phi} + dz\boldsymbol{\hat z}

d\mathbf{l} = dr\mathbf{\hat r} + rd\theta\boldsymbol{\hat \theta} + r\sin\theta d\phi\boldsymbol{\hat \phi}

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}


Differential normal area

\begin{matrix}d\mathbf{S} = &dy\,dz\,\mathbf{\hat x} + \\ 
&dx\,dz\,\mathbf{\hat y} + \\ 
&dx\,dy\,\mathbf{\hat z}\end{matrix}

\begin{matrix}
d\mathbf{S} = & s\, d\phi\, dz\,\boldsymbol{\hat s} + \\ 
& ds \,dz\,\boldsymbol{\hat \phi} + \\ 
& s \,ds d\phi \,\mathbf{\hat z}
\end{matrix}

\begin{matrix}
d\mathbf{S} = & r^2 \sin\theta \,d\theta \,d\phi \,\mathbf{\hat r} + \\
& r\sin\theta \,dr\,d\phi \,\boldsymbol{\hat \theta} + \\
& r\,dr\,d\theta\,\boldsymbol{\hat \phi}
\end{matrix}

\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}


Differential volume

d\tau = dx\,dy\,dz \,

d\tau = s\, ds\, d\phi\, dz\,

d\tau = r^2\sin\theta \,dr\,d\theta\, d\phi\,

d\tau = \left(\sigma^{2} + \tau^{2} \right) d\sigma d\tau dz,


**डेल संक्रिया के कुछ असरल नियम:**

\operatorname{div\ grad\ } f = \nabla \cdot (\nabla f) = \nabla^2 f = \Delta f (Laplacian)
\operatorname{curl\ grad\ } f = \nabla \times (\nabla f) = 0
\operatorname{div\ curl\ } \mathbf{A} = \nabla \cdot (\nabla \times \mathbf{A}) = 0
\operatorname{curl\ curl\ } \mathbf{A} = \nabla \times (\nabla \times \mathbf{A}) 
                        = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A} (using Lagrange's formula for the cross product)
\Delta f g = f \Delta g + 2 \nabla f \cdot \nabla g + g \Delta f




## इन्हें भी देखें
* सदिश कैलकुलस
* वक्रीय निर्देशांक

श्रेणी:गणित
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