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।

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।

$\varphi$ 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].

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

संक्रिया (Operation)	कार्तीय निर्देशांक (x,y,z)	बेलनाकार निर्देशांक (s,φ,z)	गोलीय निर्देशांक (r,θ,φ)	परबलीय बेलनाकार निर्देशांक (ο,τ,z)
निर्देशांकों की परिभाषा	$\begin{array}{ccc}s& =& \sqrt{x^2+y^2}\\ \varphi & =& \arctan (y/x)\\ z& =& z\end{array}$	$\begin{array}{ccc}x& =& s\cos \varphi \\ y& =& s\sin \varphi \\ z& =& z\end{array}$	$\begin{array}{ccc}x& =& r\sin \theta \cos \varphi \\ y& =& r\sin \theta \sin \varphi \\ z& =& r\cos \theta \end{array}$	$\begin{array}{ccc}x& =& \sigma \tau \\ y& =& \frac{1}{2}({\tau }^2-{\sigma }^2)\\ z& =& z\end{array}$
	$\begin{array}{ccc}r& =& \sqrt{x^2+y^2+z^2}\\ \theta & =& \arctan \left(\frac{\sqrt{x^2+y^2}}{z}\right)\\ \varphi & =& \arctan (y/x)\\ \end{array}$	$\begin{array}{ccc}r& =& \sqrt{s^2+z^2}\\ \theta & =& \arctan (s/z)\\ \varphi & =& \varphi \end{array}$	$\begin{array}{ccc}s& =& r\sin (\theta )\\ \varphi & =& \varphi \\ z& =& r\cos (\theta )\end{array}$	$\begin{array}{ccc}s\cos \varphi & =& \sigma \tau \\ s\sin \varphi & =& \frac{1}{2}({\tau }^2-{\sigma }^2)\\ z& =& z\end{array}$
सदिशों की unit परिभाषा	$\begin{array}{ccc}\hat{\mathit{s}}& =& \frac{x}{s}\hat{\mathbf{x}}+\frac{y}{s}\hat{\mathbf{y}}\\ \hat{\mathit{\varphi }}& =& -\frac{y}{s}\hat{\mathbf{x}}+\frac{x}{s}\hat{\mathbf{y}}\\ \hat{\mathbf{z}}& =& \hat{\mathbf{z}}\end{array}$	$\begin{array}{ccc}\hat{\mathbf{x}}& =& \cos \varphi \hat{\mathit{s}}-\sin \varphi \hat{\mathit{\varphi }}\\ \hat{\mathbf{y}}& =& \sin \varphi \hat{\mathit{s}}+\cos \varphi \hat{\mathit{\varphi }}\\ \hat{\mathbf{z}}& =& \hat{\mathbf{z}}\end{array}$	$\begin{array}{ccc}\hat{\mathbf{x}}& =& \sin \theta \cos \varphi \hat{\mathit{r}}+\cos \theta \cos \varphi \hat{\mathit{\theta }}-\sin \varphi \hat{\mathit{\varphi }}\\ \hat{\mathbf{y}}& =& \sin \theta \sin \varphi \hat{\mathit{r}}+\cos \theta \sin \varphi \hat{\mathit{\theta }}+\cos \varphi \hat{\mathit{\varphi }}\\ \hat{\mathbf{z}}& =& \cos \theta \hat{\mathit{r}}-\sin \theta \hat{\mathit{\theta }}\\ \end{array}$	$\begin{array}{ccc}\hat{\mathit{\sigma }}& =& \frac{\tau }{\sqrt{{\tau }^2+{\sigma }^2}}\hat{\mathbf{x}}-\frac{\sigma }{\sqrt{{\tau }^2+{\sigma }^2}}\hat{\mathbf{y}}\\ \hat{\mathit{\tau }}& =& \frac{\sigma }{\sqrt{{\tau }^2+{\sigma }^2}}\hat{\mathbf{x}}+\frac{\tau }{\sqrt{{\tau }^2+{\sigma }^2}}\hat{\mathbf{y}}\\ \hat{\mathbf{z}}& =& \hat{\mathbf{z}}\end{array}$
	$\begin{array}{ccc}\hat{\mathbf{r}}& =& \frac{x\hat{\mathbf{x}}+y\hat{\mathbf{y}}+z\hat{\mathbf{z}}}{r}\\ \hat{\mathit{\theta }}& =& \frac{xz\hat{\mathbf{x}}+yz\hat{\mathbf{y}}-s^2\hat{\mathbf{z}}}{rs}\\ \hat{\mathit{\varphi }}& =& \frac{-y\hat{\mathbf{x}}+x\hat{\mathbf{y}}}{s}\end{array}$	$\begin{array}{ccc}\hat{\mathbf{r}}& =& \frac{s}{r}\hat{\mathit{s}}+\frac{z}{r}\hat{\mathbf{z}}\\ \hat{\mathit{\theta }}& =& \frac{z}{r}\hat{\mathit{s}}-\frac{s}{r}\hat{\mathbf{z}}\\ \hat{\mathit{\varphi }}& =& \hat{\mathit{\varphi }}\end{array}$	$\begin{array}{ccc}\hat{\mathit{s}}& =& \sin \theta \hat{\mathbf{r}}+\cos \theta \hat{\mathit{\theta }}\\ \hat{\mathit{\varphi }}& =& \hat{\mathit{\varphi }}\\ \hat{\mathbf{z}}& =& \cos \theta \hat{\mathbf{r}}-\sin \theta \hat{\mathit{\theta }}\\ \end{array}$	$\begin{array}{c}\end{array}$
A vector field $\mathbf{𝐀}$	$A_x\hat{\mathbf{x}}+A_y\hat{\mathbf{y}}+A_z\hat{\mathbf{z}}$	$A_s\hat{\mathit{s}}+A_{\varphi }\hat{\mathit{\varphi }}+A_z\hat{\mathit{z}}$	$A_r\hat{\mathit{r}}+A_{\theta }\hat{\mathit{\theta }}+A_{\varphi }\hat{\mathit{\varphi }}$	$A_{\sigma }\hat{\mathit{\sigma }}+A_{\tau }\hat{\mathit{\tau }}+A_{\varphi }\hat{\mathit{z}}$
Gradient $\mathrm{\nabla }f$	$\frac{\partial f}{\partial x}\hat{\mathbf{x}}+\frac{\partial f}{\partial y}\hat{\mathbf{y}}+\frac{\partial f}{\partial z}\hat{\mathbf{z}}$	$\frac{\partial f}{\partial s}\hat{\mathit{s}}+\frac{1}{s}\frac{\partial f}{\partial \varphi }\hat{\mathit{\varphi }}+\frac{\partial f}{\partial z}\hat{\mathit{z}}$	$\frac{\partial f}{\partial r}\hat{\mathit{r}}+\frac{1}{r}\frac{\partial f}{\partial \theta }\hat{\mathit{\theta }}+\frac{1}{r\sin \theta }\frac{\partial f}{\partial \varphi }\hat{\mathit{\varphi }}$	$\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial f}{\partial \sigma }\hat{\mathit{\sigma }}+\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial f}{\partial \tau }\hat{\mathit{\tau }}+\frac{\partial f}{\partial z}\hat{\mathit{z}}$
Divergence $\mathrm{\nabla }\cdot \mathbf{𝐀}$	$\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}$	$\frac{1}{s}\frac{\partial \left(s{A}_s\right)}{\partial s}+\frac{1}{s}\frac{\partial A_{\varphi }}{\partial \varphi }+\frac{\partial A_z}{\partial z}$	$\frac{1}{r^2}\frac{\partial \left(r^2{A}_r\right)}{\partial r}+\frac{1}{r\sin \theta }\frac{\partial }{\partial \theta }(A_{\theta }\sin \theta )+\frac{1}{r\sin \theta }\frac{\partial A_{\varphi }}{\partial \varphi }$	$\frac{1}{{\sigma }^2+{\tau }^2}\frac{\partial A_{\sigma }}{\partial \sigma }+\frac{1}{{\sigma }^2+{\tau }^2}\frac{\partial A_{\tau }}{\partial \tau }+\frac{\partial A_z}{\partial z}$
Curl $\mathrm{\nabla }\times \mathbf{𝐀}$	$\begin{array}{cc}(\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z})\hat{\mathbf{x}}& +\\ (\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x})\hat{\mathbf{y}}& +\\ (\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y})\hat{\mathbf{z}}& \end{array}$	$\begin{array}{cc}(\frac{1}{s}\frac{\partial A_z}{\partial \varphi }-\frac{\partial A_{\varphi }}{\partial z})\hat{\mathit{s}}& +\\ (\frac{\partial A_s}{\partial z}-\frac{\partial A_z}{\partial s})\hat{\mathit{\varphi }}& +\\ \frac{1}{s}(\frac{\partial \left(s{A}_{\varphi }\right)}{\partial s}-\frac{\partial A_s}{\partial \varphi })\hat{\mathit{z}}& \end{array}$	$\begin{array}{cc}\frac{1}{r\sin \theta }(\frac{\partial }{\partial \theta }(A_{\varphi }\sin \theta )-\frac{\partial A_{\theta }}{\partial \varphi })\hat{\mathit{r}}& +\\ \frac{1}{r}(\frac{1}{\sin \theta }\frac{\partial A_r}{\partial \varphi }-\frac{\partial }{\partial r}\left(r{A}_{\varphi }\right))\hat{\mathit{\theta }}& +\\ \frac{1}{r}(\frac{\partial }{\partial r}\left(r{A}_{\theta }\right)-\frac{\partial A_r}{\partial \theta })\hat{\mathit{\varphi }}& \end{array}$	$\begin{array}{cc}(\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial A_z}{\partial \tau }-\frac{\partial A_{\tau }}{\partial z})\hat{\mathit{\sigma }}& -\\ (\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial A_z}{\partial \sigma }-\frac{\partial A_{\sigma }}{\partial z})\hat{\mathit{\tau }}& +\\ \frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}(\frac{\partial \left(s{A}_{\varphi }\right)}{\partial s}-\frac{\partial A_s}{\partial \varphi })\hat{\mathit{z}}& \end{array}$
Laplace operator $\mathrm{\Delta }f={\mathrm{\nabla }}^{2}f$	$\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2}+\frac{{\partial }^{2}f}{\partial z^2}$	$\frac{1}{s}\frac{\partial }{\partial s}\left(s\frac{\partial f}{\partial s}\right)+\frac{1}{s^2}\frac{{\partial }^{2}f}{\partial {\varphi }^2}+\frac{{\partial }^{2}f}{\partial z^2}$	$\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial f}{\partial r}\right)+\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }(\sin \theta \frac{\partial f}{\partial \theta })+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^{2}f}{\partial {\varphi }^2}$	$\frac{1}{{\sigma }^2+{\tau }^2}(\frac{{\partial }^{2}f}{\partial {\sigma }^2}+\frac{{\partial }^{2}f}{\partial {\tau }^2})+\frac{{\partial }^{2}f}{\partial z^2}$
Vector Laplacian $\mathrm{\Delta }\mathbf{𝐀}={\mathrm{\nabla }}^2\mathbf{𝐀}$	$\mathrm{\Delta }{A}_x\hat{\mathbf{x}}+\mathrm{\Delta }{A}_y\hat{\mathbf{y}}+\mathrm{\Delta }{A}_z\hat{\mathbf{z}}$	$\begin{array}{cc}(\mathrm{\Delta }{A}_s-\frac{A_s}{s^2}-\frac{2}{s^2}\frac{\partial A_{\varphi }}{\partial \varphi })\hat{\mathit{s}}& +\\ (\mathrm{\Delta }{A}_{\varphi }-\frac{A_{\varphi }}{s^2}+\frac{2}{s^2}\frac{\partial A_s}{\partial \varphi })\hat{\mathit{\varphi }}& +\\ \left(\mathrm{\Delta }{A}_z\right)\hat{\mathit{z}}& \end{array}$	$\begin{array}{cc}(\mathrm{\Delta }{A}_r-\frac{2A_r}{r^2}-\frac{2}{r^2\sin \theta }\frac{\partial (A_{\theta }\sin \theta )}{\partial \theta }-\frac{2}{r^2\sin \theta }\frac{\partial A_{\varphi }}{\partial \varphi })\hat{\mathit{r}}& +\\ (\mathrm{\Delta }{A}_{\theta }-\frac{A_{\theta }}{r^2{\sin }^2\theta }+\frac{2}{r^2}\frac{\partial A_r}{\partial \theta }-\frac{2\cos \theta }{r^2{\sin }^2\theta }\frac{\partial A_{\varphi }}{\partial \varphi })\hat{\mathit{\theta }}& +\\ (\mathrm{\Delta }{A}_{\varphi }-\frac{A_{\varphi }}{r^2{\sin }^2\theta }+\frac{2}{r^2\sin \theta }\frac{\partial A_r}{\partial \varphi }+\frac{2\cos \theta }{r^2{\sin }^2\theta }\frac{\partial A_{\theta }}{\partial \varphi })\hat{\mathit{\varphi }}& \end{array}$
Differential displacement	$d\mathbf{𝐥}=dx\hat{\mathbf{x}}+dy\hat{\mathbf{y}}+dz\hat{\mathbf{z}}$	$d\mathbf{𝐥}=ds\hat{\mathit{s}}+sd\varphi \hat{\mathit{\varphi }}+dz\hat{\mathit{z}}$	$d\mathbf{𝐥}=dr\hat{\mathbf{r}}+rd\theta \hat{\mathit{\theta }}+r\sin \theta d\varphi \hat{\mathit{\varphi }}$	$d\mathbf{𝐥}=\sqrt{{\sigma }^2+{\tau }^2}d\sigma \hat{\mathit{\sigma }}+\sqrt{{\sigma }^2+{\tau }^2}d\tau \hat{\mathit{\tau }}+dz\hat{\mathit{z}}$
Differential normal area	$\begin{array}{cc}d\mathbf{𝐒}=& dydz\hat{\mathbf{x}}+\\ & dxdz\hat{\mathbf{y}}+\\ & dxdy\hat{\mathbf{z}}\end{array}$	$\begin{array}{cc}d\mathbf{𝐒}=& sd\varphi dz\hat{\mathit{s}}+\\ & dsdz\hat{\mathit{\varphi }}+\\ & sdsd\varphi \hat{\mathbf{z}}\end{array}$	$\begin{array}{cc}d\mathbf{𝐒}=& r^2\sin \theta d\theta d\varphi \hat{\mathbf{r}}+\\ & r\sin \theta drd\varphi \hat{\mathit{\theta }}+\\ & rdrd\theta \hat{\mathit{\varphi }}\end{array}$	$\begin{array}{cc}d\mathbf{𝐒}=& \sqrt{{\sigma }^2+{\tau }^2},d\tau dz\hat{\mathit{\sigma }}+\\ & \sqrt{{\sigma }^2+{\tau }^2}d\sigma dz\hat{\mathit{\tau }}+\\ & {\sigma }^2+{\tau }^{2}d\sigma ,d\tau \hat{\mathbf{z}}\end{array}$
Differential volume	$d\tau =dxdydz$	$d\tau =sdsd\varphi dz$	$d\tau =r^2\sin \theta drd\theta d\varphi$	$d\tau =({\sigma }^2+{\tau }^2)d\sigma d\tau dz,$
डेल संक्रिया के कुछ असरल नियम: $\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}f=\mathrm{\nabla }\cdot (\mathrm{\nabla }f)={\mathrm{\nabla }}^{2}f=\mathrm{\Delta }f$ (Laplacian) $\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}f=\mathrm{\nabla }\times (\mathrm{\nabla }f)=0$ $\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\mathbf{𝐀}=\mathrm{\nabla }\cdot (\mathrm{\nabla }\times \mathbf{𝐀})=0$ $\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\mathbf{𝐀}=\mathrm{\nabla }\times (\mathrm{\nabla }\times \mathbf{𝐀})=\mathrm{\nabla }(\mathrm{\nabla }\cdot \mathbf{𝐀})-{\mathrm{\nabla }}^2\mathbf{𝐀}$ (using Lagrange's formula for the cross product) $\mathrm{\Delta }fg=f\mathrm{\Delta }g+2\mathrm{\nabla }f\cdot \mathrm{\nabla }g+g\mathrm{\Delta }f$

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