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=== ग्रेडिएन्ट (Gradient) ===
:<math> \operatorname{grad}~\varphi =\vec \nabla\varphi = \begin{pmatrix} \frac{\partial\varphi} {\partial x} \\[0.2cm] \frac{\partial\varphi}{\partial y} \\[0.2cm] \frac{\partial\varphi}{\partial z} \end{pmatrix}</math>
=== डाइवर्जेंस (Divergence) ===
:<math>\operatorname{rot}~\vec F = \vec \nabla\times\vec F =
\begin{pmatrix}
\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \\[0.2cm]
\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \\[0.2cm]
\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}
\end{pmatrix}</math>
:<math>\vec{\mathrm{grad}}f
= \frac{\partial f}{\partial r}\vec{u_r}
+ \frac{1}{r}\frac{\partial f}{\partial \theta}\vec{u_\theta}
+ \frac{1}{r \sin\theta}\frac{\partial f}{\partial \varphi} \vec{u_\varphi}</math>
:<math>\mathrm{div}\vec{A}
= \frac{1}{r^2}\frac{\partial}{\partial r}(r^2A_r)
+ \frac{1}{r\sin\theta}\frac{\partial} {\partial \theta}(\sin\theta A_\theta)
+ \frac{1}{r\sin\theta}\frac{\partial A_\varphi}{\partial \varphi}</math>
:<math>\vec{\mathrm{curl}}\vec{A}
= \frac{1}{r\sin\theta}\left(\frac{\partial}{\partial \theta}(\sin\theta A_\varphi)-\frac{\partial A_\theta}{\partial \varphi}\right)\vec{u_r}
+ \left(\frac{1}{r\sin\theta}\frac{\partial A_r}{\partial \varphi}-\frac{1}{r}\frac{\partial}{\partial r}(rA_\varphi)\right)\vec{u_\theta}
+ \frac{1}{r}\left(\frac{\partial}{\partial r}(rA_\theta)-\frac{\partial A_r}{\partial \theta}\right)\vec{u_\varphi}</math>
:<math>\Delta f
= \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}\left(\sin \theta\frac{\partial f}{\partial \theta}\right)
+ \frac{1}{r^2\sin^2\theta}\frac{\partial^2 f}{\partial \varphi^2}</math>
== सर्वसमिकाएँ ==
* <math>\vec \nabla \times \big(\vec \nabla \times \vec A \big) = \vec \nabla \big(\vec \nabla \cdot \vec A \big) - \nabla^2 \vec A </math>
* <math>\vec \nabla \cdot \vec x = 3</math>
* <math>\vec \nabla \times \vec x = \vec 0</math>
यदि <math>\vec \nabla \cdot \vec A = 0</math> तो <math>\vec A = \vec \nabla \times \vec B</math> जहाँ <math>\vec B</math> कोई सदिश क्षेत्र है।
== सन्दर्भ ==
* {{cite book | author = Michael J. Crowe | title = A History of Vector Analysis : The Evolution of the Idea of a Vectorial System | publisher=Dover Publications; Reprint edition | year= 1994 | id = ISBN 0-486-67910-1 }} ([http://web.archive.org/web/20040126170335/http://www.nku.edu/~curtin/crowe_oresme.doc Summary])
* {{cite book | author = H. M. Schey | title = Div Grad Curl and all that: An informal text on vector calculus | publisher=W. W. Norton & Company | year= 2005 | id = ISBN 0-393-92516-1}}
* {{cite book | author = J.E. Marsden | title = Vecor Calculus | publisher=W. H. Freeman & Company | year= 1976 | id = ISBN 0-7167-o462-5}}
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