"सदिश कलन": अवतरणों में अंतर
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अनुनाद सिंह (वार्ता | योगदान) |
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पंक्ति 55:
| <math>\iiint\limits_V\left(\nabla\cdot\mathbf{F}\right)dV=\iint\limits_{\part V}\mathbf{F}\cdot d\mathbf{S},</math> || The integral of the divergence of a vector field over some solid equals the integral of the [[flux]] through the surface bounding the solid.
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== अन्य निर्देशांकों में सदिश संक्रियाएँ ==
=== बेलनाकार निर्देशांक में ===
:<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{\partial f}{\partial z}\vec{u_z}</math>
:<math>\mathrm{div}\vec{A}=\frac{1}{r}\frac{\partial}{\partial r}\left(rA_r \right)+\frac{1}{r}\frac{\partial A_\theta}{\partial \theta}+\frac{\partial A_z}{\partial z}</math>
:<math>\vec{\mathrm{rot}}\vec{A}=\left(\frac{1}{r}\frac{\partial A_z}{\partial \theta}-\frac{\partial A_\theta}{\partial z}\right)\vec{u_r} + \left(\frac{\partial A_r}{\partial z}-\frac{\partial A_z}{\partial r}\right)\vec{u_\theta} + \frac{1}{r}\left(\frac{\partial}{\partial r}(rA_\theta)-\frac{\partial A_r}{\partial \theta}\right)\vec{u_z}</math>
:<math>\Delta f=\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial f}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 f}{\partial \theta^2} + \frac{\partial^2 f}{\partial z^2}</math>
=== गोलीय निर्देशांक में ===
:<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{rot}}\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>
==इन्हें भी देखें==
* [[सदिश कैलकुलस की सर्वसमिकाएँ]]
==सन्दर्भ==
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