"सदिश कलन": अवतरणों में अंतर

 

== अन्य निर्देशांकों में सदिश संक्रियाएँ ==

 

:<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{curl}}\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>

 

+ \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)