דל במערכות צירים שונות

באנליזה וקטורית ניתן לכתוב אופרטורים שונים, הקשורים לאופרטור דל (המסומל באמצעות הסימן נבלה), בדרכים שונות במערכות צירים שונות.

הערה: הנוסחאות שבדף זה כתובות לפי הכתיב הפיזיקלי המקובל. בקואורדינטות כדוריות, $\theta$ הזווית בין ציר Z ווקטור הרדיוס המחבר את הראשית עם הנקודה בה עוסקים. $\varphi$ היא הזווית בין היטל וקטור הרדיוס על מישור XY לבין ציר X.

	קואורדינטות קרטזיות (x,y,z)	קואורדינטות גליליות (ρ,φ,z)	קואורדינטות כדוריות (r,θ,φ)	קואורדינטות גליליות פרבוליות (σ,τ,z)
הגדרת מערכת הצירים	$\begin{array}{rl}\rho & =\sqrt{x^2+y^2}\\ \varphi & =\arctan \left(\frac{y}{x}\right)\\ z& =z\end{array}$	$\begin{array}{rl}x& =\rho \cos (\varphi )\\ y& =\rho \sin (\varphi )\\ z& =z\end{array}$	$\begin{array}{rl}x& =r\sin (\theta )\cos (\varphi )\\ y& =r\sin (\theta )\sin (\varphi )\\ z& =r\cos (\theta )\end{array}$	$\begin{array}{rl}x& =\sigma (\tau )\\ y& =\frac{{\tau }^2-{\sigma }^2}{2}\\ z& =z\end{array}$
	$\begin{array}{rl}r& =\sqrt{x^2+y^2+z^2}\\ \theta & =\arctan \left(\frac{\sqrt{x^2+y^2}}{z}\right)\\ \varphi & =\arctan \left(\frac{y}{x}\right)\end{array}$	$\begin{array}{rl}\rho & =r\sin (\theta )\\ \varphi & =\varphi \\ z& =r\cos (\theta )\end{array}$	$\begin{array}{rl}r& =\sqrt{x^2+y^2+z^2}\\ \theta & =\arctan \left(\frac{\sqrt{x^2+y^2}}{z}\right)\\ \varphi & =\arctan \left(\frac{y}{x}\right)\end{array}$	$\begin{array}{rl}\rho \cos (\varphi )& =\sigma \tau \\ \rho \sin (\varphi )& =\frac{{\tau }^2-{\sigma }^2}{2}\\ z& =z\end{array}$
הגדרת וקטורי היחידה	$\begin{array}{rl}\hat{\mathit{\rho }}& =\frac{x}{\rho }\hat{\mathbf{x}}+\frac{y}{\rho }\hat{\mathbf{y}}\\ \hat{\mathit{\varphi }}& =-\frac{y}{\rho }\hat{\mathbf{x}}+\frac{x}{\rho }\hat{\mathbf{y}}\\ \hat{\mathbf{z}}& =\hat{\mathbf{z}}\end{array}$	$\begin{array}{rl}\hat{\mathbf{x}}& =\cos (\varphi )\hat{\mathit{\rho }}-\sin (\varphi )\hat{\mathit{\varphi }}\\ \hat{\mathbf{y}}& =\sin (\varphi )\hat{\mathit{\rho }}+\cos (\varphi )\hat{\mathit{\varphi }}\\ \hat{\mathbf{z}}& =\hat{\mathbf{z}}\end{array}$	$\begin{array}{rl}\hat{\mathbf{x}}& =\sin (\theta )\cos (\varphi )\hat{\mathit{r}}+\cos (\theta )\cos (\varphi )\hat{\mathit{\theta }}-\sin (\varphi )\hat{\mathit{\varphi }}\\ \hat{\mathbf{y}}& =\sin (\theta )\sin (\varphi )\hat{\mathit{r}}+\cos (\theta )\sin (\varphi )\hat{\mathit{\theta }}+\cos (\varphi )\hat{\mathit{\varphi }}\\ \hat{\mathbf{z}}& =\cos (\theta )\hat{\mathit{r}}-\sin (\theta )\hat{\mathit{\theta }}\end{array}$	$\begin{array}{rl}\hat{\mathit{\sigma }}& =\frac{\tau }{\sqrt{{\tau }^2+{\sigma }^2}}\hat{\mathbf{x}}-\frac{\sigma }{\sqrt{{\tau }^2+{\sigma }^2}}\hat{\mathbf{y}}\\ \hat{\mathit{\tau }}& =\frac{\sigma }{\sqrt{{\tau }^2+{\sigma }^2}}\hat{\mathbf{x}}+\frac{\tau }{\sqrt{{\tau }^2+{\sigma }^2}}\hat{\mathbf{y}}\\ \hat{\mathbf{z}}& =\hat{\mathbf{z}}\end{array}$
	$\begin{array}{rl}\hat{\mathbf{r}}& =\frac{x\hat{\mathbf{x}}+y\hat{\mathbf{y}}+z\hat{\mathbf{z}}}{r}\\ \hat{\mathit{\theta }}& =\frac{xz\hat{\mathbf{x}}+yz\hat{\mathbf{y}}-{\rho }^2\hat{\mathbf{z}}}{r\rho }\\ \hat{\mathit{\varphi }}& =\frac{-y\hat{\mathbf{x}}+x\hat{\mathbf{y}}}{\rho }\end{array}$	$\begin{array}{rl}\hat{\mathit{\rho }}& =\sin (\theta )\hat{\mathbf{r}}+\cos (\theta )\hat{\mathit{\theta }}\\ \hat{\mathit{\varphi }}& =\hat{\mathit{\varphi }}\\ \hat{\mathbf{z}}& =\cos (\theta )\hat{\mathbf{r}}-\sin (\theta )\hat{\mathit{\theta }}\end{array}$	$\begin{array}{rl}\hat{\mathbf{r}}& =\frac{\rho }{r}\hat{\mathit{\rho }}+\frac{z}{r}\hat{\mathbf{z}}\\ \hat{\mathit{\theta }}& =\frac{z}{r}\hat{\mathit{\rho }}-\frac{\rho }{r}\hat{\mathbf{z}}\\ \hat{\mathit{\varphi }}& =\hat{\mathit{\varphi }}\end{array}$
שדה וקטורי $\mathbf{𝐀}$	$A_x\hat{\mathbf{x}}+A_y\hat{\mathbf{y}}+A_z\hat{\mathbf{z}}$	$A_{\rho }\hat{\mathit{\rho }}+A_{\varphi }\hat{\mathit{\varphi }}+A_z\hat{\mathit{z}}$	$A_r\hat{\mathit{r}}+A_{\theta }\hat{\mathit{\theta }}+A_{\varphi }\hat{\mathit{\varphi }}$	$A_{\sigma }\hat{\mathit{\sigma }}+A_{\tau }\hat{\mathit{\tau }}+A_{\varphi }\hat{\mathit{z}}$
גרדיאנט $\mathrm{\nabla }f$	$\frac{\partial f}{\partial x}\hat{\mathbf{x}}+\frac{\partial f}{\partial y}\hat{\mathbf{y}}+\frac{\partial f}{\partial z}\hat{\mathbf{z}}$	$\frac{\partial f}{\partial \rho }\hat{\mathit{\rho }}+\frac{1}{\rho }\frac{\partial f}{\partial \varphi }\hat{\mathit{\varphi }}+\frac{\partial f}{\partial z}\hat{\mathit{z}}$	$\frac{\partial f}{\partial r}\hat{\mathit{r}}+\frac{1}{r}\frac{\partial f}{\partial \theta }\hat{\mathit{\theta }}+\frac{1}{r\sin (\theta )}\frac{\partial f}{\partial \varphi }\hat{\mathit{\varphi }}$	$\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial f}{\partial \sigma }\hat{\mathit{\sigma }}+\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial f}{\partial \tau }\hat{\mathit{\tau }}+\frac{\partial f}{\partial z}\hat{\mathit{z}}$
דיברגנץ $\mathrm{\nabla }\cdot \mathbf{𝐀}$	$\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}$	$\frac{1}{\rho }\frac{\partial (\rho A_{\rho })}{\partial \rho }+\frac{1}{\rho }\frac{\partial A_{\varphi }}{\partial \varphi }+\frac{\partial A_z}{\partial z}$	$\frac{1}{r\sin (\theta )}[\frac{\sin (\theta )}{r}\frac{\partial (r^2{A}_r)}{\partial r}+\frac{\partial (A_{\theta }\sin (\theta ))}{\partial \theta }+\frac{\partial A_{\varphi }}{\partial \varphi }]$	$\frac{1}{{\sigma }^2+{\tau }^2}(\frac{\partial A_{\sigma }}{\partial \sigma }+\frac{\partial A_{\tau }}{\partial \tau })+\frac{\partial A_z}{\partial z}$
רוטור $\mathrm{\nabla }\times \mathbf{𝐀}$	$\begin{array}{r}(\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z})\hat{\mathbf{x}}\\ +(\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x})\hat{\mathbf{y}}\\ +(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y})\hat{\mathbf{z}}\end{array}$	$\begin{array}{r}(\frac{1}{\rho }\frac{\partial A_z}{\partial \varphi }-\frac{\partial A_{\varphi }}{\partial z})\hat{\mathit{\rho }}\\ +(\frac{\partial A_{\rho }}{\partial z}-\frac{\partial A_z}{\partial \rho })\hat{\mathit{\varphi }}\\ +\frac{1}{\rho }(\frac{\partial (\rho A_{\varphi })}{\partial \rho }-\frac{\partial A_{\rho }}{\partial \varphi })\hat{\mathit{z}}\end{array}$	$\begin{array}{r}\frac{1}{r\sin (\theta )}(\frac{\partial (A_{\varphi }\sin (\theta ))}{\partial \theta }-\frac{\partial A_{\theta }}{\partial \varphi })\hat{\mathit{r}}\\ +\frac{1}{r}(\frac{1}{\sin (\theta )}\frac{\partial A_r}{\partial \varphi }-\frac{\partial (r{A}_{\varphi })}{\partial r})\hat{\mathit{\theta }}\\ +\frac{1}{r}(\frac{\partial (r{A}_{\theta })}{\partial r}-\frac{\partial A_r}{\partial \theta })\hat{\mathit{\varphi }}\end{array}$	$\begin{array}{r}(\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial A_z}{\partial \tau }-\frac{\partial A_{\tau }}{\partial z})\hat{\mathit{\sigma }}\\ -(\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial A_z}{\partial \sigma }-\frac{\partial A_{\sigma }}{\partial z})\hat{\mathit{\tau }}\\ +\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}(\frac{\partial (\rho A_{\varphi })}{\partial \rho }-\frac{\partial A_{\rho }}{\partial \varphi })\hat{\mathit{z}}\end{array}$
לפלסיאן $\mathrm{\Delta }f={\mathrm{\nabla }}^{2}f$	$\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2}+\frac{{\partial }^{2}f}{\partial z^2}$	$\frac{1}{\rho }\frac{\partial }{\partial \rho }\left(\rho \frac{\partial f}{\partial \rho }\right)+\frac{1}{{\rho }^2}\frac{{\partial }^{2}f}{\partial {\varphi }^2}+\frac{{\partial }^{2}f}{\partial z^2}$	$\frac{1}{r^2}[\frac{\partial }{\partial r}\left(r^2\frac{\partial f}{\partial r}\right)+\frac{1}{\sin (\theta )}\frac{\partial }{\partial \theta }(\sin (\theta )\frac{\partial f}{\partial \theta })+\frac{1}{\sin (\theta {)}^2}\frac{{\partial }^{2}f}{\partial {\varphi }^2}]$	$\frac{1}{{\sigma }^2+{\tau }^2}(\frac{{\partial }^{2}f}{\partial {\sigma }^2}+\frac{{\partial }^{2}f}{\partial {\tau }^2})+\frac{{\partial }^{2}f}{\partial z^2}$
לפלסיאן וקטורי $\mathrm{\Delta }\mathbf{𝐀}={\mathrm{\nabla }}^2\mathbf{𝐀}$	$\mathrm{\Delta }{A}_x\hat{\mathbf{x}}+\mathrm{\Delta }{A}_y\hat{\mathbf{y}}+\mathrm{\Delta }{A}_z\hat{\mathbf{z}}$	$\begin{array}{rl}& (\mathrm{\Delta }{A}_{\rho }-\frac{A_{\rho }}{{\rho }^2}-\frac{2}{{\rho }^2}\frac{\partial A_{\varphi }}{\partial \varphi })\hat{\mathit{\rho }}\\ +& (\mathrm{\Delta }{A}_{\varphi }-\frac{A_{\varphi }}{{\rho }^2}+\frac{2}{{\rho }^2}\frac{\partial A_{\rho }}{\partial \varphi })\hat{\mathit{\varphi }}\\ +& \left(\mathrm{\Delta }{A}_z\right)\hat{\mathit{z}}& \end{array}$	$\begin{array}{cc}& (\mathrm{\Delta }{A}_r-\frac{2}{r^2\sin (\theta )}[\sin (\theta )A_r+\frac{\partial (A_{\theta }\sin (\theta ))}{\partial \theta }+\frac{\partial A_{\varphi }}{\partial \varphi }])\hat{\mathit{r}}\\ +& (\mathrm{\Delta }{A}_{\theta }-\frac{1}{r^2\sin (\theta {)}^2}[A_{\theta }-2\sin (\theta {)}^2\frac{\partial A_r}{\partial \theta }+2\cos (\theta )\frac{\partial A_{\varphi }}{\partial \varphi }])\hat{\mathit{\theta }}\\ +& (\mathrm{\Delta }{A}_{\varphi }-\frac{A_{\varphi }}{r^2\sin (\theta {)}^2}+\frac{2}{r^2\sin (\theta )}\frac{\partial A_r}{\partial \varphi }+\frac{2\cos (\theta )}{r^2\sin (\theta {)}^2}\frac{\partial A_{\theta }}{\partial \varphi })\hat{\mathit{\varphi }}\end{array}$
העתק אינפיניטסימלי	$d\mathbf{𝐥}=dx\hat{\mathbf{x}}+dy\hat{\mathbf{y}}+dz\hat{\mathbf{z}}$	$d\mathbf{𝐥}=d\rho \hat{\mathit{\rho }}+\rho d\varphi \hat{\mathit{\varphi }}+dz\hat{\mathit{z}}$	$d\mathbf{𝐥}=dr\hat{\mathbf{r}}+rd\theta \hat{\mathit{\theta }}+r\sin \theta d\varphi \hat{\mathit{\varphi }}$	$d\mathbf{𝐥}=\sqrt{{\sigma }^2+{\tau }^2}d\sigma \hat{\mathit{\sigma }}+\sqrt{{\sigma }^2+{\tau }^2}d\tau \hat{\mathit{\tau }}+dz\hat{\mathit{z}}$
וקטור שטח אינפיניטסימלי	$\begin{array}{cc}d\mathbf{𝐒}=& dydz\hat{\mathbf{x}}+\\ & dxdz\hat{\mathbf{y}}+\\ & dxdy\hat{\mathbf{z}}\end{array}$	$\begin{array}{cc}d\mathbf{𝐒}=& \rho d\varphi dz\hat{\mathit{\rho }}+\\ & d\rho dz\hat{\mathit{\varphi }}+\\ & \rho d\rho d\varphi \hat{\mathbf{z}}\end{array}$	$\begin{array}{cc}d\mathbf{𝐒}=& r^2\sin \theta d\theta d\varphi \hat{\mathbf{r}}+\\ & r\sin \theta drd\varphi \hat{\mathit{\theta }}+\\ & rdrd\theta \hat{\mathit{\varphi }}\end{array}$	$\begin{array}{cc}d\mathbf{𝐒}=& \sqrt{{\sigma }^2+{\tau }^2},d\tau dz\hat{\mathit{\sigma }}+\\ & \sqrt{{\sigma }^2+{\tau }^2}d\sigma dz\hat{\mathit{\tau }}+\\ & {\sigma }^2+{\tau }^{2}d\sigma ,d\tau \hat{\mathbf{z}}\end{array}$
יחידת נפח אינפיניטסימלית	$dV=dxdydz$	$dV=\rho d\rho d\varphi dz$	$dV=r^2\sin \theta drd\theta d\varphi$	$dV=({\sigma }^2+{\tau }^2)d\sigma d\tau dz$
כללים חשובים: $\text{div grad }f=\mathrm{\nabla }\cdot (\mathrm{\nabla }f)={\mathrm{\nabla }}^{2}f=\mathrm{\Delta }f$ (לפלסיאן) $\text{curl grad }f=\mathrm{\nabla }\times (\mathrm{\nabla }f)=\mathbf{𝟎}$ $\text{div curl}\mathbf{𝐀}=\mathrm{\nabla }\cdot (\mathrm{\nabla }\times \mathbf{𝐀})=0$ $\text{curl curl}\mathbf{𝐀}=\mathrm{\nabla }\times (\mathrm{\nabla }\times \mathbf{𝐀})=\mathrm{\nabla }(\mathrm{\nabla }\cdot \mathbf{𝐀})-{\mathrm{\nabla }}^2\mathbf{𝐀}$ $\mathrm{\Delta }fg=f\mathrm{\Delta }g+2\mathrm{\nabla }f\cdot \mathrm{\nabla }g+g\mathrm{\Delta }f$