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. $\phi$ 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	${\begin{aligned}\rho &={\sqrt {x^{2}+y^{2}}}\\\phi &=\arctan \left({\frac {y}{x}}\right)\\z&=z\end{aligned}}$	${\begin{aligned}x&=\rho \cos(\phi )\\y&=\rho \sin(\phi )\\z&=z\end{aligned}}$	${\begin{aligned}x&=r\sin(\theta )\cos(\phi )\\y&=r\sin(\theta )\sin(\phi )\\z&=r\cos(\theta )\end{aligned}}$	${\begin{aligned}x&=\sigma (\tau )\\y&={\frac {\tau ^{2}-\sigma ^{2}}{2}}\\z&=z\end{aligned}}$
	${\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=\arctan \left({\frac {\sqrt {x^{2}+y^{2}}}{z}}\right)\\\phi &=\arctan \left({\frac {y}{x}}\right)\end{aligned}}$	${\begin{aligned}\rho &=r\sin(\theta )\\\phi &=\phi \\z&=r\cos(\theta )\end{aligned}}$	${\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=\arctan \left({\frac {\sqrt {x^{2}+y^{2}}}{z}}\right)\\\phi &=\arctan \left({\frac {y}{x}}\right)\end{aligned}}$	${\begin{aligned}\rho \cos(\phi )&=\sigma \tau \\\rho \sin(\phi )&={\frac {\tau ^{2}-\sigma ^{2}}{2}}\\z&=z\end{aligned}}$
הגדרת וקטורי היחידה	${\begin{aligned}{\boldsymbol {\hat {\rho }}}&={\frac {x}{\rho }}\mathbf {\hat {x}} +{\frac {y}{\rho }}\mathbf {\hat {y}} \\{\boldsymbol {\hat {\phi }}}&=-{\frac {y}{\rho }}\mathbf {\hat {x}} +{\frac {x}{\rho }}\mathbf {\hat {y}} \\\mathbf {\hat {z}} &=\mathbf {\hat {z}} \end{aligned}}$	${\begin{aligned}\mathbf {\hat {x}} &=\cos(\phi ){\boldsymbol {\hat {\rho }}}-\sin(\phi ){\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {y}} &=\sin(\phi ){\boldsymbol {\hat {\rho }}}+\cos(\phi ){\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=\mathbf {\hat {z}} \end{aligned}}$	${\begin{aligned}\mathbf {\hat {x}} &=\sin(\theta )\cos(\phi ){\boldsymbol {\hat {r}}}+\cos(\theta )\cos(\phi ){\boldsymbol {\hat {\theta }}}-\sin(\phi ){\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {y}} &=\sin(\theta )\sin(\phi ){\boldsymbol {\hat {r}}}+\cos(\theta )\sin(\phi ){\boldsymbol {\hat {\theta }}}+\cos(\phi ){\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=\cos(\theta ){\boldsymbol {\hat {r}}}-\sin(\theta ){\boldsymbol {\hat {\theta }}}\end{aligned}}$	${\begin{aligned}{\boldsymbol {\hat {\sigma }}}&={\frac {\tau }{\sqrt {\tau ^{2}+\sigma ^{2}}}}\mathbf {\hat {x}} -{\frac {\sigma }{\sqrt {\tau ^{2}+\sigma ^{2}}}}\mathbf {\hat {y}} \\{\boldsymbol {\hat {\tau }}}&={\frac {\sigma }{\sqrt {\tau ^{2}+\sigma ^{2}}}}\mathbf {\hat {x}} +{\frac {\tau }{\sqrt {\tau ^{2}+\sigma ^{2}}}}\mathbf {\hat {y}} \\\mathbf {\hat {z}} &=\mathbf {\hat {z}} \end{aligned}}$
	${\begin{aligned}\mathbf {\hat {r}} &={\frac {x\mathbf {\hat {x}} +y\mathbf {\hat {y}} +z\mathbf {\hat {z}} }{r}}\\{\boldsymbol {\hat {\theta }}}&={\frac {xz\mathbf {\hat {x}} +yz\mathbf {\hat {y}} -\rho ^{2}\mathbf {\hat {z}} }{r\rho }}\\{\boldsymbol {\hat {\phi }}}&={\frac {-y\mathbf {\hat {x}} +x\mathbf {\hat {y}} }{\rho }}\end{aligned}}$	${\begin{aligned}{\boldsymbol {\hat {\rho }}}&=\sin(\theta )\mathbf {\hat {r}} +\cos(\theta ){\boldsymbol {\hat {\theta }}}\\{\boldsymbol {\hat {\phi }}}&={\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=\cos(\theta )\mathbf {\hat {r}} -\sin(\theta ){\boldsymbol {\hat {\theta }}}\end{aligned}}$	${\begin{aligned}\mathbf {\hat {r}} &={\frac {\rho }{r}}{\boldsymbol {\hat {\rho }}}+{\frac {z}{r}}\mathbf {\hat {z}} \\{\boldsymbol {\hat {\theta }}}&={\frac {z}{r}}{\boldsymbol {\hat {\rho }}}-{\frac {\rho }{r}}\mathbf {\hat {z}} \\{\boldsymbol {\hat {\phi }}}&={\boldsymbol {\hat {\phi }}}\end{aligned}}$
שדה וקטורי $\mathbf {A}$	$A_{x}\mathbf {\hat {x}} +A_{y}\mathbf {\hat {y}} +A_{z}\mathbf {\hat {z}}$	$A_{\rho }{\boldsymbol {\hat {\rho }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}+A_{z}{\boldsymbol {\hat {z}}}$	$A_{r}{\boldsymbol {\hat {r}}}+A_{\theta }{\boldsymbol {\hat {\theta }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}$	$A_{\sigma }{\boldsymbol {\hat {\sigma }}}+A_{\tau }{\boldsymbol {\hat {\tau }}}+A_{\phi }{\boldsymbol {\hat {z}}}$
גרדיאנט $\nabla f$	${\frac {\partial f}{\partial x}}\mathbf {\hat {x}} +{\frac {\partial f}{\partial y}}\mathbf {\hat {y}} +{\frac {\partial f}{\partial z}}\mathbf {\hat {z}}$	${\frac {\partial f}{\partial \rho }}{\boldsymbol {\hat {\rho }}}+{\frac {1}{\rho }}{\frac {\partial f}{\partial \phi }}{\boldsymbol {\hat {\phi }}}+{\frac {\partial f}{\partial z}}{\boldsymbol {\hat {z}}}$	${\frac {\partial f}{\partial r}}{\boldsymbol {\hat {r}}}+{\frac {1}{r}}{\frac {\partial f}{\partial \theta }}{\boldsymbol {\hat {\theta }}}+{\frac {1}{r\sin(\theta )}}{\frac {\partial f}{\partial \phi }}{\boldsymbol {\hat {\phi }}}$	${\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\frac {\partial f}{\partial \sigma }}{\boldsymbol {\hat {\sigma }}}+{\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\frac {\partial f}{\partial \tau }}{\boldsymbol {\hat {\tau }}}+{\frac {\partial f}{\partial z}}{\boldsymbol {\hat {z}}}$
דיברגנץ $\nabla \cdot \mathbf {A}$	${\frac {\partial A_{x}}{\partial x}}+{\frac {\partial A_{y}}{\partial y}}+{\frac {\partial A_{z}}{\partial z}}$	${\frac {1}{\rho }}{\frac {\partial (\rho A_{\rho })}{\partial \rho }}+{\frac {1}{\rho }}{\frac {\partial A_{\phi }}{\partial \phi }}+{\frac {\partial A_{z}}{\partial z}}$	${\frac {1}{r\sin(\theta )}}\left[{\frac {\sin(\theta )}{r}}{\frac {\partial (r^{2}A_{r})}{\partial r}}+{\frac {\partial {\bigl (}A_{\theta }\sin(\theta ){\bigr )}}{\partial \theta }}+{\frac {\partial A_{\phi }}{\partial \phi }}\right]$	${\frac {1}{\sigma ^{2}+\tau ^{2}}}\left({\frac {\partial A_{\sigma }}{\partial \sigma }}+{\frac {\partial A_{\tau }}{\partial \tau }}\right)+{\frac {\partial A_{z}}{\partial z}}$
רוטור $\nabla \times \mathbf {A}$	${\begin{aligned}\left({\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}\right)\mathbf {\hat {x}} \\+\left({\frac {\partial A_{x}}{\partial z}}-{\frac {\partial A_{z}}{\partial x}}\right)\mathbf {\hat {y}} \\+\left({\frac {\partial A_{y}}{\partial x}}-{\frac {\partial A_{x}}{\partial y}}\right)\mathbf {\hat {z}} \end{aligned}}$	${\begin{aligned}\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \phi }}-{\frac {\partial A_{\phi }}{\partial z}}\right){\boldsymbol {\hat {\rho }}}\\+\left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right){\boldsymbol {\hat {\phi }}}\\+{\frac {1}{\rho }}\left({\frac {\partial (\rho A_{\phi })}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \phi }}\right){\boldsymbol {\hat {z}}}\end{aligned}}$	${\begin{aligned}{\frac {1}{r\sin(\theta )}}\left({\frac {\partial {\bigl (}A_{\phi }\sin(\theta ){\bigr )}}{\partial \theta }}-{\frac {\partial A_{\theta }}{\partial \phi }}\right){\boldsymbol {\hat {r}}}\\+~{\frac {1}{r}}\left({\frac {1}{\sin(\theta )}}{\frac {\partial A_{r}}{\partial \phi }}-{\frac {\partial (rA_{\phi })}{\partial r}}\right){\boldsymbol {\hat {\theta }}}\\+~{\frac {1}{r}}\left({\frac {\partial (rA_{\theta })}{\partial r}}-{\frac {\partial A_{r}}{\partial \theta }}\right){\boldsymbol {\hat {\phi }}}\end{aligned}}$	${\begin{aligned}\left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\frac {\partial A_{z}}{\partial \tau }}-{\frac {\partial A_{\tau }}{\partial z}}\right){\boldsymbol {\hat {\sigma }}}\\-\left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\frac {\partial A_{z}}{\partial \sigma }}-{\frac {\partial A_{\sigma }}{\partial z}}\right){\boldsymbol {\hat {\tau }}}\\+{\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}\left({\frac {\partial (\rho A_{\phi })}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \phi }}\right){\boldsymbol {\hat {z}}}\end{aligned}}$
לפלסיאן $\Delta f=\nabla ^{2}f$	${\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}$	${\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}\left(\rho {\frac {\partial f}{\partial \rho }}\right)+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}f}{\partial \phi ^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}$	${\frac {1}{r^{2}}}\left[{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial f}{\partial r}}\right)+{\frac {1}{\sin(\theta )}}{\frac {\partial }{\partial \theta }}\left(\sin(\theta ){\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{\sin(\theta )^{2}}}{\frac {\partial ^{2}f}{\partial \phi ^{2}}}\right]$	${\frac {1}{\sigma ^{2}+\tau ^{2}}}\left({\frac {\partial ^{2}f}{\partial \sigma ^{2}}}+{\frac {\partial ^{2}f}{\partial \tau ^{2}}}\right)+{\frac {\partial ^{2}f}{\partial z^{2}}}$
לפלסיאן וקטורי $\Delta \mathbf {A} =\nabla ^{2}\mathbf {A}$	$\Delta A_{x}\mathbf {\hat {x}} +\Delta A_{y}\mathbf {\hat {y}} +\Delta A_{z}\mathbf {\hat {z}}$	${\begin{aligned}&\left(\Delta A_{\rho }-{\frac {A_{\rho }}{\rho ^{2}}}-{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\phi }}{\partial \phi }}\right){\boldsymbol {\hat {\rho }}}\\+&\left(\Delta A_{\phi }-{\frac {A_{\phi }}{\rho ^{2}}}+{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\rho }}{\partial \phi }}\right){\boldsymbol {\hat {\phi }}}\\+&\left(\Delta A_{z}\right){\boldsymbol {\hat {z}}}&\end{aligned}}$	${\begin{matrix}&\left(\Delta A_{r}-{\dfrac {2}{r^{2}\sin(\theta )}}\left[\sin(\theta )A_{r}+{\dfrac {\partial {\bigl (}A_{\theta }\sin(\theta ){\bigr )}}{\partial \theta }}+{\dfrac {\partial A_{\phi }}{\partial \phi }}\right]\right){\boldsymbol {\hat {r}}}\\+&\left(\Delta A_{\theta }-{\dfrac {1}{r^{2}\sin(\theta )^{2}}}\left[A_{\theta }-2\sin(\theta )^{2}{\dfrac {\partial A_{r}}{\partial \theta }}+2\cos(\theta ){\dfrac {\partial A_{\phi }}{\partial \phi }}\right]\right){\boldsymbol {\hat {\theta }}}\\+&\left(\Delta A_{\phi }-{\dfrac {A_{\phi }}{r^{2}\sin(\theta )^{2}}}+{\dfrac {2}{r^{2}\sin(\theta )}}{\dfrac {\partial A_{r}}{\partial \phi }}+{\dfrac {2\cos(\theta )}{r^{2}\sin(\theta )^{2}}}{\dfrac {\partial A_{\theta }}{\partial \phi }}\right){\boldsymbol {\hat {\phi }}}\end{matrix}}$
העתק אינפיניטסימלי	$d\mathbf {l} =dx\mathbf {\hat {x}} +dy\mathbf {\hat {y}} +dz\mathbf {\hat {z}}$	$d\mathbf {l} =d\rho {\boldsymbol {\hat {\rho }}}+\rho d\phi {\boldsymbol {\hat {\phi }}}+dz{\boldsymbol {\hat {z}}}$	$d\mathbf {l} =dr\mathbf {\hat {r}} +rd\theta {\boldsymbol {\hat {\theta }}}+r\sin \theta d\phi {\boldsymbol {\hat {\phi }}}$	$d\mathbf {l} ={\sqrt {\sigma ^{2}+\tau ^{2}}}d\sigma {\boldsymbol {\hat {\sigma }}}+{\sqrt {\sigma ^{2}+\tau ^{2}}}d\tau {\boldsymbol {\hat {\tau }}}+dz{\boldsymbol {\hat {z}}}$
וקטור שטח אינפיניטסימלי	${\begin{matrix}d\mathbf {S} =&dy\,dz\,\mathbf {\hat {x}} +\\&dx\,dz\,\mathbf {\hat {y}} +\\&dx\,dy\,\mathbf {\hat {z}} \end{matrix}}$	${\begin{matrix}d\mathbf {S} =&\rho \,d\phi \,dz\,{\boldsymbol {\hat {\rho }}}+\\&d\rho \,dz\,{\boldsymbol {\hat {\phi }}}+\\&\rho \,d\rho d\phi \,\mathbf {\hat {z}} \end{matrix}}$	${\begin{matrix}d\mathbf {S} =&r^{2}\sin \theta \,d\theta \,d\phi \,\mathbf {\hat {r}} +\\&r\sin \theta \,dr\,d\phi \,{\boldsymbol {\hat {\theta }}}+\\&r\,dr\,d\theta \,{\boldsymbol {\hat {\phi }}}\end{matrix}}$	${\begin{matrix}d\mathbf {S} =&{\sqrt {\sigma ^{2}+\tau ^{2}}},d\tau \,dz\,{\boldsymbol {\hat {\sigma }}}+\\&{\sqrt {\sigma ^{2}+\tau ^{2}}}d\sigma \,dz\,{\boldsymbol {\hat {\tau }}}+\\&\sigma ^{2}+\tau ^{2}d\sigma ,d\tau \,\mathbf {\hat {z}} \end{matrix}}$
יחידת נפח אינפיניטסימלית	$dV=dx\,dy\,dz$	$dV=\rho \,d\rho \,d\phi \,dz$	$dV=r^{2}\sin \theta \,dr\,d\theta \,d\phi$	$dV=\left(\sigma ^{2}+\tau ^{2}\right)d\sigma d\tau dz$
כללים חשובים: ${\text{div grad }}f=\nabla \cdot (\nabla f)=\nabla ^{2}f=\Delta f$ (לפלסיאן) ${\text{curl grad }}f=\nabla \times (\nabla f)=\mathbf {0}$ ${\text{div curl}}\mathbf {A} =\nabla \cdot (\nabla \times \mathbf {A} )=0$ ${\text{curl curl}}\mathbf {A} =\nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A}$ $\Delta fg=f\Delta g+2\nabla f\cdot \nabla g+g\Delta f$