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, <math>\theta</math> היא הזווית בין ציר z ווקטור הרדיוס המחבר את הראשית עם הנקודה בה עוסקים. <math>\phi</math> היא הזווית בין היטל וקטור הרדיוס על מישור x-y, ובין ציר x.

{| class="wikitable" style="background: white"
|-
! style="background: white" |
! style="background: white" |קואורדינטות קרטזיות (x,y,z)
! style="background: white" |קואורדינטות גליליות (ρ,φ,z)
! style="background: white" |קואורדינטות כדוריות (r,θ,φ)
! style="background: white" |קואורדינטות גליליות פרבוליות (&sigma;,τ,z)
|- align="center" 
! rowspan="2" style="background: white" |הגדרתמערכתהצירים
|<math>\begin{matrix}
 \rho & = & \sqrt{x^2+y^2} \\
 \phi & = & \arctan(y/x) \\
 z & = & z \end{matrix}</math>
|<math>\begin{matrix}
 x & = & \rho\cos\phi \\
 y & = & \rho\sin\phi \\
 z & = & z \end{matrix}</math>
|<math>\begin{matrix}
 x & = & r\sin\theta\cos\phi \\
 y & = & r\sin\theta\sin\phi \\
 z & = & r\cos\theta \end{matrix}</math>
|<math>\begin{matrix}
 x & = & \sigma \tau\\
 y & = & \frac{1}{2} \left( \tau^{2} - \sigma^{2} \right) \\
 z & = & z \end{matrix}</math>
|- align=center
|<math>\begin{matrix}
 r & = & \sqrt{x^2+y^2+z^2} \\
 \theta & = & \arctan{\left(\frac{\sqrt{x^2+y^2}}{z}\right)}\\
 \phi & = & \arctan(y/x) \\ \end{matrix}</math>
|<math>\begin{matrix}
 \rho & = & r\sin(\theta) \\
 \phi & = & \phi\\
 z & = & r\cos(\theta) \end{matrix}</math>
|<math>\begin{matrix}
 r & = & \sqrt{x^2+y^2+z^2} \\
 \theta & = & \arctan{\left(\frac{\sqrt{x^2+y^2}}{z}\right)}\\
 \phi & = & \arctan(y/x) \\ \end{matrix}</math>
|<math>\begin{matrix}
 \rho & = & \frac{\sigma^2 + \tau^2}{2} \\
 \phi & = & \arctan\left(\frac{\tau^{2} - \sigma^{2}}{2\sigma\tau} \right) \\
 z & = & z \end{matrix}</math>
|- align="center" 
! rowspan="2" style="background: white" |הגדרתוקטוריהיחידה
|<math>\begin{matrix}
 \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{matrix}</math>
|<math>\begin{matrix}
 \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{matrix}</math>
|<math>\begin{matrix}
 \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{matrix}</math>
|<math>\begin{matrix}
 \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{matrix}</math>
|- align=center
|<math>\begin{matrix}
 \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{matrix}</math>
|<math>\begin{matrix}
 \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{matrix}</math>
|<math>\begin{matrix}
 \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{matrix}</math>
|
|- align=center 
! style="background: white" |שדה וקטורי <math>\mathbf{A}</math>
|<math>A_x\mathbf{\hat x} + A_y\mathbf{\hat y} + A_z\mathbf{\hat z}</math>
|<math>A_\rho\boldsymbol{\hat \rho} + A_\phi\boldsymbol{\hat \phi} + A_z\boldsymbol{\hat z}</math>
|<math>A_r\boldsymbol{\hat r} + A_\theta\boldsymbol{\hat \theta} + A_\phi\boldsymbol{\hat \phi}</math>
|<math>A_\sigma\boldsymbol{\hat \sigma} + A_\tau\boldsymbol{\hat \tau} + A_\phi\boldsymbol{\hat z}</math>
|- align=center 
! style="background: white" |גרדיאנט <math>\nabla f</math>
|<math>{\partial f \over \partial x}\mathbf{\hat x} + {\partial f \over \partial y}\mathbf{\hat y} 
 + {\partial f \over \partial z}\mathbf{\hat z}</math>
|<math>{\partial f \over \partial \rho}\boldsymbol{\hat \rho} 
 + {1 \over \rho}{\partial f \over \partial \phi}\boldsymbol{\hat \phi}
 + {\partial f \over \partial z}\boldsymbol{\hat z}</math>
|<math>{\partial f \over \partial r}\boldsymbol{\hat r} 
 + {1 \over r}{\partial f \over \partial \theta}\boldsymbol{\hat \theta}
 + {1 \over r\sin\theta}{\partial f \over \partial \phi}\boldsymbol{\hat \phi}</math>
|<math> \frac{1}{\sqrt{\sigma^{2} + \tau^{2}}} {\partial f \over \partial \sigma}\boldsymbol{\hat \sigma} + \frac{1}{\sqrt{\sigma^{2} + \tau^{2}}} {\partial f \over \partial \tau}\boldsymbol{\hat \tau} + {\partial f \over \partial z}\boldsymbol{\hat z}</math>
|- align=center 
! style="background: white" |דיברגנץ <math>\nabla \cdot \mathbf{A}</math>
|<math>{\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z}</math>
|<math>{1 \over \rho}{\partial \left( \rho A_\rho \right) \over \partial \rho} 
 + {1 \over \rho}{\partial A_\phi \over \partial \phi}
 + {\partial A_z \over \partial z}</math>
|<math>{1 \over r^2}{\partial \left( r^2 A_r \right) \over \partial r} 
 + {1 \over r\sin\theta}{\partial \over \partial \theta} \left( A_\theta\sin\theta \right) 
 + {1 \over r\sin\theta}{\partial A_\phi \over \partial \phi}</math>
|<math> \frac{1}{\sigma^{2} + \tau^{2}}{\partial A_\sigma \over \partial \sigma} + \frac{1}{\sigma^{2} + \tau^{2}}{\partial A_\tau \over \partial \tau} + {\partial A_z \over \partial z}</math>
|- align=center 
! style="background: white" |קרל (רוטור) <math>\nabla \times \mathbf{A}</math>
|<math>\begin{matrix}
 \displaystyle\left({\partial A_z \over \partial y} - {\partial A_y \over \partial z}\right) \mathbf{\hat x} & + \\
 \displaystyle\left({\partial A_x \over \partial z} - {\partial A_z \over \partial x}\right) \mathbf{\hat y} & + \\
 \displaystyle\left({\partial A_y \over \partial x} - {\partial A_x \over \partial y}\right) \mathbf{\hat z} & \ \end{matrix}</math>
|<math>\begin{matrix}
 \displaystyle\left({1 \over \rho}{\partial A_z \over \partial \phi}
 - {\partial A_\phi \over \partial z}\right) \boldsymbol{\hat \rho} & + \\
 \displaystyle\left({\partial A_\rho \over \partial z} - {\partial A_z \over \partial \rho}\right) \boldsymbol{\hat \phi} & + \\
 \displaystyle{1 \over \rho}\left({\partial \left( \rho A_\phi \right) \over \partial \rho}
 - {\partial A_\rho \over \partial \phi}\right) \boldsymbol{\hat z} & \ \end{matrix}</math>
|<math>\begin{matrix}
 \displaystyle{1 \over r\sin\theta}\left({\partial \over \partial \theta} \left( A_\phi\sin\theta \right)
 - {\partial A_\theta \over \partial \phi}\right) \boldsymbol{\hat r} & + \\
 \displaystyle{1 \over r}\left({1 \over \sin\theta}{\partial A_r \over \partial \phi}
 - {\partial \over \partial r} \left( r A_\phi \right) \right) \boldsymbol{\hat \theta} & + \\
 \displaystyle{1 \over r}\left({\partial \over \partial r} \left( r A_\theta \right)
 - {\partial A_r \over \partial \theta}\right) \boldsymbol{\hat \phi} & \ \end{matrix}</math>
|<math>\begin{matrix}
 \displaystyle\left(\frac{1}{\sqrt{\sigma^{2} + \tau^{2}}}{\partial A_z \over \partial \tau}
 - {\partial A_\tau \over \partial z}\right) \boldsymbol{\hat \sigma} & - \\
 \displaystyle\left(\frac{1}{\sqrt{\sigma^{2} + \tau^{2}}}{\partial A_z \over \partial \sigma}- {\partial A_\sigma \over \partial z}\right) \boldsymbol{\hat \tau} & + \\
 \displaystyle\frac{1}{\sqrt{\sigma^{2} + \tau^{2}}}\left({\partial \left( \rho A_\phi \right) \over \partial \rho}
 - {\partial A_\rho \over \partial \phi}\right) \boldsymbol{\hat z} & \ \end{matrix}</math>
|- align=center 
! style="background: white" |לפלסיאן <math>\Delta f = \nabla^2 f</math>
|<math>{\partial^2 f \over \partial x^2} + {\partial^2 f \over \partial y^2} + {\partial^2 f \over \partial z^2}</math>
|<math>{1 \over \rho}{\partial \over \partial \rho}\left(\rho {\partial f \over \partial \rho}\right) 
 + {1 \over \rho^2}{\partial^2 f \over \partial \phi^2}
 + {\partial^2 f \over \partial z^2}</math>
|<math>{1 \over r^2}{\partial \over \partial r}\!\left(r^2 {\partial f \over \partial r}\right) 
 \!+\!{1 \over r^2\!\sin\theta}{\partial \over \partial \theta}\!\left(\sin\theta {\partial f \over \partial \theta}\right)
 \!+\!{1 \over r^2\!\sin^2\theta}{\partial^2 f \over \partial \phi^2}</math>
|<math> \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}}
</math>
|- align=center 
! style="background: white" |לפלסיאן וקטורי <math>\Delta \mathbf{A} = \nabla^2 \mathbf{A}</math>
|<math>\Delta A_x \mathbf{\hat x} + \Delta A_y \mathbf{\hat y} + \Delta A_z \mathbf{\hat z} </math>
|<math>\begin{matrix}
 \displaystyle\left(\Delta A_\rho - {A_\rho \over \rho^2}
 - {2 \over \rho^2}{\partial A_\phi \over \partial \phi}\right) \boldsymbol{\hat \rho} & + \\
 \displaystyle\left(\Delta A_\phi - {A_\phi \over \rho^2}
 + {2 \over \rho^2}{\partial A_\rho \over \partial \phi}\right) \boldsymbol{\hat\phi} & + \\
 \displaystyle\left(\Delta A_z \right) \boldsymbol{\hat z} & \ \end{matrix}</math>
| align=center |<math>\begin{matrix}
 \left(\Delta A_r - {2 A_r \over r^2}
 - {2 \over r^2\sin\theta}{\partial \left(A_\theta \sin\theta\right) \over \partial\theta}
 - {2 \over r^2\sin\theta}{\partial A_\phi \over \partial \phi}\right) \boldsymbol{\hat r} & + \\
 \left(\Delta A_\theta - {A_\theta \over r^2\sin^2\theta}
 + {2 \over r^2}{\partial A_r \over \partial \theta}
 - {2 \cos\theta \over r^2\sin^2\theta}{\partial A_\phi \over \partial \phi}\right) \boldsymbol{\hat\theta} & + \\
 \left(\Delta A_\phi - {A_\phi \over r^2\sin^2\theta}
 + {2 \over r^2\sin\theta}{\partial A_r \over \partial \phi}
 + {2 \cos\theta \over r^2\sin^2\theta}{\partial A_\theta \over \partial \phi}\right) \boldsymbol{\hat\phi} & \end{matrix}</math>
|- align=center 
! style="background: white" |העתק אינפיניטסימלי
|<math>d\mathbf{l} = dx\mathbf{\hat x} + dy\mathbf{\hat y} + dz\mathbf{\hat z}</math>
|<math>d\mathbf{l} = d\rho\boldsymbol{\hat \rho} + \rho d\phi\boldsymbol{\hat \phi} + dz\boldsymbol{\hat z}</math>
|<math>d\mathbf{l} = dr\mathbf{\hat r} + rd\theta\boldsymbol{\hat \theta} + r\sin\theta d\phi\boldsymbol{\hat \phi}</math>
|<math>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}</math>
|- align=center 
! style="background: white" |וקטור שטח אינפיניטסימלי
|<math>\begin{matrix}d\mathbf{S} = &dy\,dz\,\mathbf{\hat x} + \\ 
&dx\,dz\,\mathbf{\hat y} + \\ 
&dx\,dy\,\mathbf{\hat z}\end{matrix}</math>
|<math>\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}</math>
|<math>\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}</math>
|<math>\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}</math>
|- align=center 
! style="background: white" |יחידת נפח אינפיניטסימלית
|<math>dV = dx\,dy\,dz \,</math>
|<math>dV = \rho\, d\rho\, d\phi\, dz\,</math>
|<math>dV = r^2\sin\theta \,dr\,d\theta\, d\phi\,</math>
|<math>dV = \left( \sigma^{2} + \tau^{2} \right) d\sigma d\tau dz,</math>
|- 
| colspan=4 |<strong>כללים חשובים:</strong>
#<math>\operatorname{div\ grad\ } f = \nabla \cdot (\nabla f) = \nabla^2 f = \Delta f</math> (לפלסיאן)
#<math>\operatorname{curl\ grad\ } f = \nabla \times (\nabla f) = \mathbf{0}</math>
#<math>\operatorname{div\ curl\ } \mathbf{A} = \nabla \cdot (\nabla \times \mathbf{A}) = 0</math>
#<math>\operatorname{curl\ curl\ } \mathbf{A} = \nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}</math>
#<math>\Delta f g = f \Delta g + 2 \nabla f \cdot \nabla g + g \Delta f</math>
|}

קטגוריה:אנליזה וקטורית
קטגוריה:אופרטורים דיפרנציאליים