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| This is a list of some [[vector calculus]] formulae for working with common [[curvilinear coordinates|curvilinear]] [[coordinate system]]s.
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| | |
| == Notes ==
| |
| | |
| * This article uses the standard physics notation for [[spherical coordinate system|spherical coordinates]] (other sources may reverse the definitions of ''θ'' and ''ϕ''):
| |
| ** The polar angle is denoted by ''θ'': it is the angle between the ''z''-axis and the radial vector connecting the origin to the point in question.
| |
| ** The [[azimuth|azimuthal angle]] is denoted by ''ϕ'': it is the angle between the ''x''-axis and the projection of the radial vector onto the ''xy''-plane.
| |
| * The function {{nowrap|[[atan2]](''y'', ''x'')}} can be used instead of the mathematical function {{nowrap|[[arctan]](''y''/''x'')}} owing to its domain and image. The classical arctan function has an image of {{nowrap|(−π/2, +π/2)}}, whereas atan2 is defined to have an image of {{nowrap|(−π, π]}}.
| |
| <!--(The expressions for the Del in spherical coordinates may need to be corrected)--> | |
| | |
| == Formulae ==
| |
| <div style="overflow:scroll;"> | |
| {| class="wikitable" style="background: white"
| |
| |+ Table with the [[del]] operator in cylindrical, spherical and parabolic cylindrical coordinates
| |
| <!-- Header -->
| |
| |-
| |
| ! style="background: white" | Operation
| |
| ! style="background: white" | [[Cartesian coordinates]] {{math|(''x'', ''y'', ''z'')}}
| |
| ! style="background: white" | [[Cylindrical coordinates]] {{math|(''ρ'', ''ϕ'', ''z'')}}
| |
| ! style="background: white" | [[Spherical coordinates]] {{math|(''r'', ''θ'', ''ϕ'')}}
| |
| ! style="background: white" | [[Parabolic cylindrical coordinates]] {{math|(''σ'', ''τ'', ''z'')}}
| |
| | |
| <!-- Definition of coordinates -->
| |
| |- align="center"
| |
| ! rowspan="2" style="background: white" | Definition<br>of<br>coordinates
| |
| | <math>\begin{align}
| |
| \rho &= \sqrt{x^2+y^2} \\
| |
| \phi &= \arctan(y/x) \\
| |
| z &= z \end{align}</math>
| |
| | <math>\begin{align}
| |
| x &= \rho\cos\phi \\
| |
| y &= \rho\sin\phi \\
| |
| z &= z \end{align}</math>
| |
| | <math>\begin{align}
| |
| x &= r\sin\theta\cos\phi \\
| |
| y &= r\sin\theta\sin\phi \\
| |
| z &= r\cos\theta \end{align}</math>
| |
| | <math>\begin{align}
| |
| x &= \sigma \tau\\
| |
| y &= \tfrac{1}{2} \left( \tau^{2} - \sigma^{2} \right) \\
| |
| z &= z \end{align}</math>
| |
| |- align="center"
| |
| | <math>\begin{align}
| |
| r &= \sqrt{x^2+y^2+z^2} \\
| |
| \theta &= \arccos(z/r)\\
| |
| \phi &= \arctan(y/x) \end{align}</math>
| |
| | <math>\begin{align}
| |
| r &= \sqrt{\rho^2 + z^2} \\
| |
| \theta &= \arctan{(\rho/z)}\\
| |
| \phi &= \phi \end{align}</math>
| |
| | <math>\begin{align}
| |
| \rho &= r\sin\theta \\
| |
| \phi &= \phi\\
| |
| z &= r\cos\theta \end{align}</math>
| |
| | <math>\begin{align}
| |
| \rho\cos\phi &= \sigma \tau\\
| |
| \rho\sin\phi &= \tfrac{1}{2} \left( \tau^{2} - \sigma^{2} \right) \\
| |
| z &= z \end{align}</math>
| |
| | |
| <!-- Definition of unit vectors -->
| |
| |- align="center"
| |
| ! rowspan="2" style="background: white" | Definition<br>of<br>unit<br>vectors
| |
| | <math>\begin{align}
| |
| \hat{\boldsymbol\rho} &= \frac{ x \hat{\mathbf x} + y \hat{\mathbf y}}{\sqrt{x^2+y^2}} \\
| |
| \hat{\boldsymbol\phi} &= \frac{- y \hat{\mathbf x} + x \hat{\mathbf y}}{\sqrt{x^2+y^2}} \\
| |
| \hat{\mathbf z} &= \hat{\mathbf z}
| |
| \end{align}</math>
| |
| | <math>\begin{align}
| |
| \hat{\mathbf x} &= \cos\phi\hat{\boldsymbol\rho} - \sin\phi\hat{\boldsymbol\phi} \\
| |
| \hat{\mathbf y} &= \sin\phi\hat{\boldsymbol\rho} + \cos\phi\hat{\boldsymbol\phi} \\
| |
| \hat{\mathbf z} &= \hat{\mathbf z}
| |
| \end{align}</math>
| |
| | <math>\begin{align}
| |
| \hat{\mathbf x} &= \sin\theta\cos\phi\hat{\boldsymbol r} + \cos\theta\cos\phi\hat{\boldsymbol\theta}-\sin\phi\hat{\boldsymbol\phi} \\
| |
| \hat{\mathbf y} &= \sin\theta\sin\phi\hat{\boldsymbol r} + \cos\theta\sin\phi\hat{\boldsymbol\theta}+\cos\phi\hat{\boldsymbol\phi} \\
| |
| \hat{\mathbf z} &= \cos\theta \hat{\boldsymbol r} - \sin\theta \hat{\boldsymbol\theta}
| |
| \end{align}</math>
| |
| | <math>\begin{align}
| |
| \hat{\boldsymbol\sigma} &= \frac{\tau \hat{\mathbf x} - \sigma \hat{\mathbf y}}{\sqrt{\tau^2+\sigma^2}} \\
| |
| \hat{\boldsymbol\tau} &= \frac{\sigma \hat{\mathbf x} + \tau \hat{\mathbf y}}{\sqrt{\tau^2+\sigma^2}} \\
| |
| \hat{\mathbf z} &= \hat{\mathbf z}
| |
| \end{align}</math>
| |
| |- align="center"
| |
| | <math>\begin{align}
| |
| \hat{\mathbf r} &= \frac{x \hat{\mathbf x} + y \hat{\mathbf y} + z \hat{\mathbf z}}{\sqrt{x^2+y^2+z^2}} \\
| |
| \hat{\boldsymbol\theta} &= \frac{x z \hat{\mathbf x} + y z \hat{\mathbf y} - \left(x^2 + y^2\right) \hat{\mathbf z}}{\sqrt{x^2+y^2} \sqrt{x^2+y^2+z^2}} \\
| |
| \hat{\boldsymbol\phi} &= \frac{- y \hat{\mathbf x} + x \hat{\mathbf y}}{\sqrt{x^2+y^2}}
| |
| \end{align}</math>
| |
| | <math>\begin{align}
| |
| \hat{\mathbf r} &= \frac{\rho \hat{\boldsymbol\rho} + z \hat{\mathbf z}}{\sqrt{\rho^2 +z^2}} \\
| |
| \hat{\boldsymbol\theta} &= \frac{ z \hat{\boldsymbol\rho} - \rho \hat{\mathbf z}}{\sqrt{\rho^2 +z^2}} \\
| |
| \hat{\boldsymbol\phi} &= \hat{\boldsymbol\phi}
| |
| \end{align}</math>
| |
| | <math>\begin{align}
| |
| \hat{\boldsymbol\rho} &= \sin\theta \hat{\mathbf r} + \cos\theta \hat{\boldsymbol\theta} \\
| |
| \hat{\boldsymbol\phi} &= \hat{\boldsymbol\phi} \\
| |
| \hat{\mathbf z} &= \cos\theta \hat{\mathbf r} - \sin\theta \hat{\boldsymbol\theta}
| |
| \end{align}</math>
| |
| | <math>\begin{matrix}
| |
| \end{matrix}</math>
| |
| | |
| <!-- Definition of A -->
| |
| |- align="center"
| |
| ! style="background: white" | A [[vector field]] <math>\mathbf A</math>
| |
| | <math>A_x \hat{\mathbf x} + A_y \hat{\mathbf y} + A_z \hat{\mathbf z}</math>
| |
| | <math>A_\rho \hat{\boldsymbol\rho} + A_\phi \hat{\boldsymbol\phi} + A_z \hat{\mathbf z}</math>
| |
| | <math>A_r \hat{\boldsymbol r} + A_\theta \hat{\boldsymbol\theta} + A_\phi \hat{\boldsymbol\phi}</math>
| |
| | <math>A_\sigma \hat{\boldsymbol\sigma} + A_\tau \hat{\boldsymbol\tau} + A_\phi \hat{\mathbf z}</math>
| |
| | |
| <!-- grad f -->
| |
| |- align="center"
| |
| ! style="background: white" | [[Gradient]] <math>\nabla f</math>
| |
| | <math>{\partial f \over \partial x}\hat{\mathbf x} + {\partial f \over \partial y}\hat{\mathbf y}
| |
| + {\partial f \over \partial z}\hat{\mathbf z}</math>
| |
| | <math>{\partial f \over \partial \rho}\hat{\boldsymbol \rho}
| |
| + {1 \over \rho}{\partial f \over \partial \phi}\hat{\boldsymbol \phi}
| |
| + {\partial f \over \partial z}\hat{\mathbf z}</math>
| |
| | <math>{\partial f \over \partial r}\hat{\boldsymbol r}
| |
| + {1 \over r}{\partial f \over \partial \theta}\hat{\boldsymbol \theta}
| |
| + {1 \over r\sin\theta}{\partial f \over \partial \phi}\hat{\boldsymbol \phi}</math>
| |
| | <math> \frac{1}{\sqrt{\sigma^{2} + \tau^{2}}} {\partial f \over \partial \sigma}\hat{\boldsymbol \sigma} + \frac{1}{\sqrt{\sigma^{2} + \tau^{2}}} {\partial f \over \partial \tau}\hat{\boldsymbol \tau} + {\partial f \over \partial z}\hat{\mathbf z}</math>
| |
| | |
| <!-- div A -->
| |
| |- align="center"
| |
| ! style="background: white" | [[Divergence]] <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}}\left({\partial (\sqrt{\sigma^2+\tau^2} A_\sigma) \over \partial \sigma} + {\partial (\sqrt{\sigma^2+\tau^2} A_\tau) \over \partial \tau}\right) + {\partial A_z \over \partial z}</math>
| |
| | |
| <!-- curl A -->
| |
| |- align="center"
| |
| ! style="background: white" | [[Curl (mathematics)|Curl]] <math>\nabla \times \mathbf{A}</math>
| |
| | <math>\begin{align}
| |
| \left(\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}\right) &\hat{\mathbf x} + \\
| |
| + \left(\frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x}\right) &\hat{\mathbf y} + \\
| |
| + \left(\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\right) &\hat{\mathbf z}
| |
| \end{align}</math>
| |
| | <math>\begin{align}
| |
| \left(
| |
| \frac{1}{\rho} \frac{\partial A_z}{\partial \phi}
| |
| - \frac{\partial A_\phi}{\partial z}
| |
| \right) &\hat{\boldsymbol \rho} \\
| |
| + \left(
| |
| \frac{\partial A_\rho}{\partial z}
| |
| - \frac{\partial A_z}{\partial \rho}
| |
| \right) &\hat{\boldsymbol \phi} \\
| |
| + \frac{1}{\rho} \left(
| |
| \frac{\partial \left(\rho A_\phi\right)}{\partial \rho}
| |
| - \frac{\partial A_\rho}{\partial \phi}
| |
| \right) &\hat{\mathbf z}
| |
| \end{align}</math>
| |
| | <math>\begin{align}
| |
| \frac{1}{r\sin\theta} \left(
| |
| \frac{\partial}{\partial \theta} \left(A_\phi\sin\theta \right)
| |
| - \frac{\partial A_\theta}{\partial \phi}
| |
| \right) &\hat{\boldsymbol r} \\
| |
| + \frac{1}{r} \left(
| |
| \frac{1}{\sin\theta} \frac{\partial A_r}{\partial \phi}
| |
| - \frac{\partial}{\partial r} \left( r A_\phi \right)
| |
| \right) &\hat{\boldsymbol \theta} \\
| |
| + \frac{1}{r} \left(
| |
| \frac{\partial}{\partial r} \left( r A_\theta \right)
| |
| - \frac{\partial A_r}{\partial \theta}
| |
| \right) &\hat{\boldsymbol \phi}
| |
| \end{align}</math>
| |
| | <math>\begin{align}
| |
| \left(
| |
| \frac{1}{\sqrt{\sigma^2 + \tau^2}} \frac{\partial A_z}{\partial \tau}
| |
| - \frac{\partial A_\tau}{\partial z}
| |
| \right) &\hat{\boldsymbol \sigma} \\
| |
| - \left(
| |
| \frac{1}{\sqrt{\sigma^2 + \tau^2}} \frac{\partial A_z}{\partial \sigma}
| |
| - \frac{\partial A_\sigma}{\partial z}
| |
| \right) &\hat{\boldsymbol \tau} \\
| |
| + \frac{1}{\sqrt{\sigma^2 + \tau^2}} \left(
| |
| \frac{\partial \left(\sqrt{\sigma^2 + \tau^2} A_\sigma \right)}{\partial \tau}
| |
| - \frac{\partial \left(\sqrt{\sigma^2 + \tau^2} A_\tau \right)}{\partial \sigma}
| |
| \right) &\hat{\mathbf z}
| |
| \end{align}</math>
| |
| | |
| <!-- Laplacian f -->
| |
| |- align="center"
| |
| ! style="background: white" | [[Laplace operator]] <math>\Delta f \equiv \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>
| |
| | |
| <!-- vector Laplacian A -->
| |
| |- align="center"
| |
| ! style="background: white" | [[Vector Laplacian]] <math>\Delta \mathbf{A} \equiv \nabla^2 \mathbf{A}</math>
| |
| | <math>\Delta A_x \hat{\mathbf x} + \Delta A_y \hat{\mathbf y} + \Delta A_z \hat{\mathbf z} </math>
| |
| | {{Collapsible section |content =
| |
| <math>\begin{align}
| |
| \mathopen{}\left(\Delta A_\rho - \frac{A_\rho}{\rho^2} - \frac{2}{\rho^2} \frac{\partial A_\phi}{\partial \phi}\right)\mathclose{} &\hat{\boldsymbol\rho} \\
| |
| + \mathopen{}\left(\Delta A_\phi - \frac{A_\phi}{\rho^2} + \frac{2}{\rho^2} \frac{\partial A_\rho}{\partial \phi}\right)\mathclose{} &\hat{\boldsymbol\phi} \\
| |
| + \Delta A_z &\hat{\mathbf z}
| |
| \end{align}</math>
| |
| }}
| |
| | align="center" | {{Collapsible section |content =
| |
| <math>\begin{align}
| |
| \left(\Delta A_r - \frac{2 A_r}{r^2}
| |
| - \frac{2}{r^2\sin\theta} \frac{\partial \left(A_\theta \sin\theta\right)}{\partial\theta}
| |
| - \frac{2}{r^2\sin\theta}{\frac{\partial A_\phi}{\partial \phi}}\right) &\hat{\boldsymbol r} \\
| |
| + \left(\Delta A_\theta - \frac{A_\theta}{r^2\sin^2\theta}
| |
| + \frac{2}{r^2} \frac{\partial A_r}{\partial \theta}
| |
| - \frac{2 \cos\theta}{r^2\sin^2\theta} \frac{\partial A_\phi}{\partial \phi}\right) &\hat{\boldsymbol\theta} \\
| |
| + \left(\Delta A_\phi - \frac{A_\phi}{r^2\sin^2\theta}
| |
| + \frac{2}{r^2\sin\theta} \frac{\partial A_r}{\partial \phi}
| |
| + \frac{2 \cos\theta}{r^2\sin^2\theta} \frac{\partial A_\theta}{\partial \phi}\right) &\hat{\boldsymbol\phi}
| |
| \end{align}</math>
| |
| }}
| |
| | |
| <!-- Material derivative (A dot del)B -->
| |
| |- align="center"
| |
| ! style="background: white" | [[Material derivative]]<ref name="Mathworld">{{cite web |url=http://mathworld.wolfram.com/ConvectiveOperator.html|title=Convective Operator |author=Weisstein, Eric W. |date= |work=Mathworld |publisher= |accessdate=23 March 2011}}</ref>
| |
| <math>(\mathbf{A} \cdot \nabla) \mathbf{B}</math>
| |
| <!-- Cartesian -->
| |
| | {{Collapsible section |content =
| |
| <math>\begin{align}
| |
| \left(A_x \frac{\partial B_x}{\partial x} + A_y \frac{\partial B_x}{\partial y} + A_z \frac{\partial B_x}{\partial z}\right) &\hat{\mathbf{x}} \\
| |
| + \left(A_x \frac{\partial B_y}{\partial x} + A_y \frac{\partial B_y}{\partial y} + A_z \frac{\partial B_y}{\partial z}\right) &\hat{\mathbf{y}} \\
| |
| + \left(A_x \frac{\partial B_z}{\partial x} + A_y \frac{\partial B_z}{\partial y} + A_z \frac{\partial B_z}{\partial z}\right) &\hat{\mathbf{z}}
| |
| \end{align}</math>
| |
| }}
| |
| <!-- Cylindrical \frac{\partial B_}{\partial } -->
| |
| | {{Collapsible section |content =
| |
| <math>\begin{align}
| |
| \left(A_\rho \frac{\partial B_\rho}{\partial \rho}+\frac{A_\phi}{\rho}\frac{\partial B_\rho}{\partial \phi}+A_z\frac{\partial B_\rho}{\partial z}-\frac{A_\phi B_\phi}{\rho}\right)
| |
| &\hat{\boldsymbol\rho} \\
| |
| + \left(A_\rho \frac{\partial B_\phi}{\partial \rho} + \frac{A_\phi}{\rho}\frac{\partial B_\phi}{\partial \phi} + A_z\frac{\partial B_\phi}{\partial z} + \frac{A_\phi B_\rho}{\rho}\right)
| |
| &\hat{\boldsymbol\phi}\\
| |
| + \left(A_\rho \frac{\partial B_z}{\partial \rho}+\frac{A_\phi}{\rho}\frac{\partial B_z}{\partial \phi}+A_z\frac{\partial B_z}{\partial z}\right)
| |
| &\hat{\mathbf z}
| |
| \end{align}</math>
| |
| }}
| |
| <!-- Sp -->
| |
| | align="center" | {{Collapsible section |content =
| |
| <math>\begin{align}
| |
| \left(
| |
| A_r \frac{\partial B_r}{\partial r}
| |
| + \frac{A_\theta}{r} \frac{\partial B_r}{\partial \theta}
| |
| + \frac{A_\phi}{r\sin\theta} \frac{\partial B_r}{\partial \phi}
| |
| - \frac{A_\theta B_\theta + A_\phi B_\phi}{r}
| |
| \right) &\hat{\boldsymbol r} \\
| |
| + \left(
| |
| A_r \frac{\partial B_\theta}{\partial r}
| |
| + \frac{A_\theta}{r} \frac{\partial B_\theta}{\partial \theta}
| |
| + \frac{A_\phi}{r\sin\theta} \frac{\partial B_\theta}{\partial \phi}
| |
| + \frac{A_\theta B_r}{r} - \frac{A_\phi B_\phi\cot\theta}{r}
| |
| \right) &\hat{\boldsymbol\theta} \\
| |
| + \left(
| |
| A_r \frac{\partial B_\phi}{\partial r}
| |
| + \frac{A_\theta}{r} \frac{\partial B_\phi}{\partial \theta}
| |
| + \frac{A_\phi}{r\sin\theta} \frac{\partial B_\phi}{\partial \phi}
| |
| + \frac{A_\phi B_r}{r}
| |
| + \frac{A_\phi B_\theta \cot\theta}{r}
| |
| \right) &\hat{\boldsymbol\phi}
| |
| \end{align}</math>
| |
| }}
| |
| | |
| <!-- Differentials displacement -->
| |
| |- align="center"
| |
| ! style="background: white" | Differential displacement
| |
| | <math>d\mathbf{l} = dx \, \hat{\mathbf x} + dy \, \hat{\mathbf y} + dz \, \hat{\mathbf z}</math>
| |
| | <math>d\mathbf{l} = d\rho \, \hat{\boldsymbol \rho} + \rho \, d\phi \, \hat{\boldsymbol \phi} + dz \, \hat{\mathbf z}</math>
| |
| | <math>d\mathbf{l} = dr \, \hat{\mathbf r} + r \, d\theta \, \hat{\boldsymbol \theta} + r \, \sin\theta \, d\phi \, \hat{\boldsymbol \phi}</math>
| |
| | <math>d\mathbf{l} = \sqrt{\sigma^2 + \tau^2} \, d\sigma \, \hat{\boldsymbol \sigma} + \sqrt{\sigma^2 + \tau^2} \, d\tau \, \hat{\boldsymbol \tau} + dz \, \hat{\mathbf z}</math>
| |
| | |
| <!-- Differentials normal area -->
| |
| |- align="center"
| |
| ! style="background: white" | Differential normal area <math>d \mathbf S</math>
| |
| | <math>\begin{align}
| |
| dy \, dz &\hat{\mathbf x} \\
| |
| + dx \, dz &\hat{\mathbf y} \\
| |
| + dx \, dy &\hat{\mathbf z}
| |
| \end{align}</math>
| |
| | <math>\begin{align}
| |
| \rho \, d\phi \, dz &\hat{\boldsymbol\rho} \\
| |
| + d\rho \, dz &\hat{\boldsymbol\phi} \\
| |
| + \rho \, d\rho \, d\phi &\hat{\mathbf z}
| |
| \end{align}</math>
| |
| | <math>\begin{align}
| |
| r^2 \sin\theta \, d\theta \, d\phi &\hat{\mathbf r} \\
| |
| + r \sin\theta \, dr \, d\phi &\hat{\boldsymbol\theta} \\
| |
| + r \, dr \, d\theta &\hat{\boldsymbol\phi}
| |
| \end{align}</math>
| |
| | <math>\begin{align}
| |
| \sqrt{\sigma^2 + \tau^2} \, d\tau \, dz &\hat{\boldsymbol\sigma} \\
| |
| + \sqrt{\sigma^2 + \tau^2} \, d\sigma \, dz &\hat{\boldsymbol\tau} \\
| |
| + \left(\sigma^2 + \tau^2\right) \, d\sigma \, d\tau &\hat{\mathbf z}
| |
| \end{align}</math>
| |
| | |
| <!-- Differentials volume --> | |
| |- align="center"
| |
| ! style="background: white" | Differential volume <math>dV</math>
| |
| | <math>dx \, dy \, dz</math>
| |
| | <math>\rho \, d\rho \, d\phi \, dz</math>
| |
| | <math>r^2 \sin\theta \, dr \, d\theta \, d\phi</math>
| |
| | <math>\left(\sigma^2 + \tau^2\right) d\sigma \, d\tau \, dz</math>
| |
| | |
| <!-- nabla's on nabla's -->
| |
| |-
| |
| | colspan=5 | <strong>Non-trivial calculation rules:</strong>
| |
| # <math>\operatorname{div} \, \operatorname{grad} f \equiv \nabla \cdot \nabla f = \nabla^2 f \equiv \Delta f</math>
| |
| # <math>\operatorname{curl} \, \operatorname{grad} f \equiv \nabla \times \nabla f = \mathbf 0</math> | |
| # <math>\operatorname{div} \, \operatorname{curl} \mathbf{A} \equiv \nabla \cdot (\nabla \times \mathbf{A}) = 0</math>
| |
| # <math>\operatorname{curl} \, \operatorname{curl} \mathbf{A} \equiv \nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}</math> ([[Triple_product#Vector_triple_product|Lagrange's formula]] for del)
| |
| # <math>\Delta (f g) = f \Delta g + 2 \nabla f \cdot \nabla g + g \Delta f</math>
| |
| |}
| |
| </div>
| |
| | |
| == See also ==
| |
| * [[Del]]
| |
| * [[Orthogonal coordinates]]
| |
| * [[Curvilinear coordinates]]
| |
| * [[Vector fields in cylindrical and spherical coordinates]]
| |
| | |
| == References ==
| |
| {{Reflist}}
| |
| | |
| == External links ==
| |
| * [http://www.csulb.edu/~woollett/ Maxima Computer Algebra system scripts] to generate some of these operators in cylindrical and spherical coordinates.
| |
| | |
| [[Category:Vector calculus]]
| |
| [[Category:Coordinate systems]]
| |