# Coordinate systems/Derivation of formulas/Original Copy from Wikipedia
## Table from Wikipedia

* from https://en.wikipedia.org/w/index.php?title=Del_in_cylindrical_and_spherical_coordinates&oldid=612099218
* This is the backup copy. PLEASE DO NOT EDIT (Instead copy and paste onto another page)
Table with the del operator in cylindrical, spherical and parabolic cylindrical coordinates :
| Operation | Cartesian coordinates (x, y, z) | Cylindrical coordinates (ρ, ϕ, z) | Spherical coordinates (r, θ, ϕ) | Parabolic cylindrical coordinates (σ, τ, z) |
| Definition of coordinates | ${\begin{aligned}\rho &={\sqrt {x^{2}+y^{2}}}\\\phi &=\arctan(y/x)\\z&=z\end{aligned}}$ | ${\begin{aligned}x&=\rho \cos \phi \\y&=\rho \sin \phi \\z&=z\end{aligned}}$ | ${\begin{aligned}x&=r\sin \theta \cos \phi \\y&=r\sin \theta \sin \phi \\z&=r\cos \theta \end{aligned}}$ | ${\begin{aligned}x&=\sigma \tau \\y&={\tfrac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)\\z&=z\end{aligned}}$ |
| Definition of coordinates | ${\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=\arccos(z/r)\\\phi &=\arctan(y/x)\end{aligned}}$ | ${\begin{aligned}r&={\sqrt {\rho ^{2}+z^{2}}}\\\theta &=\arctan {(\rho /z)}\\\phi &=\phi \end{aligned}}$ | ${\begin{aligned}\rho &=r\sin \theta \\\phi &=\phi \\z&=r\cos \theta \end{aligned}}$ | ${\begin{aligned}\rho \cos \phi &=\sigma \tau \\\rho \sin \phi &={\tfrac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)\\z&=z\end{aligned}}$ |
| Definition of unit vectors | ${\begin{aligned}{\hat {\boldsymbol {\rho }}}&={\frac {x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\\{\hat {\boldsymbol {\phi }}}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}$ | ${\begin{aligned}{\hat {\mathbf {x} }}&=\cos \phi {\hat {\boldsymbol {\rho }}}-\sin \phi {\hat {\boldsymbol {\phi }}}\\{\hat {\mathbf {y} }}&=\sin \phi {\hat {\boldsymbol {\rho }}}+\cos \phi {\hat {\boldsymbol {\phi }}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}$ | ${\begin{aligned}{\hat {\mathbf {x} }}&=\sin \theta \cos \phi {\hat {\boldsymbol {r}}}+\cos \theta \cos \phi {\hat {\boldsymbol {\theta }}}-\sin \phi {\hat {\boldsymbol {\phi }}}\\{\hat {\mathbf {y} }}&=\sin \theta \sin \phi {\hat {\boldsymbol {r}}}+\cos \theta \sin \phi {\hat {\boldsymbol {\theta }}}+\cos \phi {\hat {\boldsymbol {\phi }}}\\{\hat {\mathbf {z} }}&=\cos \theta {\hat {\boldsymbol {r}}}-\sin \theta {\hat {\boldsymbol {\theta }}}\end{aligned}}$ | ${\begin{aligned}{\hat {\boldsymbol {\sigma }}}&={\frac {\tau {\hat {\mathbf {x} }}-\sigma {\hat {\mathbf {y} }}}{\sqrt {\tau ^{2}+\sigma ^{2}}}}\\{\hat {\boldsymbol {\tau }}}&={\frac {\sigma {\hat {\mathbf {x} }}+\tau {\hat {\mathbf {y} }}}{\sqrt {\tau ^{2}+\sigma ^{2}}}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}$ |
| Definition of unit vectors | ${\begin{aligned}{\hat {\mathbf {r} }}&={\frac {x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}+z{\hat {\mathbf {z} }}}{\sqrt {x^{2}+y^{2}+z^{2}}}}\\{\hat {\boldsymbol {\theta }}}&={\frac {xz{\hat {\mathbf {x} }}+yz{\hat {\mathbf {y} }}-\left(x^{2}+y^{2}\right){\hat {\mathbf {z} }}}{{\sqrt {x^{2}+y^{2}}}{\sqrt {x^{2}+y^{2}+z^{2}}}}}\\{\hat {\boldsymbol {\phi }}}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\end{aligned}}$ | ${\begin{aligned}{\hat {\mathbf {r} }}&={\frac {\rho {\hat {\boldsymbol {\rho }}}+z{\hat {\mathbf {z} }}}{\sqrt {\rho ^{2}+z^{2}}}}\\{\hat {\boldsymbol {\theta }}}&={\frac {z{\hat {\boldsymbol {\rho }}}-\rho {\hat {\mathbf {z} }}}{\sqrt {\rho ^{2}+z^{2}}}}\\{\hat {\boldsymbol {\phi }}}&={\hat {\boldsymbol {\phi }}}\end{aligned}}$ | ${\begin{aligned}{\hat {\boldsymbol {\rho }}}&=\sin \theta {\hat {\mathbf {r} }}+\cos \theta {\hat {\boldsymbol {\theta }}}\\{\hat {\boldsymbol {\phi }}}&={\hat {\boldsymbol {\phi }}}\\{\hat {\mathbf {z} }}&=\cos \theta {\hat {\mathbf {r} }}-\sin \theta {\hat {\boldsymbol {\theta }}}\end{aligned}}$ | ${\begin{matrix}\end{matrix}}$ |
| A vector field $\mathbf {A}$ | $A_{x}{\hat {\mathbf {x} }}+A_{y}{\hat {\mathbf {y} }}+A_{z}{\hat {\mathbf {z} }}$ | $A_{\rho }{\hat {\boldsymbol {\rho }}}+A_{\phi }{\hat {\boldsymbol {\phi }}}+A_{z}{\hat {\mathbf {z} }}$ | $A_{r}{\hat {\boldsymbol {r}}}+A_{\theta }{\hat {\boldsymbol {\theta }}}+A_{\phi }{\hat {\boldsymbol {\phi }}}$ | $A_{\sigma }{\hat {\boldsymbol {\sigma }}}+A_{\tau }{\hat {\boldsymbol {\tau }}}+A_{\phi }{\hat {\mathbf {z} }}$ |
| Gradient $\nabla f$ | ${\partial f \over \partial x}{\hat {\mathbf {x} }}+{\partial f \over \partial y}{\hat {\mathbf {y} }}+{\partial f \over \partial z}{\hat {\mathbf {z} }}$ | ${\partial f \over \partial \rho }{\hat {\boldsymbol {\rho }}}+{1 \over \rho }{\partial f \over \partial \phi }{\hat {\boldsymbol {\phi }}}+{\partial f \over \partial z}{\hat {\mathbf {z} }}$ | ${\partial f \over \partial r}{\hat {\boldsymbol {r}}}+{1 \over r}{\partial f \over \partial \theta }{\hat {\boldsymbol {\theta }}}+{1 \over r\sin \theta }{\partial f \over \partial \phi }{\hat {\boldsymbol {\phi }}}$ | ${\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\partial f \over \partial \sigma }{\hat {\boldsymbol {\sigma }}}+{\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\partial f \over \partial \tau }{\hat {\boldsymbol {\tau }}}+{\partial f \over \partial z}{\hat {\mathbf {z} }}$ |
| Divergence $\nabla \cdot \mathbf {A}$ | ${\partial A_{x} \over \partial x}+{\partial A_{y} \over \partial y}+{\partial A_{z} \over \partial z}$ | ${1 \over \rho }{\partial \left(\rho A_{\rho }\right) \over \partial \rho }+{1 \over \rho }{\partial A_{\phi } \over \partial \phi }+{\partial A_{z} \over \partial z}$ | ${1 \over r^{2}}{\partial \left(r^{2}A_{r}\right) \over \partial r}+{1 \over r\sin \theta }{\partial \over \partial \theta }\left(A_{\theta }\sin \theta \right)+{1 \over r\sin \theta }{\partial A_{\phi } \over \partial \phi }$ | ${\frac {1}{\sigma ^{2}+\tau ^{2}}}\left({\partial ({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\sigma }) \over \partial \sigma }+{\partial ({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\tau }) \over \partial \tau }\right)+{\partial A_{z} \over \partial z}$ |
| Curl $\nabla \times \mathbf {A}$ | ${\begin{aligned}\left({\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}\right)&{\hat {\mathbf {x} }}+\\+\left({\frac {\partial A_{x}}{\partial z}}-{\frac {\partial A_{z}}{\partial x}}\right)&{\hat {\mathbf {y} }}+\\+\left({\frac {\partial A_{y}}{\partial x}}-{\frac {\partial A_{x}}{\partial y}}\right)&{\hat {\mathbf {z} }}\end{aligned}}$ | ${\begin{aligned}\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \phi }}-{\frac {\partial A_{\phi }}{\partial z}}\right)&{\hat {\boldsymbol {\rho }}}\\+\left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right)&{\hat {\boldsymbol {\phi }}}\\+{\frac {1}{\rho }}\left({\frac {\partial \left(\rho A_{\phi }\right)}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \phi }}\right)&{\hat {\mathbf {z} }}\end{aligned}}$ | ${\begin{aligned}{\frac {1}{r\sin \theta }}\left({\frac {\partial }{\partial \theta }}\left(A_{\phi }\sin \theta \right)-{\frac {\partial A_{\theta }}{\partial \phi }}\right)&{\hat {\boldsymbol {r}}}\\+{\frac {1}{r}}\left({\frac {1}{\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}-{\frac {\partial }{\partial r}}\left(rA_{\phi }\right)\right)&{\hat {\boldsymbol {\theta }}}\\+{\frac {1}{r}}\left({\frac {\partial }{\partial r}}\left(rA_{\theta }\right)-{\frac {\partial A_{r}}{\partial \theta }}\right)&{\hat {\boldsymbol {\phi }}}\end{aligned}}$ | ${\begin{aligned}\left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\frac {\partial A_{z}}{\partial \tau }}-{\frac {\partial A_{\tau }}{\partial z}}\right)&{\hat {\boldsymbol {\sigma }}}\\-\left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\frac {\partial A_{z}}{\partial \sigma }}-{\frac {\partial A_{\sigma }}{\partial z}}\right)&{\hat {\boldsymbol {\tau }}}\\+{\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}\left({\frac {\partial \left({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\sigma }\right)}{\partial \tau }}-{\frac {\partial \left({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\tau }\right)}{\partial \sigma }}\right)&{\hat {\mathbf {z} }}\end{aligned}}$ |
| Laplace operator $\Delta f\equiv \nabla ^{2}f$ | ${\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}$ | ${1 \over \rho }{\partial \over \partial \rho }\left(\rho {\partial f \over \partial \rho }\right)+{1 \over \rho ^{2}}{\partial ^{2}f \over \partial \phi ^{2}}+{\partial ^{2}f \over \partial z^{2}}$ | ${1 \over r^{2}}{\partial \over \partial r}\!\left(r^{2}{\partial f \over \partial r}\right)\!+\!{1 \over r^{2}\!\sin \theta }{\partial \over \partial \theta }\!\left(\sin \theta {\partial f \over \partial \theta }\right)\!+\!{1 \over r^{2}\!\sin ^{2}\theta }{\partial ^{2}f \over \partial \phi ^{2}}$ | ${\frac {1}{\sigma ^{2}+\tau ^{2}}}\left({\frac {\partial ^{2}f}{\partial \sigma ^{2}}}+{\frac {\partial ^{2}f}{\partial \tau ^{2}}}\right)+{\frac {\partial ^{2}f}{\partial z^{2}}}$ |
| Vector Laplacian $\Delta \mathbf {A} \equiv \nabla ^{2}\mathbf {A}$ | $\Delta A_{x}{\hat {\mathbf {x} }}+\Delta A_{y}{\hat {\mathbf {y} }}+\Delta A_{z}{\hat {\mathbf {z} }}$ | click to view or hide ${\begin{aligned}{\mathopen}\left(\Delta A_{\rho }-{\frac {A_{\rho }}{\rho ^{2}}}-{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\phi }}{\partial \phi }}\right){\mathclose}&{\hat {\boldsymbol {\rho }}}\\+{\mathopen}\left(\Delta A_{\phi }-{\frac {A_{\phi }}{\rho ^{2}}}+{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\rho }}{\partial \phi }}\right){\mathclose}&{\hat {\boldsymbol {\phi }}}\\+\Delta A_{z}&{\hat {\mathbf {z} }}\end{aligned}}$ | click to view or hide ${\begin{aligned}\left(\Delta A_{r}-{\frac {2A_{r}}{r^{2}}}-{\frac {2}{r^{2}\sin \theta }}{\frac {\partial \left(A_{\theta }\sin \theta \right)}{\partial \theta }}-{\frac {2}{r^{2}\sin \theta }}{\frac {\partial A_{\phi }}{\partial \phi }}\right)&{\hat {\boldsymbol {r}}}\\+\left(\Delta A_{\theta }-{\frac {A_{\theta }}{r^{2}\sin ^{2}\theta }}+{\frac {2}{r^{2}}}{\frac {\partial A_{r}}{\partial \theta }}-{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}{\frac {\partial A_{\phi }}{\partial \phi }}\right)&{\hat {\boldsymbol {\theta }}}\\+\left(\Delta A_{\phi }-{\frac {A_{\phi }}{r^{2}\sin ^{2}\theta }}+{\frac {2}{r^{2}\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}+{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}{\frac {\partial A_{\theta }}{\partial \phi }}\right)&{\hat {\boldsymbol {\phi }}}\end{aligned}}$ |
| Material derivative $(\mathbf {A} \cdot \nabla )\mathbf {B}$ | click to view or hide ${\begin{aligned}\left(A_{x}{\frac {\partial B_{x}}{\partial x}}+A_{y}{\frac {\partial B_{x}}{\partial y}}+A_{z}{\frac {\partial B_{x}}{\partial z}}\right)&{\hat {\mathbf {x} }}\\+\left(A_{x}{\frac {\partial B_{y}}{\partial x}}+A_{y}{\frac {\partial B_{y}}{\partial y}}+A_{z}{\frac {\partial B_{y}}{\partial z}}\right)&{\hat {\mathbf {y} }}\\+\left(A_{x}{\frac {\partial B_{z}}{\partial x}}+A_{y}{\frac {\partial B_{z}}{\partial y}}+A_{z}{\frac {\partial B_{z}}{\partial z}}\right)&{\hat {\mathbf {z} }}\end{aligned}}$ | click to view or hide ${\begin{aligned}\left(A_{\rho }{\frac {\partial B_{\rho }}{\partial \rho }}+{\frac {A_{\phi }}{\rho }}{\frac {\partial B_{\rho }}{\partial \phi }}+A_{z}{\frac {\partial B_{\rho }}{\partial z}}-{\frac {A_{\phi }B_{\phi }}{\rho }}\right)&{\hat {\boldsymbol {\rho }}}\\+\left(A_{\rho }{\frac {\partial B_{\phi }}{\partial \rho }}+{\frac {A_{\phi }}{\rho }}{\frac {\partial B_{\phi }}{\partial \phi }}+A_{z}{\frac {\partial B_{\phi }}{\partial z}}+{\frac {A_{\phi }B_{\rho }}{\rho }}\right)&{\hat {\boldsymbol {\phi }}}\\+\left(A_{\rho }{\frac {\partial B_{z}}{\partial \rho }}+{\frac {A_{\phi }}{\rho }}{\frac {\partial B_{z}}{\partial \phi }}+A_{z}{\frac {\partial B_{z}}{\partial z}}\right)&{\hat {\mathbf {z} }}\end{aligned}}$ | click to view or hide ${\begin{aligned}\left(A_{r}{\frac {\partial B_{r}}{\partial r}}+{\frac {A_{\theta }}{r}}{\frac {\partial B_{r}}{\partial \theta }}+{\frac {A_{\phi }}{r\sin \theta }}{\frac {\partial B_{r}}{\partial \phi }}-{\frac {A_{\theta }B_{\theta }+A_{\phi }B_{\phi }}{r}}\right)&{\hat {\boldsymbol {r}}}\\+\left(A_{r}{\frac {\partial B_{\theta }}{\partial r}}+{\frac {A_{\theta }}{r}}{\frac {\partial B_{\theta }}{\partial \theta }}+{\frac {A_{\phi }}{r\sin \theta }}{\frac {\partial B_{\theta }}{\partial \phi }}+{\frac {A_{\theta }B_{r}}{r}}-{\frac {A_{\phi }B_{\phi }\cot \theta }{r}}\right)&{\hat {\boldsymbol {\theta }}}\\+\left(A_{r}{\frac {\partial B_{\phi }}{\partial r}}+{\frac {A_{\theta }}{r}}{\frac {\partial B_{\phi }}{\partial \theta }}+{\frac {A_{\phi }}{r\sin \theta }}{\frac {\partial B_{\phi }}{\partial \phi }}+{\frac {A_{\phi }B_{r}}{r}}+{\frac {A_{\phi }B_{\theta }\cot \theta }{r}}\right)&{\hat {\boldsymbol {\phi }}}\end{aligned}}$ |
| Differential displacement | $d\mathbf {l} =dx\,{\hat {\mathbf {x} }}+dy\,{\hat {\mathbf {y} }}+dz\,{\hat {\mathbf {z} }}$ | $d\mathbf {l} =d\rho \,{\hat {\boldsymbol {\rho }}}+\rho \,d\phi \,{\hat {\boldsymbol {\phi }}}+dz\,{\hat {\mathbf {z} }}$ | $d\mathbf {l} =dr\,{\hat {\mathbf {r} }}+r\,d\theta \,{\hat {\boldsymbol {\theta }}}+r\,\sin \theta \,d\phi \,{\hat {\boldsymbol {\phi }}}$ | $d\mathbf {l} ={\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\sigma \,{\hat {\boldsymbol {\sigma }}}+{\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\tau \,{\hat {\boldsymbol {\tau }}}+dz\,{\hat {\mathbf {z} }}$ |
| Differential normal area $d\mathbf {S}$ | ${\begin{aligned}dy\,dz&{\hat {\mathbf {x} }}\\+dx\,dz&{\hat {\mathbf {y} }}\\+dx\,dy&{\hat {\mathbf {z} }}\end{aligned}}$ | ${\begin{aligned}\rho \,d\phi \,dz&{\hat {\boldsymbol {\rho }}}\\+d\rho \,dz&{\hat {\boldsymbol {\phi }}}\\+\rho \,d\rho \,d\phi &{\hat {\mathbf {z} }}\end{aligned}}$ | ${\begin{aligned}r^{2}\sin \theta \,d\theta \,d\phi &{\hat {\mathbf {r} }}\\+r\sin \theta \,dr\,d\phi &{\hat {\boldsymbol {\theta }}}\\+r\,dr\,d\theta &{\hat {\boldsymbol {\phi }}}\end{aligned}}$ | ${\begin{aligned}{\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\tau \,dz&{\hat {\boldsymbol {\sigma }}}\\+{\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\sigma \,dz&{\hat {\boldsymbol {\tau }}}\\+\left(\sigma ^{2}+\tau ^{2}\right)\,d\sigma \,d\tau &{\hat {\mathbf {z} }}\end{aligned}}$ |
| Differential volume $dV$ | $dx\,dy\,dz$ | $\rho \,d\rho \,d\phi \,dz$ | $r^{2}\sin \theta \,dr\,d\theta \,d\phi$ | $\left(\sigma ^{2}+\tau ^{2}\right)d\sigma \,d\tau \,dz$ |
| Non-trivial calculation rules: $\operatorname {div} \,\operatorname {grad} f\equiv \nabla \cdot \nabla f=\nabla ^{2}f\equiv \Delta f$ $\operatorname {curl} \,\operatorname {grad} f\equiv \nabla \times \nabla f=\mathbf {0}$ $\operatorname {div} \,\operatorname {curl} \mathbf {A} \equiv \nabla \cdot (\nabla \times \mathbf {A} )=0$ $\operatorname {curl} \,\operatorname {curl} \mathbf {A} \equiv \nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A}$ (Lagrange's formula for del) $\Delta (fg)=f\Delta g+2\nabla f\cdot \nabla g+g\Delta f$ |||||

## Original Wikipedia Table

* from https://en.wikipedia.org/w/index.php?title=Del_in_cylindrical_and_spherical_coordinates&oldid=612099218
* Please do not edit or modify this table without very good reason
Table with the del operator in cylindrical, spherical and parabolic cylindrical coordinates :
| Operation | Cartesian coordinates (x, y, z) | Cylindrical coordinates (ρ, ϕ, z) | Spherical coordinates (r, θ, ϕ) | Parabolic cylindrical coordinates (σ, τ, z) |
| Definition of coordinates | ${\begin{aligned}\rho &={\sqrt {x^{2}+y^{2}}}\\\phi &=\arctan(y/x)\\z&=z\end{aligned}}$ | ${\begin{aligned}x&=\rho \cos \phi \\y&=\rho \sin \phi \\z&=z\end{aligned}}$ | ${\begin{aligned}x&=r\sin \theta \cos \phi \\y&=r\sin \theta \sin \phi \\z&=r\cos \theta \end{aligned}}$ | ${\begin{aligned}x&=\sigma \tau \\y&={\tfrac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)\\z&=z\end{aligned}}$ |
| Definition of coordinates | ${\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=\arccos(z/r)\\\phi &=\arctan(y/x)\end{aligned}}$ | ${\begin{aligned}r&={\sqrt {\rho ^{2}+z^{2}}}\\\theta &=\arctan {(\rho /z)}\\\phi &=\phi \end{aligned}}$ | ${\begin{aligned}\rho &=r\sin \theta \\\phi &=\phi \\z&=r\cos \theta \end{aligned}}$ | ${\begin{aligned}\rho \cos \phi &=\sigma \tau \\\rho \sin \phi &={\tfrac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)\\z&=z\end{aligned}}$ |
| Definition of unit vectors | ${\begin{aligned}{\hat {\boldsymbol {\rho }}}&={\frac {x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\\{\hat {\boldsymbol {\phi }}}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}$ | ${\begin{aligned}{\hat {\mathbf {x} }}&=\cos \phi {\hat {\boldsymbol {\rho }}}-\sin \phi {\hat {\boldsymbol {\phi }}}\\{\hat {\mathbf {y} }}&=\sin \phi {\hat {\boldsymbol {\rho }}}+\cos \phi {\hat {\boldsymbol {\phi }}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}$ | ${\begin{aligned}{\hat {\mathbf {x} }}&=\sin \theta \cos \phi {\hat {\boldsymbol {r}}}+\cos \theta \cos \phi {\hat {\boldsymbol {\theta }}}-\sin \phi {\hat {\boldsymbol {\phi }}}\\{\hat {\mathbf {y} }}&=\sin \theta \sin \phi {\hat {\boldsymbol {r}}}+\cos \theta \sin \phi {\hat {\boldsymbol {\theta }}}+\cos \phi {\hat {\boldsymbol {\phi }}}\\{\hat {\mathbf {z} }}&=\cos \theta {\hat {\boldsymbol {r}}}-\sin \theta {\hat {\boldsymbol {\theta }}}\end{aligned}}$ | ${\begin{aligned}{\hat {\boldsymbol {\sigma }}}&={\frac {\tau {\hat {\mathbf {x} }}-\sigma {\hat {\mathbf {y} }}}{\sqrt {\tau ^{2}+\sigma ^{2}}}}\\{\hat {\boldsymbol {\tau }}}&={\frac {\sigma {\hat {\mathbf {x} }}+\tau {\hat {\mathbf {y} }}}{\sqrt {\tau ^{2}+\sigma ^{2}}}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}$ |
| Definition of unit vectors | ${\begin{aligned}{\hat {\mathbf {r} }}&={\frac {x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}+z{\hat {\mathbf {z} }}}{\sqrt {x^{2}+y^{2}+z^{2}}}}\\{\hat {\boldsymbol {\theta }}}&={\frac {xz{\hat {\mathbf {x} }}+yz{\hat {\mathbf {y} }}-\left(x^{2}+y^{2}\right){\hat {\mathbf {z} }}}{{\sqrt {x^{2}+y^{2}}}{\sqrt {x^{2}+y^{2}+z^{2}}}}}\\{\hat {\boldsymbol {\phi }}}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\end{aligned}}$ | ${\begin{aligned}{\hat {\mathbf {r} }}&={\frac {\rho {\hat {\boldsymbol {\rho }}}+z{\hat {\mathbf {z} }}}{\sqrt {\rho ^{2}+z^{2}}}}\\{\hat {\boldsymbol {\theta }}}&={\frac {z{\hat {\boldsymbol {\rho }}}-\rho {\hat {\mathbf {z} }}}{\sqrt {\rho ^{2}+z^{2}}}}\\{\hat {\boldsymbol {\phi }}}&={\hat {\boldsymbol {\phi }}}\end{aligned}}$ | ${\begin{aligned}{\hat {\boldsymbol {\rho }}}&=\sin \theta {\hat {\mathbf {r} }}+\cos \theta {\hat {\boldsymbol {\theta }}}\\{\hat {\boldsymbol {\phi }}}&={\hat {\boldsymbol {\phi }}}\\{\hat {\mathbf {z} }}&=\cos \theta {\hat {\mathbf {r} }}-\sin \theta {\hat {\boldsymbol {\theta }}}\end{aligned}}$ | ${\begin{matrix}\end{matrix}}$ |
| A vector field $\mathbf {A}$ | $A_{x}{\hat {\mathbf {x} }}+A_{y}{\hat {\mathbf {y} }}+A_{z}{\hat {\mathbf {z} }}$ | $A_{\rho }{\hat {\boldsymbol {\rho }}}+A_{\phi }{\hat {\boldsymbol {\phi }}}+A_{z}{\hat {\mathbf {z} }}$ | $A_{r}{\hat {\boldsymbol {r}}}+A_{\theta }{\hat {\boldsymbol {\theta }}}+A_{\phi }{\hat {\boldsymbol {\phi }}}$ | $A_{\sigma }{\hat {\boldsymbol {\sigma }}}+A_{\tau }{\hat {\boldsymbol {\tau }}}+A_{\phi }{\hat {\mathbf {z} }}$ |
| Gradient $\nabla f$ | ${\partial f \over \partial x}{\hat {\mathbf {x} }}+{\partial f \over \partial y}{\hat {\mathbf {y} }}+{\partial f \over \partial z}{\hat {\mathbf {z} }}$ | ${\partial f \over \partial \rho }{\hat {\boldsymbol {\rho }}}+{1 \over \rho }{\partial f \over \partial \phi }{\hat {\boldsymbol {\phi }}}+{\partial f \over \partial z}{\hat {\mathbf {z} }}$ | ${\partial f \over \partial r}{\hat {\boldsymbol {r}}}+{1 \over r}{\partial f \over \partial \theta }{\hat {\boldsymbol {\theta }}}+{1 \over r\sin \theta }{\partial f \over \partial \phi }{\hat {\boldsymbol {\phi }}}$ | ${\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\partial f \over \partial \sigma }{\hat {\boldsymbol {\sigma }}}+{\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\partial f \over \partial \tau }{\hat {\boldsymbol {\tau }}}+{\partial f \over \partial z}{\hat {\mathbf {z} }}$ |
| Divergence $\nabla \cdot \mathbf {A}$ | ${\partial A_{x} \over \partial x}+{\partial A_{y} \over \partial y}+{\partial A_{z} \over \partial z}$ | ${1 \over \rho }{\partial \left(\rho A_{\rho }\right) \over \partial \rho }+{1 \over \rho }{\partial A_{\phi } \over \partial \phi }+{\partial A_{z} \over \partial z}$ | ${1 \over r^{2}}{\partial \left(r^{2}A_{r}\right) \over \partial r}+{1 \over r\sin \theta }{\partial \over \partial \theta }\left(A_{\theta }\sin \theta \right)+{1 \over r\sin \theta }{\partial A_{\phi } \over \partial \phi }$ | ${\frac {1}{\sigma ^{2}+\tau ^{2}}}\left({\partial ({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\sigma }) \over \partial \sigma }+{\partial ({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\tau }) \over \partial \tau }\right)+{\partial A_{z} \over \partial z}$ |
| Curl $\nabla \times \mathbf {A}$ | ${\begin{aligned}\left({\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}\right)&{\hat {\mathbf {x} }}+\\+\left({\frac {\partial A_{x}}{\partial z}}-{\frac {\partial A_{z}}{\partial x}}\right)&{\hat {\mathbf {y} }}+\\+\left({\frac {\partial A_{y}}{\partial x}}-{\frac {\partial A_{x}}{\partial y}}\right)&{\hat {\mathbf {z} }}\end{aligned}}$ | ${\begin{aligned}\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \phi }}-{\frac {\partial A_{\phi }}{\partial z}}\right)&{\hat {\boldsymbol {\rho }}}\\+\left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right)&{\hat {\boldsymbol {\phi }}}\\+{\frac {1}{\rho }}\left({\frac {\partial \left(\rho A_{\phi }\right)}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \phi }}\right)&{\hat {\mathbf {z} }}\end{aligned}}$ | ${\begin{aligned}{\frac {1}{r\sin \theta }}\left({\frac {\partial }{\partial \theta }}\left(A_{\phi }\sin \theta \right)-{\frac {\partial A_{\theta }}{\partial \phi }}\right)&{\hat {\boldsymbol {r}}}\\+{\frac {1}{r}}\left({\frac {1}{\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}-{\frac {\partial }{\partial r}}\left(rA_{\phi }\right)\right)&{\hat {\boldsymbol {\theta }}}\\+{\frac {1}{r}}\left({\frac {\partial }{\partial r}}\left(rA_{\theta }\right)-{\frac {\partial A_{r}}{\partial \theta }}\right)&{\hat {\boldsymbol {\phi }}}\end{aligned}}$ | ${\begin{aligned}\left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\frac {\partial A_{z}}{\partial \tau }}-{\frac {\partial A_{\tau }}{\partial z}}\right)&{\hat {\boldsymbol {\sigma }}}\\-\left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\frac {\partial A_{z}}{\partial \sigma }}-{\frac {\partial A_{\sigma }}{\partial z}}\right)&{\hat {\boldsymbol {\tau }}}\\+{\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}\left({\frac {\partial \left({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\sigma }\right)}{\partial \tau }}-{\frac {\partial \left({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\tau }\right)}{\partial \sigma }}\right)&{\hat {\mathbf {z} }}\end{aligned}}$ |
| Laplace operator $\Delta f\equiv \nabla ^{2}f$ | ${\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}$ | ${1 \over \rho }{\partial \over \partial \rho }\left(\rho {\partial f \over \partial \rho }\right)+{1 \over \rho ^{2}}{\partial ^{2}f \over \partial \phi ^{2}}+{\partial ^{2}f \over \partial z^{2}}$ | ${1 \over r^{2}}{\partial \over \partial r}\!\left(r^{2}{\partial f \over \partial r}\right)\!+\!{1 \over r^{2}\!\sin \theta }{\partial \over \partial \theta }\!\left(\sin \theta {\partial f \over \partial \theta }\right)\!+\!{1 \over r^{2}\!\sin ^{2}\theta }{\partial ^{2}f \over \partial \phi ^{2}}$ | ${\frac {1}{\sigma ^{2}+\tau ^{2}}}\left({\frac {\partial ^{2}f}{\partial \sigma ^{2}}}+{\frac {\partial ^{2}f}{\partial \tau ^{2}}}\right)+{\frac {\partial ^{2}f}{\partial z^{2}}}$ |
| Vector Laplacian $\Delta \mathbf {A} \equiv \nabla ^{2}\mathbf {A}$ | $\Delta A_{x}{\hat {\mathbf {x} }}+\Delta A_{y}{\hat {\mathbf {y} }}+\Delta A_{z}{\hat {\mathbf {z} }}$ | Template:Collapsible section | Template:Collapsible section |
| Material derivative $(\mathbf {A} \cdot \nabla )\mathbf {B}$ | Template:Collapsible section | Template:Collapsible section | Template:Collapsible section |
| Differential displacement | $d\mathbf {l} =dx\,{\hat {\mathbf {x} }}+dy\,{\hat {\mathbf {y} }}+dz\,{\hat {\mathbf {z} }}$ | $d\mathbf {l} =d\rho \,{\hat {\boldsymbol {\rho }}}+\rho \,d\phi \,{\hat {\boldsymbol {\phi }}}+dz\,{\hat {\mathbf {z} }}$ | $d\mathbf {l} =dr\,{\hat {\mathbf {r} }}+r\,d\theta \,{\hat {\boldsymbol {\theta }}}+r\,\sin \theta \,d\phi \,{\hat {\boldsymbol {\phi }}}$ | $d\mathbf {l} ={\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\sigma \,{\hat {\boldsymbol {\sigma }}}+{\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\tau \,{\hat {\boldsymbol {\tau }}}+dz\,{\hat {\mathbf {z} }}$ |
| Differential normal area $d\mathbf {S}$ | ${\begin{aligned}dy\,dz&{\hat {\mathbf {x} }}\\+dx\,dz&{\hat {\mathbf {y} }}\\+dx\,dy&{\hat {\mathbf {z} }}\end{aligned}}$ | ${\begin{aligned}\rho \,d\phi \,dz&{\hat {\boldsymbol {\rho }}}\\+d\rho \,dz&{\hat {\boldsymbol {\phi }}}\\+\rho \,d\rho \,d\phi &{\hat {\mathbf {z} }}\end{aligned}}$ | ${\begin{aligned}r^{2}\sin \theta \,d\theta \,d\phi &{\hat {\mathbf {r} }}\\+r\sin \theta \,dr\,d\phi &{\hat {\boldsymbol {\theta }}}\\+r\,dr\,d\theta &{\hat {\boldsymbol {\phi }}}\end{aligned}}$ | ${\begin{aligned}{\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\tau \,dz&{\hat {\boldsymbol {\sigma }}}\\+{\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\sigma \,dz&{\hat {\boldsymbol {\tau }}}\\+\left(\sigma ^{2}+\tau ^{2}\right)\,d\sigma \,d\tau &{\hat {\mathbf {z} }}\end{aligned}}$ |
| Differential volume $dV$ | $dx\,dy\,dz$ | $\rho \,d\rho \,d\phi \,dz$ | $r^{2}\sin \theta \,dr\,d\theta \,d\phi$ | $\left(\sigma ^{2}+\tau ^{2}\right)d\sigma \,d\tau \,dz$ |
| Non-trivial calculation rules: $\operatorname {div} \,\operatorname {grad} f\equiv \nabla \cdot \nabla f=\nabla ^{2}f\equiv \Delta f$ $\operatorname {curl} \,\operatorname {grad} f\equiv \nabla \times \nabla f=\mathbf {0}$ $\operatorname {div} \,\operatorname {curl} \mathbf {A} \equiv \nabla \cdot (\nabla \times \mathbf {A} )=0$ $\operatorname {curl} \,\operatorname {curl} \mathbf {A} \equiv \nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A}$ (Lagrange's formula for del) $\Delta (fg)=f\Delta g+2\nabla f\cdot \nabla g+g\Delta f$ |||||
* 1 2 Weisstein, Eric W. "Convective Operator". Mathworld. Retrieved 23 March 2011.