Оператор
Прямоугольные координаты x, y, z Цилиндрические координаты ρ, φ, z )
Сферические координаты r , θ, φПараболические координаты σ, τ, z )
Формулы преобразования координат
ρ
=
x
2
+
y
2
φ
=
arctg
(
y
/
x
)
z
=
z
{\displaystyle {\begin{matrix}\rho &=&{\sqrt {x^{2}+y^{2}}}\\\varphi &=&\operatorname {arctg} (y/x)\\z&=&z\end{matrix}}}
x
=
ρ
cos
φ
y
=
ρ
sin
φ
z
=
z
{\displaystyle {\begin{matrix}x&=&\rho \cos \varphi \\y&=&\rho \sin \varphi \\z&=&z\end{matrix}}}
x
=
r
sin
θ
cos
φ
y
=
r
sin
θ
sin
φ
z
=
r
cos
θ
{\displaystyle {\begin{matrix}x&=&r\sin \theta \cos \varphi \\y&=&r\sin \theta \sin \varphi \\z&=&r\cos \theta \end{matrix}}}
x
=
σ
τ
y
=
1
2
(
τ
2
−
σ
2
)
z
=
z
{\displaystyle {\begin{matrix}x&=&\sigma \tau \\y&=&{\frac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)\\z&=&z\end{matrix}}}
r
=
x
2
+
y
2
+
z
2
θ
=
arccos
(
z
/
r
)
φ
=
arctg
(
y
/
x
)
{\displaystyle {\begin{matrix}r&=&{\sqrt {{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}}\\\theta &=&\arccos \left(z/r\right)\\\varphi &=&\operatorname {arctg} (y/x)\\\end{matrix}}}
r
=
ρ
2
+
z
2
θ
=
arctg
(
ρ
/
z
)
φ
=
φ
{\displaystyle {\begin{matrix}r&=&{\sqrt {\rho ^{2}+z^{2}}}\\\theta &=&\operatorname {arctg} {(\rho /z)}\\\varphi &=&\varphi \end{matrix}}}
ρ
=
r
sin
θ
φ
=
φ
z
=
r
cos
θ
{\displaystyle {\begin{matrix}\rho &=&r\sin {\theta }\\\varphi &=&\varphi \\z&=&r\cos {\theta }\end{matrix}}}
ρ
cos
φ
=
σ
τ
ρ
sin
φ
=
1
2
(
τ
2
−
σ
2
)
z
=
z
{\displaystyle {\begin{matrix}\rho \cos \varphi &=&\sigma \tau \\\rho \sin \varphi &=&{\frac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)\\z&=&z\end{matrix}}}
Радиус-вектор произвольной точки
x
x
^
+
y
y
^
+
z
z
^
{\displaystyle x\mathbf {\hat {x}} +y\mathbf {\hat {y}} +z\mathbf {\hat {z}} }
ρ
ρ
^
+
z
z
^
{\displaystyle \rho {\boldsymbol {\hat {\rho }}}+z{\boldsymbol {\hat {z}}}}
r
r
^
{\displaystyle r{\boldsymbol {\hat {r}}}}
1
2
σ
2
+
τ
2
σ
σ
^
+
1
2
σ
2
+
τ
2
τ
τ
^
+
z
z
^
{\displaystyle {\frac {1}{2}}{\sqrt {\sigma ^{2}+\tau ^{2}}}\sigma {\boldsymbol {\hat {\sigma }}}+{\frac {1}{2}}{\sqrt {\sigma ^{2}+\tau ^{2}}}\tau {\boldsymbol {\hat {\tau }}}+z\mathbf {\hat {z}} }
Связь единичных векторов
ρ
^
=
x
ρ
x
^
+
y
ρ
y
^
φ
^
=
−
y
ρ
x
^
+
x
ρ
y
^
z
^
=
z
^
{\displaystyle {\begin{matrix}{\boldsymbol {\hat {\rho }}}&=&{\frac {x}{\rho }}\mathbf {\hat {x}} +{\frac {y}{\rho }}\mathbf {\hat {y}} \\{\boldsymbol {\hat {\varphi }}}&=&-{\frac {y}{\rho }}\mathbf {\hat {x}} +{\frac {x}{\rho }}\mathbf {\hat {y}} \\\mathbf {\hat {z}} &=&\mathbf {\hat {z}} \end{matrix}}}
x
^
=
cos
φ
ρ
^
−
sin
φ
φ
^
y
^
=
sin
φ
ρ
^
+
cos
φ
φ
^
z
^
=
z
^
{\displaystyle {\begin{matrix}\mathbf {\hat {x}} &=&\cos \varphi {\boldsymbol {\hat {\rho }}}-\sin \varphi {\boldsymbol {\hat {\varphi }}}\\\mathbf {\hat {y}} &=&\sin \varphi {\boldsymbol {\hat {\rho }}}+\cos \varphi {\boldsymbol {\hat {\varphi }}}\\\mathbf {\hat {z}} &=&\mathbf {\hat {z}} \end{matrix}}}
x
^
=
sin
θ
cos
φ
r
^
+
cos
θ
cos
φ
θ
^
−
sin
φ
φ
^
y
^
=
sin
θ
sin
φ
r
^
+
cos
θ
sin
φ
θ
^
+
cos
φ
φ
^
z
^
=
cos
θ
r
^
−
sin
θ
θ
^
{\displaystyle {\begin{matrix}\mathbf {\hat {x}} &=&\sin \theta \cos \varphi {\boldsymbol {\hat {r}}}+\cos \theta \cos \varphi {\boldsymbol {\hat {\theta }}}-\sin \varphi {\boldsymbol {\hat {\varphi }}}\\\mathbf {\hat {y}} &=&\sin \theta \sin \varphi {\boldsymbol {\hat {r}}}+\cos \theta \sin \varphi {\boldsymbol {\hat {\theta }}}+\cos \varphi {\boldsymbol {\hat {\varphi }}}\\\mathbf {\hat {z}} &=&\cos \theta {\boldsymbol {\hat {r}}}-\sin \theta {\boldsymbol {\hat {\theta }}}\\\end{matrix}}}
σ
^
=
τ
τ
2
+
σ
2
x
^
−
σ
τ
2
+
σ
2
y
^
τ
^
=
σ
τ
2
+
σ
2
x
^
+
τ
τ
2
+
σ
2
y
^
z
^
=
z
^
{\displaystyle {\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}}}
r
^
=
x
x
^
+
y
y
^
+
z
z
^
r
θ
^
=
x
z
x
^
+
y
z
y
^
−
ρ
2
z
^
r
ρ
φ
^
=
−
y
x
^
+
x
y
^
ρ
{\displaystyle {\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 {\varphi }}}&=&{\frac {-y\mathbf {\hat {x}} +x\mathbf {\hat {y}} }{\rho }}\end{matrix}}}
r
^
=
ρ
r
ρ
^
+
z
r
z
^
θ
^
=
z
r
ρ
^
−
ρ
r
z
^
φ
^
=
φ
^
{\displaystyle {\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 {\varphi }}}&=&{\boldsymbol {\hat {\varphi }}}\end{matrix}}}
ρ
^
=
sin
θ
r
^
+
cos
θ
θ
^
φ
^
=
φ
^
z
^
=
cos
θ
r
^
−
sin
θ
θ
^
{\displaystyle {\begin{matrix}{\boldsymbol {\hat {\rho }}}&=&\sin \theta \mathbf {\hat {r}} +\cos \theta {\boldsymbol {\hat {\theta }}}\\{\boldsymbol {\hat {\varphi }}}&=&{\boldsymbol {\hat {\varphi }}}\\\mathbf {\hat {z}} &=&\cos \theta \mathbf {\hat {r}} -\sin \theta {\boldsymbol {\hat {\theta }}}\\\end{matrix}}}
{\displaystyle {\begin{matrix}\end{matrix}}}
Векторное поле
A
{\displaystyle \mathbf {A} }
A
x
x
^
+
A
y
y
^
+
A
z
z
^
{\displaystyle A_{x}\mathbf {\hat {x}} +A_{y}\mathbf {\hat {y}} +A_{z}\mathbf {\hat {z}} }
A
ρ
ρ
^
+
A
φ
φ
^
+
A
z
z
^
{\displaystyle A_{\rho }{\boldsymbol {\hat {\rho }}}+A_{\varphi }{\boldsymbol {\hat {\varphi }}}+A_{z}{\boldsymbol {\hat {z}}}}
A
r
r
^
+
A
θ
θ
^
+
A
φ
φ
^
{\displaystyle A_{r}{\boldsymbol {\hat {r}}}+A_{\theta }{\boldsymbol {\hat {\theta }}}+A_{\varphi }{\boldsymbol {\hat {\varphi }}}}
A
σ
σ
^
+
A
τ
τ
^
+
A
φ
z
^
{\displaystyle A_{\sigma }{\boldsymbol {\hat {\sigma }}}+A_{\tau }{\boldsymbol {\hat {\tau }}}+A_{\varphi }{\boldsymbol {\hat {z}}}}
Градиент
∇
f
{\displaystyle \nabla f}
∂
f
∂
x
x
^
+
∂
f
∂
y
y
^
+
∂
f
∂
z
z
^
{\displaystyle {\partial f \over \partial x}\mathbf {\hat {x}} +{\partial f \over \partial y}\mathbf {\hat {y}} +{\partial f \over \partial z}\mathbf {\hat {z}} }
∂
f
∂
ρ
ρ
^
+
1
ρ
∂
f
∂
φ
φ
^
+
∂
f
∂
z
z
^
{\displaystyle {\partial f \over \partial \rho }{\boldsymbol {\hat {\rho }}}+{1 \over \rho }{\partial f \over \partial \varphi }{\boldsymbol {\hat {\varphi }}}+{\partial f \over \partial z}{\boldsymbol {\hat {z}}}}
∂
f
∂
r
r
^
+
1
r
∂
f
∂
θ
θ
^
+
1
r
sin
θ
∂
f
∂
φ
φ
^
{\displaystyle {\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 \varphi }{\boldsymbol {\hat {\varphi }}}}
1
σ
2
+
τ
2
∂
f
∂
σ
σ
^
+
1
σ
2
+
τ
2
∂
f
∂
τ
τ
^
+
∂
f
∂
z
z
^
{\displaystyle {\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}}}}
Дивергенция
∇
⋅
A
{\displaystyle \nabla \cdot \mathbf {A} }
∂
A
x
∂
x
+
∂
A
y
∂
y
+
∂
A
z
∂
z
{\displaystyle {\partial A_{x} \over \partial x}+{\partial A_{y} \over \partial y}+{\partial A_{z} \over \partial z}}
1
ρ
∂
(
ρ
A
ρ
)
∂
ρ
+
1
ρ
∂
A
φ
∂
φ
+
∂
A
z
∂
z
{\displaystyle {1 \over \rho }{\partial \left(\rho A_{\rho }\right) \over \partial \rho }+{1 \over \rho }{\partial A_{\varphi } \over \partial \varphi }+{\partial A_{z} \over \partial z}}
1
r
2
∂
(
r
2
A
r
)
∂
r
+
1
r
sin
θ
∂
∂
θ
(
A
θ
sin
θ
)
+
1
r
sin
θ
∂
A
φ
∂
φ
{\displaystyle {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_{\varphi } \over \partial \varphi }}
1
σ
2
+
τ
2
∂
A
σ
∂
σ
+
1
σ
2
+
τ
2
∂
A
τ
∂
τ
+
∂
A
z
∂
z
{\displaystyle {\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}}
Ротор
∇
×
A
{\displaystyle \nabla \times \mathbf {A} }
(
∂
A
z
∂
y
−
∂
A
y
∂
z
)
x
^
+
(
∂
A
x
∂
z
−
∂
A
z
∂
x
)
y
^
+
(
∂
A
y
∂
x
−
∂
A
x
∂
y
)
z
^
{\displaystyle {\begin{matrix}\left({\partial A_{z} \over \partial y}-{\partial A_{y} \over \partial z}\right)\mathbf {\hat {x}} &+\\\left({\partial A_{x} \over \partial z}-{\partial A_{z} \over \partial x}\right)\mathbf {\hat {y}} &+\\\left({\partial A_{y} \over \partial x}-{\partial A_{x} \over \partial y}\right)\mathbf {\hat {z}} &\ \end{matrix}}}
(
1
ρ
∂
A
z
∂
φ
−
∂
A
φ
∂
z
)
ρ
^
+
(
∂
A
ρ
∂
z
−
∂
A
z
∂
ρ
)
φ
^
+
1
ρ
(
∂
(
ρ
A
φ
)
∂
ρ
−
∂
A
ρ
∂
φ
)
z
^
{\displaystyle {\begin{matrix}\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \varphi }}-{\frac {\partial A_{\varphi }}{\partial z}}\right){\boldsymbol {\hat {\rho }}}&+\\\left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right){\boldsymbol {\hat {\varphi }}}&+\\{\frac {1}{\rho }}\left({\frac {\partial (\rho A_{\varphi })}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \varphi }}\right){\boldsymbol {\hat {z}}}&\ \end{matrix}}}
1
r
sin
θ
(
∂
∂
θ
(
A
φ
sin
θ
)
−
∂
A
θ
∂
φ
)
r
^
+
1
r
(
1
sin
θ
∂
A
r
∂
φ
−
∂
∂
r
(
r
A
φ
)
)
θ
^
+
1
r
(
∂
∂
r
(
r
A
θ
)
−
∂
A
r
∂
θ
)
φ
^
{\displaystyle {\begin{matrix}{1 \over r\sin \theta }\left({\partial \over \partial \theta }\left(A_{\varphi }\sin \theta \right)-{\partial A_{\theta } \over \partial \varphi }\right){\boldsymbol {\hat {r}}}&+\\{1 \over r}\left({1 \over \sin \theta }{\partial A_{r} \over \partial \varphi }-{\partial \over \partial r}\left(rA_{\varphi }\right)\right){\boldsymbol {\hat {\theta }}}&+\\{1 \over r}\left({\partial \over \partial r}\left(rA_{\theta }\right)-{\partial A_{r} \over \partial \theta }\right){\boldsymbol {\hat {\varphi }}}&\ \end{matrix}}}
(
1
σ
2
+
τ
2
∂
A
z
∂
τ
−
∂
A
τ
∂
z
)
σ
^
−
(
1
σ
2
+
τ
2
∂
A
z
∂
σ
−
∂
A
σ
∂
z
)
τ
^
+
1
σ
2
+
τ
2
(
∂
(
s
A
φ
)
∂
s
−
∂
A
s
∂
φ
)
z
^
{\displaystyle {\begin{matrix}\left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\partial A_{z} \over \partial \tau }-{\partial A_{\tau } \over \partial z}\right){\boldsymbol {\hat {\sigma }}}&-\\\left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\partial A_{z} \over \partial \sigma }-{\partial A_{\sigma } \over \partial z}\right){\boldsymbol {\hat {\tau }}}&+\\{\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}\left({\partial \left(sA_{\varphi }\right) \over \partial s}-{\partial A_{s} \over \partial \varphi }\right){\boldsymbol {\hat {z}}}&\ \end{matrix}}}
Оператор Лапласа
Δ
f
=
∇
2
f
{\displaystyle \Delta f=\nabla ^{2}f}
∂
2
f
∂
x
2
+
∂
2
f
∂
y
2
+
∂
2
f
∂
z
2
{\displaystyle {\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}}
1
ρ
∂
∂
ρ
(
ρ
∂
f
∂
ρ
)
+
1
ρ
2
∂
2
f
∂
φ
2
+
∂
2
f
∂
z
2
{\displaystyle {1 \over \rho }{\partial \over \partial \rho }\left(\rho {\partial f \over \partial \rho }\right)+{1 \over \rho ^{2}}{\partial ^{2}f \over \partial \varphi ^{2}}+{\partial ^{2}f \over \partial z^{2}}}
1
r
2
∂
∂
r
(
r
2
∂
f
∂
r
)
+
1
r
2
sin
θ
∂
∂
θ
(
sin
θ
∂
f
∂
θ
)
+
1
r
2
sin
2
θ
∂
2
f
∂
φ
2
{\displaystyle {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 \varphi ^{2}}}
1
σ
2
+
τ
2
(
∂
2
f
∂
σ
2
+
∂
2
f
∂
τ
2
)
+
∂
2
f
∂
z
2
{\displaystyle {\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}}}}
шаблон не поддерживает такой синтаксис векторной функции
Δ
A
=
∇
2
A
{\displaystyle \Delta \mathbf {A} =\nabla ^{2}\mathbf {A} }
Δ
A
x
x
^
+
Δ
A
y
y
^
+
Δ
A
z
z
^
{\displaystyle \Delta A_{x}\mathbf {\hat {x}} +\Delta A_{y}\mathbf {\hat {y}} +\Delta A_{z}\mathbf {\hat {z}} }
(
Δ
A
ρ
−
A
ρ
ρ
2
−
2
ρ
2
∂
A
φ
∂
φ
)
ρ
^
+
(
Δ
A
φ
−
A
φ
ρ
2
+
2
ρ
2
∂
A
ρ
∂
φ
)
φ
^
+
(
Δ
A
z
)
z
^
{\displaystyle {\begin{matrix}\left(\Delta A_{\rho }-{A_{\rho } \over \rho ^{2}}-{2 \over \rho ^{2}}{\partial A_{\varphi } \over \partial \varphi }\right){\boldsymbol {\hat {\rho }}}&+\\\left(\Delta A_{\varphi }-{A_{\varphi } \over \rho ^{2}}+{2 \over \rho ^{2}}{\partial A_{\rho } \over \partial \varphi }\right){\boldsymbol {\hat {\varphi }}}&+\\\left(\Delta A_{z}\right){\boldsymbol {\hat {z}}}&\ \end{matrix}}}
(
Δ
A
r
−
2
A
r
r
2
−
2
r
2
sin
θ
∂
(
A
θ
sin
θ
)
∂
θ
−
2
r
2
sin
θ
∂
A
φ
∂
φ
)
r
^
+
(
Δ
A
θ
−
A
θ
r
2
sin
2
θ
+
2
r
2
∂
A
r
∂
θ
−
2
cos
θ
r
2
sin
2
θ
∂
A
φ
∂
φ
)
θ
^
+
(
Δ
A
φ
−
A
φ
r
2
sin
2
θ
+
2
r
2
sin
θ
∂
A
r
∂
φ
+
2
cos
θ
r
2
sin
2
θ
∂
A
θ
∂
φ
)
φ
^
{\displaystyle {\begin{matrix}\left(\Delta A_{r}-{2A_{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_{\varphi } \over \partial \varphi }\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_{\varphi } \over \partial \varphi }\right){\boldsymbol {\hat {\theta }}}&+\\\left(\Delta A_{\varphi }-{A_{\varphi } \over r^{2}\sin ^{2}\theta }+{2 \over r^{2}\sin \theta }{\partial A_{r} \over \partial \varphi }+{2\cos \theta \over r^{2}\sin ^{2}\theta }{\partial A_{\theta } \over \partial \varphi }\right){\boldsymbol {\hat {\varphi }}}&\end{matrix}}}
?
Элемент длины
d
l
=
d
x
x
^
+
d
y
y
^
+
d
z
z
^
{\displaystyle d\mathbf {l} =dx\mathbf {\hat {x}} +dy\mathbf {\hat {y}} +dz\mathbf {\hat {z}} }
d
l
=
d
ρ
ρ
^
+
ρ
d
φ
φ
^
+
d
z
z
^
{\displaystyle d\mathbf {l} =d\rho {\boldsymbol {\hat {\rho }}}+\rho d\varphi {\boldsymbol {\hat {\varphi }}}+dz{\boldsymbol {\hat {z}}}}
d
l
=
d
r
r
^
+
r
d
θ
θ
^
+
r
sin
θ
d
φ
φ
^
{\displaystyle d\mathbf {l} =dr\mathbf {\hat {r}} +rd\theta {\boldsymbol {\hat {\theta }}}+r\sin \theta d\varphi {\boldsymbol {\hat {\varphi }}}}
d
l
=
σ
2
+
τ
2
d
σ
σ
^
+
σ
2
+
τ
2
d
τ
τ
^
+
d
z
z
^
{\displaystyle 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}}}}
Элемент ориентированной площади
d
S
=
d
y
d
z
x
^
+
d
x
d
z
y
^
+
d
x
d
y
z
^
{\displaystyle {\begin{matrix}d\mathbf {S} =&dy\,dz\,\mathbf {\hat {x}} +\\&dx\,dz\,\mathbf {\hat {y}} +\\&dx\,dy\,\mathbf {\hat {z}} \end{matrix}}}
d
S
=
ρ
d
φ
d
z
ρ
^
+
d
ρ
d
z
φ
^
+
ρ
d
ρ
d
φ
z
^
{\displaystyle {\begin{matrix}d\mathbf {S} =&\rho \,d\varphi \,dz\,{\boldsymbol {\hat {\rho }}}+\\&d\rho \,dz\,{\boldsymbol {\hat {\varphi }}}+\\&\rho \,d\rho d\varphi \,\mathbf {\hat {z}} \end{matrix}}}
d
S
=
r
2
sin
θ
d
θ
d
φ
r
^
+
r
sin
θ
d
r
d
φ
θ
^
+
r
d
r
d
θ
φ
^
{\displaystyle {\begin{matrix}d\mathbf {S} =&r^{2}\sin \theta \,d\theta \,d\varphi \,\mathbf {\hat {r}} +\\&r\sin \theta \,dr\,d\varphi \,{\boldsymbol {\hat {\theta }}}+\\&r\,dr\,d\theta \,{\boldsymbol {\hat {\varphi }}}\end{matrix}}}
d
S
=
σ
2
+
τ
2
,
d
τ
d
z
σ
^
+
σ
2
+
τ
2
d
σ
d
z
τ
^
+
σ
2
+
τ
2
d
σ
,
d
τ
z
^
{\displaystyle {\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}}}
Элемент объёма
d
τ
=
d
x
d
y
d
z
{\displaystyle d\tau =dx\,dy\,dz\,}
d
τ
=
ρ
d
ρ
d
φ
d
z
{\displaystyle d\tau =\rho \,d\rho \,d\varphi \,dz\,}
d
τ
=
r
2
sin
θ
d
r
d
θ
d
φ
{\displaystyle d\tau =r^{2}\sin \theta \,dr\,d\theta \,d\varphi \,}
d
τ
=
(
σ
2
+
τ
2
)
d
σ
d
τ
d
z
,
{\displaystyle d\tau =\left(\sigma ^{2}+\tau ^{2}\right)d\sigma d\tau dz,}
