Оператор Прямоугольные координаты (x, y, z ) Цилиндрические координаты (ρ, φ, z ) Сферические координаты (r , θ, φ ) Параболические 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}}} . 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}}}} Векторный оператор Лапласа Δ A {\displaystyle \Delta \mathbf {A} } Δ A x x ^ + Δ A y y ^ + Δ A z z ^ = ( ∂ 2 A x ∂ x 2 + ∂ 2 A x ∂ y 2 + ∂ 2 A x ∂ z 2 ) x ^ + ( ∂ 2 A y ∂ x 2 + ∂ 2 A y ∂ y 2 + ∂ 2 A y ∂ z 2 ) y ^ + ( ∂ 2 A z ∂ x 2 + ∂ 2 A z ∂ y 2 + ∂ 2 A z ∂ z 2 ) z ^ {\displaystyle {\begin{matrix}\Delta A_{x}\mathbf {\hat {x}} +\Delta A_{y}\mathbf {\hat {y}} +\Delta A_{z}\mathbf {\hat {z}} =\\{\biggl (}{\partial ^{2}A_{x} \over \partial x^{2}}+{\partial ^{2}A_{x} \over \partial y^{2}}+{\partial ^{2}A_{x} \over \partial z^{2}}{\biggr )}\mathbf {\hat {x}} +\\{\biggl (}{\partial ^{2}A_{y} \over \partial x^{2}}+{\partial ^{2}A_{y} \over \partial y^{2}}+{\partial ^{2}A_{y} \over \partial z^{2}}{\biggr )}\mathbf {\hat {y}} +\\{\biggl (}{\partial ^{2}A_{z} \over \partial x^{2}}+{\partial ^{2}A_{z} \over \partial y^{2}}+{\partial ^{2}A_{z} \over \partial z^{2}}{\biggr )}\mathbf {\hat {z}} \ \end{matrix}}} ( 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 V = d x d y d z {\displaystyle dV=dx\,dy\,dz} d V = ρ d ρ d φ d z {\displaystyle dV=\rho \,d\rho \,d\varphi \,dz} d V = r 2 sin θ d r d θ d φ {\displaystyle dV=r^{2}\sin \theta \,dr\,d\theta \,d\varphi } d V = ( σ 2 + τ 2 ) d σ d τ d z {\displaystyle dV=\left(\sigma ^{2}+\tau ^{2}\right)d\sigma d\tau dz}