在圆柱和球坐标系中的del

注释

• 本文对球坐标使用标准符号ISO 80000-2，它取代了ISO 31-11，（部分其他来源可能有着颠倒θ和φ的定义）：
• 极角表示为θ：它是在z轴与连接原点和目标点的径向向量之间的角度。
• 方位角表示为φ：它是在x轴与径向向量在xy面上的投影之间的角度。
• 函数atan2(y, x)可以用于替代数学函数arctan(y/x)。这是由于它的定义域的缘故，经典arctan函数的像为(−π/2, +π/2)，而atan2定义的像为(−π, π]

坐标转换

{\displaystyle {\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=\arctan \left({\frac {\sqrt {x^{2}+y^{2}}}{z}}\right)\\\varphi &=\arctan \left({\frac {y}{x}}\right)\end{aligned}}} {\displaystyle {\begin{aligned}r&={\sqrt {\rho ^{2}+z^{2}}}\\\theta &=\arctan {\left({\frac {\rho }{z}}\right)}\\\varphi &=\varphi \end{aligned}}} {\displaystyle {\begin{aligned}r&=r\\\theta &=\theta \\\varphi &=\varphi \end{aligned}}}

单位向量转换

{\displaystyle {\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 {\left(x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}\right)z-\left(x^{2}+y^{2}\right){\hat {\mathbf {z} }}}{{\sqrt {x^{2}+y^{2}+z^{2}}}{\sqrt {x^{2}+y^{2}}}}}\\{\hat {\boldsymbol {\varphi }}}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\end{aligned}}} {\displaystyle {\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 {\varphi }}}&={\hat {\boldsymbol {\varphi }}}\end{aligned}}} 不適用

{\displaystyle {\begin{aligned}{\hat {\mathbf {r} }}&=\sin \theta \left(\cos \varphi {\hat {\mathbf {x} }}+\sin \varphi {\hat {\mathbf {y} }}\right)+\cos \theta {\hat {\mathbf {z} }}\\{\hat {\boldsymbol {\theta }}}&=\cos \theta \left(\cos \varphi {\hat {\mathbf {x} }}+\sin \varphi {\hat {\mathbf {y} }}\right)-\sin \theta {\hat {\mathbf {z} }}\\{\hat {\boldsymbol {\varphi }}}&=-\sin \varphi {\hat {\mathbf {x} }}+\cos \varphi {\hat {\mathbf {y} }}\end{aligned}}} {\displaystyle {\begin{aligned}{\hat {\mathbf {r} }}&=\sin \theta {\hat {\boldsymbol {\rho }}}+\cos \theta {\hat {\mathbf {z} }}\\{\hat {\boldsymbol {\theta }}}&=\cos \theta {\hat {\boldsymbol {\rho }}}-\sin \theta {\hat {\mathbf {z} }}\\{\hat {\boldsymbol {\varphi }}}&={\hat {\boldsymbol {\varphi }}}\end{aligned}}} 不適用

Del公式

非平凡的演算规则

1. ${\displaystyle \operatorname {div} \,\operatorname {grad} f\equiv \nabla \cdot \nabla f\equiv \nabla ^{2}f}$
2. ${\displaystyle \operatorname {curl} \,\operatorname {grad} f\equiv \nabla \times \nabla f=\mathbf {0} }$
3. ${\displaystyle \operatorname {div} \,\operatorname {curl} \mathbf {A} \equiv \nabla \cdot (\nabla \times \mathbf {A} )=0}$
4. ${\displaystyle \operatorname {curl} \,\operatorname {curl} \mathbf {A} \equiv \nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A} }$（del的拉格朗日公式
5. ${\displaystyle \nabla ^{2}(fg)=f\nabla ^{2}g+2\nabla f\cdot \nabla g+g\nabla ^{2}f}$

直角坐标系推导

{\displaystyle {\begin{aligned}\operatorname {div} \mathbf {A} =\lim _{V\to 0}{\frac {\iint _{\partial V}\mathbf {A} \cdot d\mathbf {S} }{\iiint _{V}dV}}&={\frac {A_{x}(x+dx)dydz-A_{x}(x)dydz+A_{y}(y+dy)dxdz-A_{y}(y)dxdz+A_{z}(z+dz)dxdy-A_{z}(z)dxdy}{dxdydz}}\\&={\frac {\partial A_{x}}{\partial x}}+{\frac {\partial A_{y}}{\partial y}}+{\frac {\partial A_{z}}{\partial z}}\end{aligned}}}

{\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{x}=\lim _{S^{\perp \mathbf {\hat {x}} }\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}&={\frac {A_{z}(y+dy)dz-A_{z}(y)dz+A_{y}(z)dy-A_{y}(z+dz)dy}{dydz}}\\&={\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}\end{aligned}}}

${\displaystyle (\operatorname {curl} \mathbf {A} )_{y}}$${\displaystyle (\operatorname {curl} \mathbf {A} )_{z}}$的表达式可以同理得出。

圆柱坐标系推导

{\displaystyle {\begin{aligned}\operatorname {div} \mathbf {A} &=\lim _{V\to 0}{\frac {\iint _{\partial V}\mathbf {A} \cdot d\mathbf {S} }{\iiint _{V}dV}}\\&={\frac {A_{\rho }(\rho +d\rho )(\rho +d\rho )d\phi dz-A_{\rho }(\rho )\rho d\phi dz+A_{\phi }(\phi +d\phi )d\rho dz-A_{\phi }(\phi )d\rho dz+A_{z}(z+dz)d\rho (\rho +d\rho /2)d\phi -A_{z}(z)d\rho (\rho +d\rho /2)d\phi }{\rho d\phi d\rho dz}}\\&={\frac {1}{\rho }}{\frac {\partial (\rho A_{\rho })}{\partial \rho }}+{\frac {1}{\rho }}{\frac {\partial A_{\phi }}{\partial \phi }}+{\frac {\partial A_{z}}{\partial z}}\end{aligned}}}
{\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{\rho }&=\lim _{S^{\perp {\boldsymbol {\hat {\rho }}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}\\&={\frac {A_{\phi }(z)(\rho +d\rho )d\phi -A_{\phi }(z+dz)(\rho +d\rho )d\phi +A_{z}(\phi +d\phi )dz-A_{z}(\phi )dz}{(\rho +d\rho )d\phi dz}}\\&=-{\frac {\partial A_{\phi }}{\partial z}}+{\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \phi }}\end{aligned}}}
{\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{\phi }&=\lim _{S^{\perp {\boldsymbol {\hat {\phi }}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}\\&={\frac {A_{z}(\rho )dz-A_{z}(\rho +d\rho )dz+A_{\rho }(z+dz)d\rho -A_{\rho }(z)d\rho }{d\rho dz}}\\&=-{\frac {\partial A_{z}}{\partial \rho }}+{\frac {\partial A_{\rho }}{\partial z}}\end{aligned}}}
{\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{z}&=\lim _{S^{\perp {\boldsymbol {\hat {z}}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}\\&={\frac {A_{\rho }(\phi )d\rho -A_{\rho }(\phi +d\phi )d\rho +A_{\phi }(\rho +d\rho )(\rho +d\rho )d\phi -A_{\phi }(\rho )\rho d\phi }{\rho d\rho d\phi }}\\&=-{\frac {1}{\rho }}{\frac {\partial A_{\rho }}{\partial \phi }}+{\frac {1}{\rho }}{\frac {\partial (\rho A_{\phi })}{\partial \rho }}\end{aligned}}}
{\displaystyle {\begin{aligned}\operatorname {curl} \mathbf {A} &=(\operatorname {curl} \mathbf {A} )_{\rho }{\hat {\boldsymbol {\rho }}}+(\operatorname {curl} \mathbf {A} )_{\phi }{\hat {\boldsymbol {\phi }}}+(\operatorname {curl} \mathbf {A} )_{z}{\hat {\boldsymbol {z}}}\\&=\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 (\rho A_{\phi })}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \phi }}\right){\hat {\boldsymbol {z}}}\end{aligned}}}

球坐标系推导

{\displaystyle {\begin{aligned}\operatorname {div} \mathbf {A} &=\lim _{V\to 0}{\frac {\iint _{\partial V}\mathbf {A} \cdot d\mathbf {S} }{\iiint _{V}dV}}\\&={\frac {A_{r}(r+dr)(r+dr)d\theta \,(r+dr)\sin \theta d\phi -A_{r}(r)rd\theta \,r\sin \theta d\phi +A_{\theta }(\theta +d\theta )\sin(\theta +d\theta )\,rdrd\phi -A_{\theta }(\theta )\sin(\theta )\,rdrd\phi +A_{\phi }(\phi +d\phi )(r+dr/2)drd\theta -A_{\phi }(\phi )(r+dr/2)drd\theta }{dr\,rd\theta \,r\sin \theta d\phi }}\\&={\frac {1}{r^{2}}}{\frac {\partial (r^{2}A_{r})}{\partial r}}+{\frac {1}{r\sin \theta }}{\frac {\partial (A_{\theta }\sin \theta )}{\partial \theta }}+{\frac {1}{r\sin \theta }}{\frac {\partial A_{\phi }}{\partial \phi }}\end{aligned}}}

{\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{r}=\lim _{S^{\perp {\boldsymbol {\hat {r}}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}&={\frac {A_{\theta }(\phi )\,rd\theta +A_{\phi }(\theta +d\theta )\,r\sin(\theta +d\theta )d\phi -A_{\theta }(\phi +d\phi )\,rd\theta -A_{\phi }(\theta )\,r\sin(\theta )d\phi }{rd\theta \,r\sin \theta d\phi }}\\&={\frac {1}{r\sin \theta }}{\frac {\partial (A_{\phi }\sin \theta )}{\partial \theta }}-{\frac {1}{r\sin \theta }}{\frac {\partial A_{\theta }}{\partial \phi }}\end{aligned}}}

{\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{\theta }=\lim _{S^{\perp {\boldsymbol {\hat {\theta }}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}&={\frac {A_{\phi }(r)\,r\sin \theta d\phi +A_{r}(\phi +d\phi )dr-A_{\phi }(r+dr)(r+dr)\sin \theta d\phi -A_{r}(\phi )dr}{dr\,r\sin \theta d\phi }}\\&={\frac {1}{r\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}-{\frac {1}{r}}{\frac {\partial (rA_{\phi })}{\partial r}}\end{aligned}}}

{\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{\phi }=\lim _{S^{\perp {\boldsymbol {\hat {\phi }}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}&={\frac {A_{r}(\theta )dr+A_{\theta }(r+dr)(r+dr)d\theta -A_{r}(\theta +d\theta )dr-A_{\theta }(r)\,rd\theta }{(r+dr/2)drd\theta }}\\&={\frac {1}{r}}{\frac {\partial (rA_{\theta })}{\partial r}}-{\frac {1}{r}}{\frac {\partial A_{r}}{\partial \theta }}\end{aligned}}}

${\displaystyle \operatorname {curl} \mathbf {A} =(\operatorname {curl} \mathbf {A} )_{r}\,{\hat {\boldsymbol {r}}}+(\operatorname {curl} \mathbf {A} )_{\theta }\,{\hat {\boldsymbol {\theta }}}+(\operatorname {curl} \mathbf {A} )_{\phi }\,{\hat {\boldsymbol {\phi }}}={\frac {1}{r\sin \theta }}\left({\frac {\partial (A_{\phi }\sin \theta )}{\partial \theta }}-{\frac {\partial A_{\theta }}{\partial \phi }}\right){\hat {\boldsymbol {r}}}+{\frac {1}{r}}\left({\frac {1}{\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}-{\frac {\partial (rA_{\phi })}{\partial r}}\right){\hat {\boldsymbol {\theta }}}+{\frac {1}{r}}\left({\frac {\partial (rA_{\theta })}{\partial r}}-{\frac {\partial A_{r}}{\partial \theta }}\right){\hat {\boldsymbol {\phi }}}}$

单位向量转换公式

${\displaystyle {\partial {\boldsymbol {\vec {r}}} \over \partial u}={\partial {s} \over \partial u}{\boldsymbol {\hat {u}}}}$

${\displaystyle d{\boldsymbol {\vec {r}}}=\sum _{i}{\partial {\boldsymbol {\vec {r}}} \over \partial u_{i}}du_{i}=\sum _{i}{\partial {s} \over \partial u_{i}}{\boldsymbol {\hat {u_{i}}}}du_{i}=\sum _{j}{\partial {s} \over \partial v_{j}}{\boldsymbol {\hat {v_{j}}}}dv_{j}=\sum _{j}{\partial {s} \over \partial v_{j}}{\boldsymbol {\hat {v_{j}}}}\sum _{i}{\partial {v_{j}} \over \partial u_{i}}du_{i}=\sum _{i}\sum _{j}{\partial {s} \over \partial v_{j}}{\partial {v_{j}} \over \partial u_{i}}{\boldsymbol {\hat {v_{j}}}}du_{i}}$

${\displaystyle {\partial {s} \over \partial u_{i}}{\boldsymbol {\hat {u_{i}}}}=\sum _{j}{\partial {s} \over \partial v_{j}}{\partial {v_{j}} \over \partial u_{i}}{\boldsymbol {\hat {v_{j}}}}}$

引用

1. Griffiths, David J. Introduction to Electrodynamics. Pearson. 2012. ISBN 978-0-321-85656-2.
2. ^ Weisstein, Eric W. Convective Operator. Mathworld. [23 March 2011]. （原始内容存档于2016-03-03）.