在圆柱和球坐标系中的del, 下面是常用于正交曲线坐标系, 英语, curvilinear, coordinates, 中的一些向量微积分公式, 目录, 注释, 坐标转换, 单位向量转换, del公式, 非平凡的演算规则, 直角坐标系推导, 圆柱坐标系推导, 球坐标系推导, 单位向量转换公式, 参见, 引用, 外部链接注释, 编辑本文对球坐标使用标准符号iso, 80000, 它取代了iso, 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. 下面是常用于正交曲线坐标系 英语 Curvilinear coordinates 中的一些向量微积分公式 目录 1 注释 2 坐标转换 3 单位向量转换 4 Del公式 4 1 非平凡的演算规则 5 直角坐标系推导 6 圆柱坐标系推导 7 球坐标系推导 8 单位向量转换公式 9 参见 10 引用 11 外部链接注释 编辑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8和f的定义 极角表示为8 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f 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和像的缘故 经典arctan函数的像为 p 2 p 2 而atan2定义的像为 p p 坐标转换 编辑在直角 圆柱和球坐标间的变换 1 从直角 圆柱 球到 直角 x x y y z z displaystyle begin aligned x amp x y amp y z amp z end aligned x r cos f y r sin f z z displaystyle begin aligned x amp rho cos varphi y amp rho sin varphi z amp z end aligned x r sin 8 cos f y r sin 8 sin f z r cos 8 displaystyle begin aligned x amp r sin theta cos varphi y amp r sin theta sin varphi z amp r cos theta end aligned 圆柱 r x 2 y 2 f arctan y x z z displaystyle begin aligned rho amp sqrt x 2 y 2 varphi amp arctan left frac y x right z amp z end aligned r r f f z z displaystyle begin aligned rho amp rho varphi amp varphi z amp z end aligned r r sin 8 f f z r cos 8 displaystyle begin aligned rho amp r sin theta varphi amp varphi z amp r cos theta end aligned 球 r x 2 y 2 z 2 8 arctan x 2 y 2 z f arctan y x displaystyle begin aligned r amp sqrt x 2 y 2 z 2 theta amp arctan left frac sqrt x 2 y 2 z right varphi amp arctan left frac y x right end aligned r r 2 z 2 8 arctan r z f f displaystyle begin aligned r amp sqrt rho 2 z 2 theta amp arctan left frac rho z right varphi amp varphi end aligned r r 8 8 f f displaystyle begin aligned r amp r theta amp theta varphi amp varphi end aligned 单位向量转换 编辑在直角 圆柱和球坐标系间的单位向量转换 从目的坐标的角度 1 直角 圆柱 球直角 不適用 x cos f r sin f f y sin f r cos f f z z displaystyle begin aligned hat mathbf x amp cos varphi hat boldsymbol rho sin varphi hat boldsymbol varphi hat mathbf y amp sin varphi hat boldsymbol rho cos varphi hat boldsymbol varphi hat mathbf z amp hat mathbf z end aligned x sin 8 cos f r cos 8 cos f 8 sin f f y sin 8 sin f r cos 8 sin f 8 cos f f z cos 8 r sin 8 8 displaystyle begin aligned hat mathbf x amp sin theta cos varphi hat mathbf r cos theta cos varphi hat boldsymbol theta sin varphi hat boldsymbol varphi hat mathbf y amp sin theta sin varphi hat mathbf r cos theta sin varphi hat boldsymbol theta cos varphi hat boldsymbol varphi hat mathbf z amp cos theta hat mathbf r sin theta hat boldsymbol theta end aligned 圆柱 r x x y y x 2 y 2 f y x x y x 2 y 2 z z displaystyle begin aligned hat boldsymbol rho amp frac x hat mathbf x y hat mathbf y sqrt x 2 y 2 hat boldsymbol varphi amp frac y hat mathbf x x hat mathbf y sqrt x 2 y 2 hat mathbf z amp hat mathbf z end aligned 不適用 r sin 8 r cos 8 8 f f z cos 8 r sin 8 8 displaystyle begin aligned hat boldsymbol rho amp sin theta hat mathbf r cos theta hat boldsymbol theta hat boldsymbol varphi amp hat boldsymbol varphi hat mathbf z amp cos theta hat mathbf r sin theta hat boldsymbol theta end aligned 球 r x x y y z z x 2 y 2 z 2 8 x x y y z x 2 y 2 z x 2 y 2 z 2 x 2 y 2 f y x x y x 2 y 2 displaystyle begin aligned hat mathbf r amp frac x hat mathbf x y hat mathbf y z hat mathbf z sqrt x 2 y 2 z 2 hat boldsymbol theta amp 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 amp frac y hat mathbf x x hat mathbf y sqrt x 2 y 2 end aligned r r r z z r 2 z 2 8 z r r z r 2 z 2 f f displaystyle begin aligned hat mathbf r amp frac rho hat boldsymbol rho z hat mathbf z sqrt rho 2 z 2 hat boldsymbol theta amp frac z hat boldsymbol rho rho hat mathbf z sqrt rho 2 z 2 hat boldsymbol varphi amp hat boldsymbol varphi end aligned 不適用在直角 圆柱和球坐标系间的单位向量转换 从源坐标的角度 直角 圆柱 球直角 不適用 x x r y f x 2 y 2 y y r x f x 2 y 2 z z displaystyle begin aligned hat mathbf x amp frac x hat boldsymbol rho y hat boldsymbol varphi sqrt x 2 y 2 hat mathbf y amp frac y hat boldsymbol rho x hat boldsymbol varphi sqrt x 2 y 2 hat mathbf z amp hat mathbf z end aligned x x x 2 y 2 r z 8 y x 2 y 2 z 2 f x 2 y 2 x 2 y 2 z 2 y y x 2 y 2 r z 8 x x 2 y 2 z 2 f x 2 y 2 x 2 y 2 z 2 z z r x 2 y 2 8 x 2 y 2 z 2 displaystyle begin aligned hat mathbf x amp frac x left sqrt x 2 y 2 hat mathbf r z hat boldsymbol theta right y sqrt x 2 y 2 z 2 hat boldsymbol varphi sqrt x 2 y 2 sqrt x 2 y 2 z 2 hat mathbf y amp frac y left sqrt x 2 y 2 hat mathbf r z hat boldsymbol theta right x sqrt x 2 y 2 z 2 hat boldsymbol varphi sqrt x 2 y 2 sqrt x 2 y 2 z 2 hat mathbf z amp frac z hat mathbf r sqrt x 2 y 2 hat boldsymbol theta sqrt x 2 y 2 z 2 end aligned 圆柱 r cos f x sin f y f sin f x cos f y z z displaystyle begin aligned hat boldsymbol rho amp cos varphi hat mathbf x sin varphi hat mathbf y hat boldsymbol varphi amp sin varphi hat mathbf x cos varphi hat mathbf y hat mathbf z amp hat mathbf z end aligned 不適用 r r r z 8 r 2 z 2 f f z z r r 8 r 2 z 2 displaystyle begin aligned hat boldsymbol rho amp frac rho hat mathbf r z hat boldsymbol theta sqrt rho 2 z 2 hat boldsymbol varphi amp hat boldsymbol varphi hat mathbf z amp frac z hat mathbf r rho hat boldsymbol theta sqrt rho 2 z 2 end aligned 球 r sin 8 cos f x sin f y cos 8 z 8 cos 8 cos f x sin f y sin 8 z f sin f x cos f y displaystyle begin aligned hat mathbf r amp sin theta left cos varphi hat mathbf x sin varphi hat mathbf y right cos theta hat mathbf z hat boldsymbol theta amp cos theta left cos varphi hat mathbf x sin varphi hat mathbf y right sin theta hat mathbf z hat boldsymbol varphi amp sin varphi hat mathbf x cos varphi hat mathbf y end aligned r sin 8 r cos 8 z 8 cos 8 r sin 8 z f f displaystyle begin aligned hat mathbf r amp sin theta hat boldsymbol rho cos theta hat mathbf z hat boldsymbol theta amp cos theta hat boldsymbol rho sin theta hat mathbf z hat boldsymbol varphi amp hat boldsymbol varphi end aligned 不適用Del公式 编辑在直角 圆柱和球坐标下的del算子的表格 运算 直角坐标 x y z 圆柱坐标 r f z 球坐标 r 8 f 这里的8是极角而f 是方位角a向量场 A A x x A y y A z z displaystyle A x hat mathbf x A y hat mathbf y A z hat mathbf z A r r A f f A z z displaystyle A rho hat boldsymbol rho A varphi hat boldsymbol varphi A z hat mathbf z A r r A 8 8 A f f displaystyle A r hat mathbf r A theta hat boldsymbol theta A varphi hat boldsymbol varphi 梯度 f 1 f x x f y y f z z displaystyle partial f over partial x hat mathbf x partial f over partial y hat mathbf y partial f over partial z hat mathbf z f r r 1 r f f f f z z displaystyle partial f over partial rho hat boldsymbol rho 1 over rho partial f over partial varphi hat boldsymbol varphi partial f over partial z hat mathbf z f r r 1 r f 8 8 1 r sin 8 f f f displaystyle partial f over partial r hat mathbf r 1 over r partial f over partial theta hat boldsymbol theta 1 over r sin theta partial f over partial varphi hat boldsymbol varphi 散度 A 1 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 r r A r r 1 r A f f 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 8 8 A 8 sin 8 1 r sin 8 A f f 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 旋度 A 1 A z y A y z x A x z A z x y A y x A x y z displaystyle begin aligned left frac partial A z partial y frac partial A y partial z right amp hat mathbf x left frac partial A x partial z frac partial A z partial x right amp hat mathbf y left frac partial A y partial x frac partial A x partial y right amp hat mathbf z end aligned 1 r A z f A f z r A r z A z r f 1 r r A f r A r f z displaystyle begin aligned left frac 1 rho frac partial A z partial varphi frac partial A varphi partial z right amp hat boldsymbol rho left frac partial A rho partial z frac partial A z partial rho right amp hat boldsymbol varphi frac 1 rho left frac partial left rho A varphi right partial rho frac partial A rho partial varphi right amp hat mathbf z end aligned 1 r sin 8 8 A f sin 8 A 8 f r 1 r 1 sin 8 A r f r r A f 8 1 r r r A 8 A r 8 f displaystyle begin aligned frac 1 r sin theta left frac partial partial theta left A varphi sin theta right frac partial A theta partial varphi right amp hat mathbf r frac 1 r left frac 1 sin theta frac partial A r partial varphi frac partial partial r left rA varphi right right amp hat boldsymbol theta frac 1 r left frac partial partial r left rA theta right frac partial A r partial theta right amp hat boldsymbol varphi end aligned 拉普拉斯算子 2f f 1 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 r r r f r 1 r 2 2 f f 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 8 8 sin 8 f 8 1 r 2 sin 2 8 2 f f 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 向量拉普拉斯算子 2A A 2 A x x 2 A y y 2 A z z displaystyle nabla 2 A x hat mathbf x nabla 2 A y hat mathbf y nabla 2 A z hat mathbf z 点击查看 显示 2 A r A r r 2 2 r 2 A f f r 2 A f A f r 2 2 r 2 A r f f 2 A z z displaystyle begin aligned mathopen left nabla 2 A rho frac A rho rho 2 frac 2 rho 2 frac partial A varphi partial varphi right mathclose amp hat boldsymbol rho mathopen left nabla 2 A varphi frac A varphi rho 2 frac 2 rho 2 frac partial A rho partial varphi right mathclose amp hat boldsymbol varphi nabla 2 A z amp hat mathbf z end aligned 点击查看 显示 2 A r 2 A r r 2 2 r 2 sin 8 A 8 sin 8 8 2 r 2 sin 8 A f f r 2 A 8 A 8 r 2 sin 2 8 2 r 2 A r 8 2 cos 8 r 2 sin 2 8 A f f 8 2 A f A f r 2 sin 2 8 2 r 2 sin 8 A r f 2 cos 8 r 2 sin 2 8 A 8 f f displaystyle begin aligned left nabla 2 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 varphi partial varphi right amp hat mathbf r left nabla 2 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 varphi partial varphi right amp hat boldsymbol theta left nabla 2 A varphi frac A varphi r 2 sin 2 theta frac 2 r 2 sin theta frac partial A r partial varphi frac 2 cos theta r 2 sin 2 theta frac partial A theta partial varphi right amp hat boldsymbol varphi end aligned 物质导数a 2 A B A B x x A B y y A B z z displaystyle mathbf A cdot nabla B x hat mathbf x mathbf A cdot nabla B y hat mathbf y mathbf A cdot nabla B z hat mathbf z A r B r r A f r B r f A z B r z A f B f r r A r B f r A f r B f f A z B f z A f B r r f A r B z r A f r B z f A z B z z z displaystyle begin aligned left A rho frac partial B rho partial rho frac A varphi rho frac partial B rho partial varphi A z frac partial B rho partial z frac A varphi B varphi rho right amp hat boldsymbol rho left A rho frac partial B varphi partial rho frac A varphi rho frac partial B varphi partial varphi A z frac partial B varphi partial z frac A varphi B rho rho right amp hat boldsymbol varphi left A rho frac partial B z partial rho frac A varphi rho frac partial B z partial varphi A z frac partial B z partial z right amp hat mathbf z end aligned 点击查看 显示 A r B r r A 8 r B r 8 A f r sin 8 B r f A 8 B 8 A f B f r r A r B 8 r A 8 r B 8 8 A f r sin 8 B 8 f A 8 B r r A f B f cot 8 r 8 A r B f r A 8 r B f 8 A f r sin 8 B f f A f B r r A f B 8 cot 8 r f displaystyle begin aligned left A r frac partial B r partial r frac A theta r frac partial B r partial theta frac A varphi r sin theta frac partial B r partial varphi frac A theta B theta A varphi B varphi r right amp hat mathbf r left A r frac partial B theta partial r frac A theta r frac partial B theta partial theta frac A varphi r sin theta frac partial B theta partial varphi frac A theta B r r frac A varphi B varphi cot theta r right amp hat boldsymbol theta left A r frac partial B varphi partial r frac A theta r frac partial B varphi partial theta frac A varphi r sin theta frac partial B varphi partial varphi frac A varphi B r r frac A varphi B theta cot theta r right amp hat boldsymbol varphi end aligned 张量散度 T 点击查看 显示 T x x x T y x y T z x z x T x y x T y y y T z y z y T x z x T y z y T z z z z displaystyle begin aligned left frac partial T xx partial x frac partial T yx partial y frac partial T zx partial z right amp hat mathbf x left frac partial T xy partial x frac partial T yy partial y frac partial T zy partial z right amp hat mathbf y left frac partial T xz partial x frac partial T yz partial y frac partial T zz partial z right amp hat mathbf z end aligned 点击查看 显示 T r r r 1 r T f r f T z r z 1 r T r r T f f r T r f r 1 r T f f f T z f z 1 r T r f T f r f T r z r 1 r T f z f T z z z T r z r z displaystyle begin aligned left frac partial T rho rho partial rho frac 1 rho frac partial T varphi rho partial varphi frac partial T z rho partial z frac 1 rho T rho rho T varphi varphi right amp hat boldsymbol rho left frac partial T rho varphi partial rho frac 1 rho frac partial T varphi varphi partial varphi frac partial T z varphi partial z frac 1 rho T rho varphi T varphi rho right amp hat boldsymbol varphi left frac partial T rho z partial rho frac 1 rho frac partial T varphi z partial varphi frac partial T zz partial z frac T rho z rho right amp hat mathbf z end aligned 点击查看 显示 T r r r 2 T r r r 1 r T 8 r 8 cot 8 r T 8 r 1 r sin 8 T f r f 1 r T 8 8 T f f r T r 8 r 2 T r 8 r 1 r T 8 8 8 cot 8 r T 8 8 1 r sin 8 T f 8 f T 8 r r cot 8 r T f f 8 T r f r 2 T r f r 1 r T 8 f 8 1 r sin 8 T f f f T f r r cot 8 r T 8 f T f 8 f displaystyle begin aligned left frac partial T rr partial r 2 frac T rr r frac 1 r frac partial T theta r partial theta frac cot theta r T theta r frac 1 r sin theta frac partial T varphi r partial varphi frac 1 r T theta theta T varphi varphi right amp hat mathbf r left frac partial T r theta partial r 2 frac T r theta r frac 1 r frac partial T theta theta partial theta frac cot theta r T theta theta frac 1 r sin theta frac partial T varphi theta partial varphi frac T theta r r frac cot theta r T varphi varphi right amp hat boldsymbol theta left frac partial T r varphi partial r 2 frac T r varphi r frac 1 r frac partial T theta varphi partial theta frac 1 r sin theta frac partial T varphi varphi partial varphi frac T varphi r r frac cot theta r T theta varphi T varphi theta right amp hat boldsymbol varphi end aligned 微分位移 dℓ 1 d x x d y y d z z displaystyle dx hat mathbf x dy hat mathbf y dz hat mathbf z d r r r d f f d z z displaystyle d rho hat boldsymbol rho rho d varphi hat boldsymbol varphi dz hat mathbf z d r r r d 8 8 r sin 8 d f f displaystyle dr hat mathbf r r d theta hat boldsymbol theta r sin theta d varphi hat boldsymbol varphi 微分正规面积 dS d y d z x d x d z y d x d y z displaystyle begin aligned dy dz amp hat mathbf x dx dz amp hat mathbf y dx dy amp hat mathbf z end aligned r d f d z r d r d z f r d r d f z displaystyle begin aligned rho d varphi dz amp hat boldsymbol rho d rho dz amp hat boldsymbol varphi rho d rho d varphi amp hat mathbf z end aligned r 2 sin 8 d 8 d f r r sin 8 d r d f 8 r d r d 8 f displaystyle begin aligned r 2 sin theta d theta d varphi amp hat mathbf r r sin theta dr d varphi amp hat boldsymbol theta r dr d theta amp hat boldsymbol varphi end aligned 微分体积 dV 1 d x d y d z displaystyle dx dy dz r d r d f d z displaystyle rho d rho d varphi dz r 2 sin 8 d r d 8 d f displaystyle r 2 sin theta dr d theta d varphi a 本页对极角采用8 displaystyle theta 对方位角采用f displaystyle varphi 这是在物理学中常用的符号 某些来源在这些公式中对方位角采用8 displaystyle theta 对极角采用f displaystyle varphi 这是常用数学符号 如果需要这种数学公式 可对换上表公式中的8 displaystyle theta 和f displaystyle varphi 非平凡的演算规则 编辑 div grad f f 2 f displaystyle operatorname div operatorname grad f equiv nabla cdot nabla f equiv nabla 2 f curl grad f f 0 displaystyle operatorname curl operatorname grad f equiv nabla times nabla f mathbf 0 div curl A A 0 displaystyle operatorname div operatorname curl mathbf A equiv nabla cdot nabla times mathbf A 0 curl curl A A A 2 A 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 2 f g f 2 g 2 f g g 2 f 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 编辑 div A lim V 0 V A d S V d V A x x d x d y d z A x x d y d z A y y d y d x d z A y y d x d z A z z d z d x d y A z z d x d y d x d y d z A x x A y y A z z displaystyle begin aligned operatorname div mathbf A lim V to 0 frac iint partial V mathbf A cdot d mathbf S iiint V dV amp 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 amp frac partial A x partial x frac partial A y partial y frac partial A z partial z end aligned curl A x lim S x 0 S A d ℓ S d S A z y d y d z A z y d z A y z d y A y z d z d y d y d z A z y A y z 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 amp frac A z y dy dz A z y dz A y z dy A y z dz dy dydz amp frac partial A z partial y frac partial A y partial z end aligned curl A y displaystyle operatorname curl mathbf A y 和 curl A z displaystyle operatorname curl mathbf A z 的表达式可以同理得出 註 第一式中的A x x d x displaystyle A x x dx 是A x displaystyle A x 在x d x displaystyle x dx 時的量值 並非A x displaystyle A x 值乘上x d x displaystyle x dx 以下圓柱座標 球座標的推導中亦然 圆柱坐标系推导 编辑 div A lim V 0 V A d S V d V A r r d r r d r d ϕ d z A r r r d ϕ d z A ϕ ϕ d ϕ d r d z A ϕ ϕ d r d z A z z d z d r r d r 2 d ϕ A z z d r r d r 2 d ϕ r d ϕ d r d z 1 r r A r r 1 r A ϕ ϕ A z z displaystyle begin aligned operatorname div mathbf A amp lim V to 0 frac iint partial V mathbf A cdot d mathbf S iiint V dV amp 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 amp 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 curl A r lim S r 0 S A d ℓ S d S A ϕ z r d r d ϕ A ϕ z d z r d r d ϕ A z ϕ d ϕ d z A z ϕ d z r d r d ϕ d z A ϕ z 1 r A z ϕ displaystyle begin aligned operatorname curl mathbf A rho amp lim S perp boldsymbol hat rho to 0 frac int partial S mathbf A cdot d mathbf ell iint S dS amp 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 amp frac partial A phi partial z frac 1 rho frac partial A z partial phi end aligned curl A ϕ lim S ϕ 0 S A d ℓ S d S A z r d z A z r d r d z A r z d z d r A r z d r d r d z A z r A r z displaystyle begin aligned operatorname curl mathbf A phi amp lim S perp boldsymbol hat phi to 0 frac int partial S mathbf A cdot d mathbf ell iint S dS amp 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 amp frac partial A z partial rho frac partial A rho partial z end aligned curl A z lim S z 0 S A d ℓ S d S A r ϕ d r A r ϕ d ϕ d r A ϕ r d r r d r d ϕ A ϕ r r d ϕ r d r d ϕ 1 r A r ϕ 1 r r A ϕ r displaystyle begin aligned operatorname curl mathbf A z amp lim S perp boldsymbol hat z to 0 frac int partial S mathbf A cdot d mathbf ell iint S dS amp 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 amp frac 1 rho frac partial A rho partial phi frac 1 rho frac partial rho A phi partial rho end aligned curl A curl A r r curl A ϕ ϕ curl A z z 1 r A z ϕ A ϕ z r A r z A z r ϕ 1 r r A ϕ r A r ϕ z displaystyle begin aligned operatorname curl mathbf A amp operatorname curl mathbf A rho hat boldsymbol rho operatorname curl mathbf A phi hat boldsymbol phi operatorname curl mathbf A z hat boldsymbol z amp 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 编辑 div A lim V 0 V A d S V d V A r r d r r d r d 8 r d r sin 8 d ϕ A r r r d 8 r sin 8 d ϕ A 8 8 d 8 sin 8 d 8 r d r d ϕ A 8 8 sin 8 r d r d ϕ A ϕ ϕ d ϕ r d r 2 d r d 8 A ϕ ϕ r d r 2 d r d 8 d r r d 8 r sin 8 d ϕ 1 r 2 r 2 A r r 1 r sin 8 A 8 sin 8 8 1 r sin 8 A ϕ ϕ displaystyle begin aligned operatorname div mathbf A amp lim V to 0 frac iint partial V mathbf A cdot d mathbf S iiint V dV amp 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 amp 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 curl A r lim S r 0 S A d ℓ S d S A 8 ϕ r d 8 A ϕ 8 d 8 r sin 8 d 8 d ϕ A 8 ϕ d ϕ r d 8 A ϕ 8 r sin 8 d ϕ r d 8 r sin 8 d ϕ 1 r sin 8 A ϕ sin 8 8 1 r sin 8 A 8 ϕ 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 amp 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 amp 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 curl A 8 lim S 8 0 S A d ℓ S d S A ϕ r r sin 8 d ϕ A r ϕ d ϕ d r A ϕ r d r r d r sin 8 d ϕ A r ϕ d r d r r sin 8 d ϕ 1 r sin 8 A r ϕ 1 r r A ϕ r 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 amp 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 amp frac 1 r sin theta frac partial A r partial phi frac 1 r frac partial rA phi partial r end aligned curl A ϕ lim S ϕ 0 S A d ℓ S d S A r 8 d r A 8 r d r r d r d 8 A r 8 d 8 d r A 8 r r d 8 r d r 2 d r d 8 1 r r A 8 r 1 r A r 8 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 amp 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 amp frac 1 r frac partial rA theta partial r frac 1 r frac partial A r partial theta end aligned curl A curl A r r curl A 8 8 curl A ϕ ϕ 1 r sin 8 A ϕ sin 8 8 A 8 ϕ r 1 r 1 sin 8 A r ϕ r A ϕ r 8 1 r r A 8 r A r 8 ϕ 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 编辑坐标参数u的单位向量以如下方式定义 u的小的正值改变导致位置向量r displaystyle boldsymbol vec r 在u displaystyle boldsymbol hat u 方向上的改变 因此 r u s u u displaystyle partial boldsymbol vec r over partial u partial s over partial u boldsymbol hat u 这里的s是弧长参数 对于两组坐标系u i displaystyle u i 和v j displaystyle v j 依据链式法则 d r i r u i d u i i s u i u i d u i j s v j v j d v j j s v j v j i v j u i d u i i j s v j v j u i v j d u i 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 img, 维基百科,wiki,书籍,书籍,图书馆,