下面是常用於正交曲線坐標系中的一些向量微積分公式。



注釋


* 本文對球坐標使用標準符號 ISO 80000-2，它取代了 ISO 31-11，（部分其他來源可能有著顛倒 θ 和 φ 的定義）：
  
* 極角表示為 θ：它是在 z 軸與連接原點和目標點的徑向向量之間的角度。
  
* 方位角表示為 φ：它是在 x 軸與徑向向量在 xy 面上的投影之間的角度。

* 函數 atan2(y, x) 可以用於替代數學函數 arctan(y/x)。這是由於它的定義域和像的緣故，經典 arctan 函數的像為 (−π/2, +π/2)，而 atan2 定義的像為 (−π, π]。



坐標轉換



單位向量轉換



Del 公式

　
: ^α 本頁對極角採用 $\theta $ 對方位角採用 $\varphi $ ，這是在物理學中常用的符號。某些來源在這些公式中對方位角採用 $\theta $ 對極角採用 $\varphi $ ，這是常用數學符號，如果需要這種數學公式，可對換上表公式中的 $\theta $ 和 $\varphi $ 。

非平凡的演算規則

 1. $\operatorname {div} \,\operatorname {grad} f\equiv \nabla \cdot \nabla f\equiv \nabla ^{2}f$
2. $\operatorname {curl} \,\operatorname {grad} f\equiv \nabla \times \nabla f=\mathbf {0} $
3. $\operatorname {div} \,\operatorname {curl} \mathbf {A} \equiv \nabla \cdot (\nabla \times \mathbf {A} )=0$
4. $\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. $\nabla ^{2}(fg)=f\nabla ^{2}g+2\nabla f\cdot \nabla g+g\nabla ^{2}f$



直角坐標系推導

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

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

 $(\operatorname {curl} \mathbf {A} )_{y}$ 和 $(\operatorname {curl} \mathbf {A} )_{z}$ 的表達式可以同理得出。 

註：第一式中的 $A_{x}(x+dx)$ 是 $A_{x}$ 在 $x+dx$ 時的量值，並非 $A_{x}$ 值乘上 $x+dx$ 。以下圓柱座標、球座標的推導中亦然。



圓柱坐標系推導

　
: ${\begin{aligned}\operatorname {div} \mathbf {A} &=\lim _{V\to 0}{\frac {\iint _{\partial V}\mathbf {A} \cdot d\mathbf {S} }{\iiint _{V}dV}}\\&={\frac {A_{\rho }(\rho +d\rho )(\rho +d\rho )d\phi dz-A_{\rho }(\rho )\rho d\phi dz+A_{\phi }(\phi +d\phi )d\rho dz-A_{\phi }(\phi )d\rho dz+A_{z}(z+dz)d\rho (\rho +d\rho /2)d\phi -A_{z}(z)d\rho (\rho +d\rho /2)d\phi }{\rho d\phi d\rho dz}}\\&={\frac {1}{\rho }}{\frac {\partial (\rho A_{\rho })}{\partial \rho }}+{\frac {1}{\rho }}{\frac {\partial A_{\phi }}{\partial \phi }}+{\frac {\partial A_{z}}{\partial z}}\end{aligned}}$

　
: ${\begin{aligned}(\operatorname {curl} \mathbf {A} )_{\rho }&=\lim _{S^{\perp {\boldsymbol {\hat {\rho }}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}\\&={\frac {A_{\phi }(z)(\rho +d\rho )d\phi -A_{\phi }(z+dz)(\rho +d\rho )d\phi +A_{z}(\phi +d\phi )dz-A_{z}(\phi )dz}{(\rho +d\rho )d\phi dz}}\\&=-{\frac {\partial A_{\phi }}{\partial z}}+{\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \phi }}\end{aligned}}$

　
: ${\begin{aligned}(\operatorname {curl} \mathbf {A} )_{\phi }&=\lim _{S^{\perp {\boldsymbol {\hat {\phi }}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}\\&={\frac {A_{z}(\rho )dz-A_{z}(\rho +d\rho )dz+A_{\rho }(z+dz)d\rho -A_{\rho }(z)d\rho }{d\rho dz}}\\&=-{\frac {\partial A_{z}}{\partial \rho }}+{\frac {\partial A_{\rho }}{\partial z}}\end{aligned}}$

　
: ${\begin{aligned}(\operatorname {curl} \mathbf {A} )_{z}&=\lim _{S^{\perp {\boldsymbol {\hat {z}}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}\\&={\frac {A_{\rho }(\phi )d\rho -A_{\rho }(\phi +d\phi )d\rho +A_{\phi }(\rho +d\rho )(\rho +d\rho )d\phi -A_{\phi }(\rho )\rho d\phi }{\rho d\rho d\phi }}\\&=-{\frac {1}{\rho }}{\frac {\partial A_{\rho }}{\partial \phi }}+{\frac {1}{\rho }}{\frac {\partial (\rho A_{\phi })}{\partial \rho }}\end{aligned}}$

　
: ${\begin{aligned}\operatorname {curl} \mathbf {A} &=(\operatorname {curl} \mathbf {A} )_{\rho }{\hat {\boldsymbol {\rho }}}+(\operatorname {curl} \mathbf {A} )_{\phi }{\hat {\boldsymbol {\phi }}}+(\operatorname {curl} \mathbf {A} )_{z}{\hat {\boldsymbol {z}}}\\&=\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \phi }}-{\frac {\partial A_{\phi }}{\partial z}}\right){\hat {\boldsymbol {\rho }}}+\left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right){\hat {\boldsymbol {\phi }}}+{\frac {1}{\rho }}\left({\frac {\partial (\rho A_{\phi })}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \phi }}\right){\hat {\boldsymbol {z}}}\end{aligned}}$



球坐標系推導

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

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

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

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

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



單位向量轉換公式

坐標參數 _u_ 的單位向量以如下方式定義，_u_ 的小的正值改變導致位置向量 ${\boldsymbol {\vec {r}}}$ 在 ${\boldsymbol {\hat {u}}}$ 方向上的改變。因此：

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

這裡的 _s_ 是弧長參數。

對於兩組坐標系 $u_{i}$ 和 $v_{j}$ ，依據鏈式法則：

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

現在，使除了一個之外的所有 $du_{i}=0$ 並在兩邊除以對應的坐標參數的微分，得到：

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



參見


* Del 算子

* 正交坐標系

* 曲線坐標系

* 在圓柱和球坐標中的向量場



引用



外部連結


* Maxima Computer Algebra system scripts（頁面存檔備份，存於網際網路檔案館） to generate some of these operators in cylindrical and spherical coordinates.

* Curl, Divergence, and Gradient in Cylindrical and Spherical Coordinate Systems.