ベクトル解析の公式の一覧
円柱座標 ( r , θ , z ) {\displaystyle (r,\theta ,z)} と直交座標 ( x , y , z ) {\displaystyle (x,y,z)} の変換
{ x = r cos  θ y = r sin  θ z = z { r = x 2 + y 2 θ = tan − 1  y x z = z {\displaystyle \left\{{\begin{array}{l}x=r\cos \theta \\y=r\sin \theta \\z=z\end{array}}\right.\ \ \ \ \left\{{\begin{array}{l}r={\sqrt {x^{2}+y^{2}}}\\\theta =\tan ^{-1}{\frac {y}{x}}\\z=z\end{array}}\right.}
単位基底ベクトル
e r = cos  θ e x + sin  θ e y {\displaystyle \mathbf {e} _{r}=\cos \theta \mathbf {e} _{x}+\sin \theta \mathbf {e} _{y}}
e θ = − sin  θ e x + cos  θ e y {\displaystyle \mathbf {e} _{\theta }=-\sin \theta \mathbf {e} _{x}+\cos \theta \mathbf {e} _{y}}
e z = e z {\displaystyle \mathbf {e} _{z}=\mathbf {e} _{z}}
発散
∇ ⋅ A = ∑ i = 1 3 1 h ∂ ∂ x i ( h h i A i ) {\displaystyle \mathbf {\nabla } \cdot \mathbf {A} =\sum _{i=1}^{3}{\frac {1}{h}}{\frac {\partial }{\partial x^{i}}}\left({\frac {h}{h_{i}}}A_{i}\right)}
回転
∇ × A = ∑ i = 1 3 e i ∑ j = 1 3 ∑ k = 1 3 ε i j k h i h ∂ ( h k A k ) ∂ x j {\displaystyle \mathbf {\nabla } \times \mathbf {A} =\sum _{i=1}^{3}\mathbf {e} _{i}\sum _{j=1}^{3}\sum _{k=1}^{3}\epsilon _{ijk}{\frac {h_{i}}{h}}{\frac {\partial (h_{k}A_{k})}{\partial x^{j}}}}
ラプラシアン (スカラー場)