球面座標系
d S = ∂ x ∂ u × x ∂ v d u ∧ d v = e r r 2 sin  θ d θ ∧ d φ + e θ r sin  θ d φ ∧ d r + e φ r d r ∧ d θ {\displaystyle {\begin{aligned}d{\boldsymbol {S}}&={\frac {\partial {\boldsymbol {x}}}{\partial u}}\times {\frac {\boldsymbol {x}}{\partial v}}\,du\wedge dv\\&={\boldsymbol {e}}_{r}\,r^{2}\sin \theta \,d\theta \wedge d\phi +{\boldsymbol {e}}_{\theta }\,r\sin \theta \,d\phi \wedge dr+{\boldsymbol {e}}_{\phi }\,r\,dr\wedge d\theta \\\end{aligned}}}
で与えられる。
任意のベクトル場 A は
A r = e r ⋅ A , A θ = e θ ⋅ A , A φ = e φ ⋅ A {\displaystyle A_{r}={\boldsymbol {e}}_{r}\cdot {\boldsymbol {A}},~A_{\theta }={\boldsymbol {e}}_{\theta }\cdot {\boldsymbol {A}},~A_{\phi }={\boldsymbol {e}}_{\phi }\cdot {\boldsymbol {A}}}
A = A r e r + A θ e θ + A φ e φ {\displaystyle {\boldsymbol {A}}=A_{r}{\boldsymbol {e}}_{r}+A_{\theta }{\boldsymbol {e}}_{\theta }+A_{\phi }{\boldsymbol {e}}_{\phi }}