球面座標系
となる。ベクトル微分演算子を
∇ = e r ∂ ∂ r + e θ r ∂ ∂ θ + e φ r sin  θ ∂ ∂ φ {\displaystyle \nabla ={\boldsymbol {e}}_{r}{\frac {\partial }{\partial r}}+{\frac {{\boldsymbol {e}}_{\theta }}{r}}{\frac {\partial }{\partial \theta }}+{\frac {{\boldsymbol {e}}_{\phi }}{r\sin \theta }}{\frac {\partial }{\partial \phi }}}
で定めれば
g r a d f = ∇ f {\displaystyle \mathrm {grad} \,f=\nabla f}
と書ける。
( rot  A ) r r 2 sin  θ d θ ∧ d φ + ( rot  A ) θ r sin  θ d φ ∧ d r + ( rot  A ) φ r d r ∧ d θ = A r d r + r A θ d θ + r A φ sin  θ d φ = [ ∂ ∂ θ ( r A φ sin  θ ) − ∂ ( r A θ ) ∂ φ ] d θ ∧ d φ + [ ∂ A r ∂ φ − ∂ ∂ r ( r A φ sin  θ ) ] d φ ∧ d r + [ ∂ ( r A θ ) ∂ r − ∂ A r ∂ θ ] d r ∧ d θ {\displaystyle {\begin{aligned}(\operatorname {rot} {\boldsymbol {A}})_{r}\,&r^{2}\sin \theta \,d\theta \wedge d\phi +(\operatorname {rot} {\boldsymbol {A}})_{\theta }\,r\sin \theta \,d\phi \wedge dr+(\operatorname {rot} {\boldsymbol {A}})_{\phi }\,r\,dr\wedge d\theta \\&=A_{r}\,dr+rA_{\theta }\,d\theta +rA_{\phi }\sin \theta \,d\phi \\&=\left[{\frac {\partial }{\partial \theta }}(rA_{\phi }\sin \theta )-{\frac {\partial (rA_{\theta })}{\partial \phi }}\right]d\theta \wedge d\phi +\left[{\frac {\partial A_{r}}{\partial \phi }}-{\frac {\partial }{\partial r}}(rA_{\phi }\sin \theta )\right]d\phi \wedge dr+\left[{\frac {\partial (rA_{\theta })}{\partial r}}-{\frac {\partial A_{r}}{\partial \theta }}\right]dr\wedge d\theta \end{aligned}}}