球面座標系
∂ ( x , y , z ) ∂ ( r , θ , φ ) = ( sin  θ cos  φ r cos  θ cos  φ − r sin  θ sin  φ sin  θ sin  φ r cos  θ sin  φ r sin  θ cos  φ cos  θ − r sin  θ 0 ) = ( cos  φ − sin  φ 0 sin  φ cos  φ 0 0 0 1 ) ( sin  θ cos  θ 0 0 0 1 cos  θ − sin  θ 0 ) ( 1 0 0 0 r 0 0 0 r sin  θ ) {\displaystyle {\begin{aligned}{\frac {\partial (x,y,z)}{\partial (r,\theta ,\phi )}}&={\begin{pmatrix}\sin \theta \,\cos \phi &r\cos \theta \,\cos \phi &-r\sin \theta \,\sin \phi \\\sin \theta \,\sin \phi &r\cos \theta \,\sin \phi &r\sin \theta \,\cos \phi \\\cos \theta &-r\sin \theta &0\\\end{pmatrix}}\\&={\begin{pmatrix}\cos \phi &-\sin \phi &0\\\sin \phi &\cos \phi &0\\0&0&1\\\end{pmatrix}}{\begin{pmatrix}\sin \theta &\cos \theta &0\\0&0&1\\\cos \theta &-\sin \theta &0\\\end{pmatrix}}{\begin{pmatrix}1&0&0\\0&r&0\\0&0&r\sin \theta \\\end{pmatrix}}\\\end{aligned}}}
∂ A ∂ φ = ( ∂ A r ∂ φ − A φ sin  θ ) e r + ( ∂ A θ ∂ φ − A φ cos  θ ) e θ + ( ∂ A φ ∂ φ + A r sin  θ + A θ cos  θ ) e φ {\displaystyle {\frac {\partial {\boldsymbol {A}}}{\partial \phi }}=\left({\frac {\partial A_{r}}{\partial \phi }}-A_{\phi }\sin \theta \right){\boldsymbol {e}}_{r}+\left({\frac {\partial A_{\theta }}{\partial \phi }}-A_{\phi }\cos \theta \right){\boldsymbol {e}}_{\theta }+\left({\frac {\partial A_{\phi }}{\partial \phi }}+A_{r}\sin \theta +A_{\theta }\cos \theta \right){\boldsymbol {e}}_{\phi }}
で与えられる。