ベクトル解析の公式の一覧
e θ = cos  θ cos  φ e x + cos  θ sin  φ e y − sin  θ e z {\displaystyle \mathbf {e} _{\theta }=\cos \theta \cos \phi \mathbf {e} _{x}+\cos \theta \sin \phi \mathbf {e} _{y}-\sin \theta \mathbf {e} _{z}}
e φ = − sin  φ e x + cos  φ e y {\displaystyle \mathbf {e} _{\phi }=-\sin \phi \mathbf {e} _{x}+\cos \phi \mathbf {e} _{y}}
計量
d s 2 = d r 2 + r 2 ( d θ 2 + sin 2  θ d φ 2 ) {\displaystyle ds^{2}=dr^{2}+r^{2}(d\theta ^{2}+\sin ^{2}\theta d\phi ^{2})}
体積要素
d V = r 2 sin  θ d r d θ d φ {\displaystyle dV=r^{2}\sin \theta \,dr\,d\theta \,d\phi }
勾配
∇ f = ∂ f ∂ r e r + 1 r ∂ f ∂ θ e θ + 1 r sin  θ ∂ f ∂ φ e φ {\displaystyle \mathbf {\nabla } f={\frac {\partial f}{\partial r}}\mathbf {e} _{r}+{\frac {1}{r}}{\frac {\partial f}{\partial \theta }}\mathbf {e} _{\theta }+{\frac {1}{r\sin \theta }}{\frac {\partial f}{\partial \phi }}\mathbf {e} _{\phi }}