Φ = E → ⋅ A → {\displaystyle \Phi ={\vec {E}}\cdot {\vec {A}}} → ∫ E → ⋅ d A → = ∫ E → ⋅ n ^ d A {\displaystyle \to \int {\vec {E}}\cdot d{\vec {A}}=\int {\vec {E}}\cdot {\hat {n}}\,dA} = electric flux
q e n c l o s e d = ε 0 ∮ E → ⋅ d A → {\displaystyle q_{enclosed}=\varepsilon _{0}\oint {\vec {E}}\cdot d{\vec {A}}}
d Vol = d x d y d z = r 2 d r d A {\displaystyle d\,{\text{Vol}}=dxdydz=r^{2}drdA} where d A = r 2 d ϕ d θ {\displaystyle dA=r^{2}d\phi d\theta }
A sphere = r 2 ∫ 0 π sin θ d θ ∫ 0 2 π d ϕ = 4 π r 2 {\displaystyle A_{\text{sphere}}=r^{2}\int _{0}^{\pi }\sin \theta d\theta \int _{0}^{2\pi }d\phi =4\pi r^{2}}
Calculating ∫ f d A {\displaystyle \int fdA} and ∫ f d V {\displaystyle \int fdV} with angular symmetry Cyndrical: d A = 2 π r d z ; d V = d A d r {\displaystyle dA=2\pi r\,dz;\,dV=dA\,dr} . Spherical: ∫ d A = 4 π r 2 , d V = 4 π r 2 d r {\displaystyle \int dA=4\pi r^{2},\;dV=4\pi r^{2}\,dr}
