ベクトル解析の公式の一覧

ベクトル解析の公式の一覧（ベクトルかいせきのこうしきのいちらん）では、3次元空間におけるベクトル解析の公式の一覧を与える。 ここで A {\displaystyle \mathbf {A} } , B {\displaystyle \mathbf {B} } , C {\displaystyle \mathbf {C} } は任意のベクトルである。また重複添え字については和を取る。 ϵ i j k {\displaystyle \epsilon _{ijk}} はレヴィ＝チヴィタ記号、 θ {\displaystyle \theta } は A {\displaystyle \mathbf {A} } , B {\displaystyle \mathbf {B} } がなす角である。 内積 A ⋅ B = A i B i = A x B x + A y B y + A z B z = | A | | B | cos ⁡ θ {\displaystyle \mathbf {A} \cdot \mathbf {B} =A_{i}B_{i}=A_{x}B_{x}+A_{y}B_{y}+A_{z}B_{z}=|\mathbf {A} ||\mathbf {B} |\cos \theta } A ⋅ B = B ⋅ A {\displaystyle \mathbf {A} \cdot \mathbf {B} =\mathbf {B} \cdot \mathbf {A} } 外積 A × B = ( ϵ i j k A j B k ) e i = ( A y B z − A z B y ) e x + ( A z B x − A x B z ) e y + ( A x B y − A y B x ) e z {\displaystyle \mathbf {A} \times \mathbf {B} =(\epsilon _{ijk}A_{j}B_{k})\mathbf {e} _{i}=(A_{y}B_{z}-A_{z}B_{y})\mathbf {e} _{x}+(A_{z}B_{x}-A_{x}B_{z})\mathbf {e} _{y}+(A_{x}B_{y}-A_{y}B_{x})\mathbf {e} _{z}} A × B = − B × A {\displaystyle \mathbf {A} \times \mathbf {B} =-\mathbf {B} \times \mathbf {A} } | A × B | = | A | B | sin ⁡ θ {\displaystyle |\mathbf {A} \times \mathbf {B} |=|\mathbf {A} |\mathbf {B} |\sin \theta } スカラー三重積 A ⋅ ( B × C ) = B ⋅ ( C × A ) = C ⋅ ( A × B ) {\displaystyle \mathbf {A} \cdot (\mathbf {B} \times \mathbf {C} )=\mathbf {B} \cdot (\mathbf {C} \times \mathbf {A} )=\mathbf {C} \cdot (\mathbf {A} \times \mathbf {B} )} ベクトル三重積 A × ( B × C ) = ( A ⋅ C ) B − ( A ⋅ B ) C {\displaystyle \mathbf {A} \times (\mathbf {B} \times \mathbf {C} )=(\mathbf {A} \cdot \mathbf {C} )\mathbf {B} -(\mathbf {A} \cdot \mathbf {B} )\mathbf {C} } ヤコビ恒等式 A × ( B × C ) + B × ( C × A ) + C × ( A × B ) = 0 {\displaystyle \mathbf {A} \times (\mathbf {B} \times \mathbf {C} )+\mathbf {B} \times (\mathbf {C} \times \mathbf {A} )+\mathbf {C} \times (\mathbf {A} \times \mathbf {B} )=0} 四重積 ( A × B ) ⋅ ( C × D ) = ( A ⋅ C ) ( B ⋅ D ) − ( A ⋅ D ) ( B ⋅ C ) {\displaystyle (\mathbf {A} \times \mathbf {B} )\cdot (\mathbf {C} \times \mathbf {D} )=(\mathbf {A} \cdot \mathbf {C} )(\mathbf {B} \cdot \mathbf {D} )-(\mathbf {A} \cdot \mathbf {D} )(\mathbf {B} \cdot \mathbf {C} )} ( A × B ) × ( C × D ) = [ A ⋅ ( C × D ) ] B − [ B ⋅ ( C × D ) ] A {\displaystyle (\mathbf {A} \times \mathbf {B} )\times (\mathbf {C} \times \mathbf {D} )=[\mathbf {A} \cdot (\mathbf {C} \times \mathbf {D} )]\mathbf {B} -[\mathbf {B} \cdot (\mathbf {C} \times \mathbf {D} )]\mathbf {A} } ここで A {\displaystyle \mathbf {A} } , B {\displaystyle \mathbf {B} } は任意のベクトル場, f {\displaystyle f} は任意のスカラー場である。 ∇ ⋅ ( f A ) = ∇ f ⋅ A + f ∇ ⋅ A {\displaystyle \mathbf {\nabla } \cdot (f\mathbf {A} )=\mathbf {\nabla } f\cdot \mathbf {A} +f\mathbf {\nabla } \cdot \mathbf {A} } ∇ ( A ⋅ B ) = ( B ⋅ ∇ ) A + ( A ⋅ ∇ ) B + A × ( ∇ × B ) + B × ( ∇ × A ) {\displaystyle \mathbf {\nabla } (\mathbf {A} \cdot \mathbf {B} )=(\mathbf {B} \cdot \mathbf {\nabla } )\mathbf {A} +(\mathbf {A} \cdot \mathbf {\nabla } )\mathbf {B} +\mathbf {A} \times (\mathbf {\nabla } \times \mathbf {B} )+\mathbf {B} \times (\mathbf {\nabla } \times \mathbf {A} )} ∇ ⋅ ( A × B ) = B ⋅ ( ∇ × A ) − A ⋅ ( ∇ × B ) {\displaystyle \mathbf {\nabla } \cdot (\mathbf {A} \times \mathbf {B} )=\mathbf {B} \cdot (\mathbf {\nabla } \times \mathbf {A} )-\mathbf {A} \cdot (\mathbf {\nabla } \times \mathbf {B} )} ∇ × ( f A ) = ∇ f × A + f ∇ × A {\displaystyle \mathbf {\nabla } \times (f\mathbf {A} )=\mathbf {\nabla } f\times \mathbf {A} +f\mathbf {\nabla } \times \mathbf {A} } ∇ × ( A × B ) = ( B ⋅ ∇ ) A − ( A ⋅ ∇ ) B + A ( ∇ ⋅ B ) − B ( ∇ ⋅ A ) {\displaystyle \mathbf {\nabla } \times (\mathbf {A} \times \mathbf {B} )=(\mathbf {B} \cdot \mathbf {\nabla } )\mathbf {A} -(\mathbf {A} \cdot \mathbf {\nabla } )\mathbf {B} +\mathbf {A} (\mathbf {\nabla } \cdot \mathbf {B} )-\mathbf {B} (\mathbf {\nabla } \cdot \mathbf {A} )} ∇ × ∇ f = 0 {\displaystyle \mathbf {\nabla } \times \mathbf {\nabla } f=0} ∇ ⋅ ( ∇ × A ) = 0 {\displaystyle \mathbf {\nabla } \cdot (\mathbf {\nabla } \times \mathbf {A} )=0} ∇ × ( ∇ × A ) = ∇ ( ∇ ⋅ A ) − ∇ 2 A {\displaystyle \mathbf {\nabla } \times (\mathbf {\nabla } \times \mathbf {A} )=\mathbf {\nabla } (\mathbf {\nabla } \cdot \mathbf {A} )-\mathbf {\nabla } ^{2}\mathbf {A} } ヘルムホルツ分解 B = ∇ f + ∇ × A {\displaystyle \mathbf {B} =\mathbf {\nabla } f+\mathbf {\nabla } \times \mathbf {A} } ここで A {\displaystyle \mathbf {A} } , B {\displaystyle \mathbf {B} } , C {\displaystyle \mathbf {C} } は任意のベクトル場, f {\displaystyle f} , g {\displaystyle g} は任意のスカラー場である。また, V {\displaystyle V} は空間領域, ∂ V {\displaystyle \partial V} はその境界, S {\displaystyle S} は面, n {\displaystyle \mathbf {n} } はその法線ベクトル ( S = ∂ V {\displaystyle S=\partial V} の場合 n {\displaystyle \mathbf {n} } は外向きに取る), d S = n d S {\displaystyle d\mathbf {S} =\mathbf {n} dS} は面要素ベクトルである。閉曲線 ∂ S {\displaystyle \partial S} に関する線積分 d r {\displaystyle d\mathbf {r} } は法線 n {\displaystyle \mathbf {n} } に対応する向きとする。 ガウスの発散定理および関連する公式（最後の等式はグリーンの定理である） ∫ V ∇ ⋅ A d V = ∮ ∂ V A ⋅ d S {\displaystyle \int _{V}\mathbf {\nabla } \cdot \mathbf {A} \,dV=\oint _{\partial V}\mathbf {A} \cdot d\mathbf {S} } ∫ V ∇ f d V = ∮ ∂ V f d S {\displaystyle \int _{V}\mathbf {\nabla } f\,dV=\oint _{\partial V}f\,d\mathbf {S} } ∫ V ∇ × A d V = ∮ ∂ V d S × A {\displaystyle \int _{V}\mathbf {\nabla } \times \mathbf {A} \,dV=\oint _{\partial V}d\mathbf {S} \times \mathbf {A} } ∫ V ( f ∇ 2 g − g ∇ 2 f ) d V = ∮ ∂ V ( f ∇ g − g ∇ f ) ⋅ d S {\displaystyle \int _{V}(f\mathbf {\nabla } ^{2}g-g\mathbf {\nabla } ^{2}f)dV=\oint _{\partial V}(f\mathbf {\nabla } g-g\mathbf {\nabla } f)\cdot d\mathbf {S} } ストークスの定理および関連する公式 ∫ S ( ∇ × A ) ⋅ d S = ∮ ∂ S A ⋅ d r {\displaystyle \int _{S}(\mathbf {\nabla } \times \mathbf {A} )\cdot d\mathbf {S} =\oint _{\partial S}\mathbf {A} \cdot d\mathbf {r} } ∫ S d S × ∇ f = ∮ ∂ S f d r {\displaystyle \int _{S}d\mathbf {S} \times \mathbf {\nabla } f=\oint _{\partial S}fd\mathbf {r} } ∫ S ( d S × ∇ ) × A = ∮ ∂ S d r × A {\displaystyle \int _{S}(d\mathbf {S} \times \mathbf {\nabla } )\times \mathbf {A} =\oint _{\partial S}d\mathbf {r} \times \mathbf {A} } 曲線座標における勾配、発散、回転、ラプラシアン、物質微分の公式。 円柱座標 ( r , θ , z ) {\displaystyle (r,\theta ,z)} と直交座標 ( x , y , z ) {\displaystyle (x,y,z)} の変換 { x = r cos ⁡ θ y = r sin ⁡ θ z = z         { r = x 2 + y 2 θ = tan − 1 ⁡ y x z = z {\displaystyle \left\{{\begin{array}{l}x=r\cos \theta \\y=r\sin \theta \\z=z\end{array}}\right.\ \ \ \ \left\{{\begin{array}{l}r={\sqrt {x^{2}+y^{2}}}\\\theta =\tan ^{-1}{\frac {y}{x}}\\z=z\end{array}}\right.} 単位基底ベクトル e r = cos ⁡ θ e x + sin ⁡ θ e y {\displaystyle \mathbf {e} _{r}=\cos \theta \mathbf {e} _{x}+\sin \theta \mathbf {e} _{y}} e θ = − sin ⁡ θ e x + cos ⁡ θ e y {\displaystyle \mathbf {e} _{\theta }=-\sin \theta \mathbf {e} _{x}+\cos \theta \mathbf {e} _{y}} e z = e z {\displaystyle \mathbf {e} _{z}=\mathbf {e} _{z}} 計量 d s 2 = d r 2 + r 2 d θ 2 + d z 2 {\displaystyle ds^{2}=dr^{2}+r^{2}d\theta ^{2}+dz^{2}} 体積要素 d V = r d r d θ d z {\displaystyle dV=r\,dr\,d\theta \,dz} 勾配 ∇ f = ∂ f ∂ r e r + 1 r ∂ f ∂ θ e θ + ∂ f ∂ z e z {\displaystyle \mathbf {\nabla } f={\frac {\partial f}{\partial r}}\mathbf {e} _{r}+{\frac {1}{r}}{\frac {\partial f}{\partial \theta }}\mathbf {e} _{\theta }+{\frac {\partial f}{\partial z}}\mathbf {e} _{z}} 発散 ∇ ⋅ A = 1 r ∂ ∂ r ( r A r ) + 1 r ∂ A θ ∂ θ + ∂ A z ∂ z {\displaystyle \mathbf {\nabla } \cdot \mathbf {A} ={\frac {1}{r}}{\frac {\partial }{\partial r}}(rA_{r})+{\frac {1}{r}}{\frac {\partial A_{\theta }}{\partial \theta }}+{\frac {\partial A_{z}}{\partial z}}} 回転 ∇ × A = ( 1 r ∂ A z ∂ θ − ∂ A θ ∂ z ) e r + ( ∂ A r ∂ z − ∂ A z ∂ r ) e θ + [ 1 r ∂ ∂ r ( r A θ ) − 1 r ∂ A r ∂ θ ] e z {\displaystyle \mathbf {\nabla } \times \mathbf {A} =\left({\frac {1}{r}}{\frac {\partial A_{z}}{\partial \theta }}-{\frac {\partial A_{\theta }}{\partial z}}\right)\mathbf {e} _{r}+\left({\frac {\partial A_{r}}{\partial z}}-{\frac {\partial A_{z}}{\partial r}}\right)\mathbf {e} _{\theta }+\left[{\frac {1}{r}}{\frac {\partial }{\partial r}}(rA_{\theta })-{\frac {1}{r}}{\frac {\partial A_{r}}{\partial \theta }}\right]\mathbf {e} _{z}} ラプラシアン (スカラー場) ∇ 2 f = 1 r ∂ ∂ r ( r ∂ f ∂ r ) + 1 r 2 ∂ 2 f ∂ θ 2 + ∂ 2 f ∂ z 2 {\displaystyle \mathbf {\nabla } ^{2}f={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \theta ^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}} ラプラシアン (ベクトル場) [ ∇ 2 A ] r = ∇ 2 A r − 2 r 2 ∂ A θ ∂ θ − A r r 2 {\displaystyle [\mathbf {\nabla } ^{2}\mathbf {A} ]_{r}=\mathbf {\nabla } ^{2}A_{r}-{\frac {2}{r^{2}}}{\frac {\partial A_{\theta }}{\partial \theta }}-{\frac {A_{r}}{r^{2}}}} [ ∇ 2 A ] θ = ∇ 2 A θ + 2 r 2 ∂ A r ∂ θ − A θ r 2 {\displaystyle [\mathbf {\nabla } ^{2}\mathbf {A} ]_{\theta }=\mathbf {\nabla } ^{2}A_{\theta }+{\frac {2}{r^{2}}}{\frac {\partial A_{r}}{\partial \theta }}-{\frac {A_{\theta }}{r^{2}}}} [ ∇ 2 A ] z = ∇ 2 A z {\displaystyle [\mathbf {\nabla } ^{2}\mathbf {A} ]_{z}=\mathbf {\nabla } ^{2}A_{z}} 物質微分 [ ( A ⋅ ∇ ) B ] r = ( A ⋅ ∇ ) B r − A θ B θ r {\displaystyle [(\mathbf {A} \cdot \mathbf {\nabla } )\mathbf {B} ]_{r}=(\mathbf {A} \cdot \mathbf {\nabla } )B_{r}-{\frac {A_{\theta }B_{\theta }}{r}}} [ ( A ⋅ ∇ ) B ] θ = ( A ⋅ ∇ ) B θ + A θ B r r {\displaystyle [(\mathbf {A} \cdot \mathbf {\nabla } )\mathbf {B} ]_{\theta }=(\mathbf {A} \cdot \mathbf {\nabla } )B_{\theta }+{\frac {A_{\theta }B_{r}}{r}}} [ ( A ⋅ ∇ ) B ] z = ( A ⋅ ∇ ) B z {\displaystyle [(\mathbf {A} \cdot \mathbf {\nabla } )\mathbf {B} ]_{z}=(\mathbf {A} \cdot \mathbf {\nabla } )B_{z}} 球座標 ( r , θ , ϕ ) {\displaystyle (r,\theta ,\phi )} と直交座標 ( x , y , z ) {\displaystyle (x,y,z)} の変換 { x = r sin ⁡ θ cos ⁡ ϕ y = r sin ⁡ θ sin ⁡ ϕ z = r cos ⁡ θ         { r = x 2 + y 2 + z 2 θ = cos − 1 ⁡ z r ϕ = tan − 1 ⁡ y x {\displaystyle \left\{{\begin{array}{l}x=r\sin \theta \cos \phi \\y=r\sin \theta \sin \phi \\z=r\cos \theta \end{array}}\right.\ \ \ \ \left\{{\begin{array}{l}r={\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta =\cos ^{-1}{\frac {z}{r}}\\\phi =\tan ^{-1}{\frac {y}{x}}\end{array}}\right.} 単位基底ベクトル e r = sin ⁡ θ cos ⁡ θ e x + sin ⁡ θ cos ⁡ ϕ e y + cos ⁡ θ e z {\displaystyle \mathbf {e} _{r}=\sin \theta \cos \theta \mathbf {e} _{x}+\sin \theta \cos \phi \mathbf {e} _{y}+\cos \theta \mathbf {e} _{z}} 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 }} 発散 ∇ ⋅ A = 1 r 2 ∂ ∂ r ( r 2 A r ) + 1 r sin ⁡ θ ∂ ∂ θ ( sin ⁡ θ A θ ) + 1 r sin ⁡ θ ∂ A ϕ ∂ ϕ {\displaystyle \mathbf {\nabla } \cdot \mathbf {A} ={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}(r^{2}A_{r})+{\frac {1}{r\sin \theta }}{\frac {\partial }{\partial \theta }}(\sin \theta A_{\theta })+{\frac {1}{r\sin \theta }}{\frac {\partial A_{\phi }}{\partial \phi }}} 回転 ∇ × A = 1 r sin ⁡ θ [ ∂ ∂ θ ( sin ⁡ θ A ϕ ) − ∂ A θ ∂ ϕ ] e r + 1 r [ 1 sin ⁡ θ ∂ A r ∂ ϕ − ∂ ∂ r ( r A ϕ ) ] e θ + 1 r [ ∂ ∂ r ( r A θ ) − ∂ A r ∂ θ ] e ϕ {\displaystyle \mathbf {\nabla } \times \mathbf {A} ={\frac {1}{r\sin \theta }}\left[{\frac {\partial }{\partial \theta }}(\sin \theta A_{\phi })-{\frac {\partial A_{\theta }}{\partial \phi }}\right]\mathbf {e} _{r}+{\frac {1}{r}}\left[{\frac {1}{\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}-{\frac {\partial }{\partial r}}(rA_{\phi })\right]\mathbf {e} _{\theta }+{\frac {1}{r}}\left[{\frac {\partial }{\partial r}}(rA_{\theta })-{\frac {\partial A_{r}}{\partial \theta }}\right]\mathbf {e} _{\phi }} ラプラシアン (スカラー場) ∇ 2 f = 1 r 2 ∂ ∂ r ( r 2 ∂ f ∂ r ) + 1 r 2 sin ⁡ θ ∂ ∂ θ ( sin ⁡ θ ∂ f ∂ θ ) + 1 r 2 sin 2 ⁡ θ ∂ 2 f ∂ ϕ 2 {\displaystyle \mathbf {\nabla } ^{2}f={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \phi ^{2}}}} ラプラシアン (ベクトル場) [ ∇ 2 A ] r = ∇ 2 A r − 2 r 2 ∂ A θ ∂ θ − 2 r 2 sin ⁡ θ ∂ A ϕ ∂ ϕ − 2 A r r 2 − 2 cot ⁡ θ A θ r 2 {\displaystyle [\mathbf {\nabla } ^{2}\mathbf {A} ]_{r}=\mathbf {\nabla } ^{2}A_{r}-{\frac {2}{r^{2}}}{\frac {\partial A_{\theta }}{\partial \theta }}-{\frac {2}{r^{2}\sin \theta }}{\frac {\partial A_{\phi }}{\partial \phi }}-{\frac {2A_{r}}{r^{2}}}-{\frac {2\cot \theta A_{\theta }}{r^{2}}}} [ ∇ 2 A ] θ = ∇ 2 A θ + 2 r 2 ∂ A r ∂ θ − 2 cot ⁡ θ r 2 sin ⁡ θ ∂ A ϕ ∂ ϕ − A θ r 2 sin 2 ⁡ θ {\displaystyle [\mathbf {\nabla } ^{2}\mathbf {A} ]_{\theta }=\mathbf {\nabla } ^{2}A_{\theta }+{\frac {2}{r^{2}}}{\frac {\partial A_{r}}{\partial \theta }}-{\frac {2\cot \theta }{r^{2}\sin \theta }}{\frac {\partial A_{\phi }}{\partial \phi }}-{\frac {A_{\theta }}{r^{2}\sin ^{2}\theta }}} [ ∇ 2 A ] ϕ = ∇ 2 A ϕ + 2 r 2 sin ⁡ θ ∂ A r ∂ ϕ + 2 cot ⁡ θ r 2 sin ⁡ θ ∂ A θ ∂ ϕ − A ϕ r 2 sin 2 ⁡ θ {\displaystyle [\mathbf {\nabla } ^{2}\mathbf {A} ]_{\phi }=\mathbf {\nabla } ^{2}A_{\phi }+{\frac {2}{r^{2}\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}+{\frac {2\cot \theta }{r^{2}\sin \theta }}{\frac {\partial A_{\theta }}{\partial \phi }}-{\frac {A_{\phi }}{r^{2}\sin ^{2}\theta }}} 物質微分 [ ( A ⋅ ∇ ) B ] r = ( A ⋅ ∇ ) B r − A θ B θ + A ϕ B ϕ r {\displaystyle [(\mathbf {A} \cdot \mathbf {\nabla } )\mathbf {B} ]_{r}=(\mathbf {A} \cdot \mathbf {\nabla } )B_{r}-{\frac {A_{\theta }B_{\theta }+A_{\phi }B_{\phi }}{r}}} [ ( A ⋅ ∇ ) B ] θ = ( A ⋅ ∇ ) B θ + A θ B r r − A ϕ B ϕ cot ⁡ θ r {\displaystyle [(\mathbf {A} \cdot \mathbf {\nabla } )\mathbf {B} ]_{\theta }=(\mathbf {A} \cdot \mathbf {\nabla } )B_{\theta }+{\frac {A_{\theta }B_{r}}{r}}-{\frac {A_{\phi }B_{\phi }\cot \theta }{r}}} [ ( A ⋅ ∇ ) B ] ϕ = ( A ⋅ ∇ ) B z + A ϕ B r r + A ϕ B θ cot ⁡ θ r {\displaystyle [(\mathbf {A} \cdot \mathbf {\nabla } )\mathbf {B} ]_{\phi }=(\mathbf {A} \cdot \mathbf {\nabla } )B_{z}+{\frac {A_{\phi }B_{r}}{r}}+{\frac {A_{\phi }B_{\theta }\cot \theta }{r}}} 3次元ユークリッド空間 R 3 {\displaystyle \mathbb {R} ^{3}} の曲線座標 x i {\displaystyle x^{i}} について、その座標系で計量が d s 2 = ∑ i = 1 3 h i ( x ) 2 ( d x i ) 2 {\displaystyle ds^{2}=\sum _{i=1}^{3}h_{i}(x)^{2}(dx^{i})^{2}} という対角形になるとき、これを直交曲線座標と呼ぶ。この座標系に付随する規格化された基底ベクトルを e i {\displaystyle \mathbf {e} _{i}} とする。 体積要素 d V = h d x 1 d x 2 d x 3 ,     h = h 1 h 2 h 3 {\displaystyle dV=hdx^{1}dx^{2}dx^{3},\ \ h=h_{1}h_{2}h_{3}} 勾配 ∇ f = ∑ i = 1 3 1 h i ∂ f ∂ x i e i {\displaystyle \mathbf {\nabla } f=\sum _{i=1}^{3}{\frac {1}{h_{i}}}{\frac {\partial f}{\partial x^{i}}}\mathbf {e} _{i}} 発散 ∇ ⋅ A = ∑ i = 1 3 1 h ∂ ∂ x i ( h h i A i ) {\displaystyle \mathbf {\nabla } \cdot \mathbf {A} =\sum _{i=1}^{3}{\frac {1}{h}}{\frac {\partial }{\partial x^{i}}}\left({\frac {h}{h_{i}}}A_{i}\right)} 回転 ∇ × A = ∑ i = 1 3 e i ∑ j = 1 3 ∑ k = 1 3 ϵ i j k h i h ∂ ( h k A k ) ∂ x j {\displaystyle \mathbf {\nabla } \times \mathbf {A} =\sum _{i=1}^{3}\mathbf {e} _{i}\sum _{j=1}^{3}\sum _{k=1}^{3}\epsilon _{ijk}{\frac {h_{i}}{h}}{\frac {\partial (h_{k}A_{k})}{\partial x^{j}}}} ラプラシアン (スカラー場) ∇ 2 f = ∑ i = 1 3 1 h ∂ ∂ x i ( h h i 2 ∂ f ∂ x i ) {\displaystyle \mathbf {\nabla } ^{2}f=\sum _{i=1}^{3}{\frac {1}{h}}{\frac {\partial }{\partial x^{i}}}\left({\frac {h}{h_{i}^{2}}}{\frac {\partial f}{\partial x^{i}}}\right)} 物質微分 [ ( A ⋅ ∇ ) B ] i = ∑ k = 1 3 [ A k h k ∂ B i ∂ x k + ( A i ∂ h i ∂ x k − A k ∂ h k ∂ x i ) B k h k h i ] {\displaystyle [(\mathbf {A} \cdot \mathbf {\nabla } )\mathbf {B} ]_{i}=\sum _{k=1}^{3}\left[{\frac {A_{k}}{h_{k}}}{\frac {\partial B_{i}}{\partial x_{k}}}+\left(A_{i}{\frac {\partial h_{i}}{\partial x_{k}}}-A_{k}{\frac {\partial h_{k}}{\partial x_{i}}}\right){\frac {B_{k}}{h_{k}h_{i}}}\right]}