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。
内積と外積

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（アインシュタインの縮約記法）。 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[1] 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[1] 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[2][3] 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[4][3] 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[3] 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[3] ( 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} }
微分公式

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。[3] ∇ ⋅ ( 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} }