ナブラ
div  ( curl  v ) = ∇ ⋅ ∇ × v = 0 {\displaystyle \operatorname {div} (\operatorname {curl} {\boldsymbol {v}})=\nabla \cdot \nabla \times {\boldsymbol {v}}=0}
が成り立ち、また二つは常に等しい:
div  ( grad  f ) = ∇ ⋅ ( ∇ f ) = ∇ 2 f = Δ f . {\displaystyle \operatorname {div} (\operatorname {grad} f)=\nabla \cdot (\nabla f)=\nabla ^{2}f=\Delta f.}
残る三種のベクトル微分の間には等式
∇ × ∇ × v = ∇ ( ∇ ⋅ v ) − ∇ 2 v {\displaystyle \nabla \times \nabla \times {\boldsymbol {v}}=\nabla (\nabla \cdot {\boldsymbol {v}})-\nabla ^{2}{\boldsymbol {v}}}
が成り立ち、さらに一つはテンソル積を用いて表すことができて、素性の良い函数に対して
∇ ( ∇ ⋅ v ) = ∇ ⋅ ( ∇ ⊗ v ) {\displaystyle \nabla (\nabla \cdot {\boldsymbol {v}})=\nabla \cdot (\nabla \otimes {\boldsymbol {v}})}
が成り立つ。