ヘルムホルツの定理
F L ( x ) = − ∇ φ ( x ) {\displaystyle \mathbf {F} _{\mathrm {L} }(\mathbf {x} )=-\nabla \phi (\mathbf {x} )}
F T ( x ) = ∇ × A ( x ) {\displaystyle \mathbf {F} _{\mathrm {T} }(\mathbf {x} )=\nabla \times \mathbf {A} (\mathbf {x} )}
とすると
∇ × F L ( x ) = rot  F L ( x ) = 0 {\displaystyle \nabla \times \mathbf {F} _{\mathrm {L} }(\mathbf {x} )=\operatorname {rot} \mathbf {F} _{\mathrm {L} }(\mathbf {x} )=\mathbf {0} }
∇ ⋅ F T ( x ) = div  F T ( x ) = 0 {\displaystyle \nabla \cdot \mathbf {F} _{\mathrm {T} }(\mathbf {x} )=\operatorname {div} \mathbf {F} _{\mathrm {T} }(\mathbf {x} )=0}