物質微分
曲線直交座標系 r = r ( q 1 , q 2 , q 3 ) {\displaystyle {\boldsymbol {r}}={\boldsymbol {r}}(q^{1},q^{2},q^{3})} における対流項 v ⋅ ∇ A {\displaystyle {\boldsymbol {v}}\cdot \nabla {\boldsymbol {A}}} の j {\displaystyle j} 成分は以下のように与えられる。
[ v ⋅ ∇ A ] j = ∑ k { v k h k ∂ A j ∂ q k + A k h k h j ( v j ∂ h j ∂ q k − v k ∂ h k ∂ q j ) } {\displaystyle [{\boldsymbol {v}}\cdot \nabla {\boldsymbol {A}}]_{j}=\sum _{k}\left\{{\frac {v_{k}}{h_{k}}}{\frac {\partial A_{j}}{\partial q^{k}}}+{\frac {A_{k}}{h_{k}h_{j}}}\left(v_{j}{\frac {\partial h_{j}}{\partial q^{k}}}-v_{k}{\frac {\partial h_{k}}{\partial q^{j}}}\right)\right\}}
ただし、
h k = | ∂ r ∂ q k | = g k k {\displaystyle h_{k}=\left|{\partial {\boldsymbol {r}} \over \partial q^{k}}\right|={\sqrt {g_{kk}}}}