Systems of first-order equations
Systems of first-order equations and characteristic surfaces [ edit ] The classification of partial differential equations can be extended to systems of first-order equations, where the unknown u is now a vector with m components, and the coefficient matrices A ν are m by m matrices for ν = 1, 2, …, n . The partial differential equation takes the form � � = ∑ � = 1 � � � ∂ � ∂ � � + � = 0 , where the coefficient matrices A ν and the vector B may depend upon x and u . If a hypersurface S is given in the implicit form � ( � 1 , � 2 , … , � � ) = 0 , where φ has a non-zero gradient, then S is a characteristic surface for the operator L at a given point if the characteristic form vanishes: � ( ∂ � ∂ � 1 , … , ∂ � ∂ � � ) = det [ ∑ � = 1 � � � ∂ � ∂ � � ] = 0. The geometric interpretation ...