Editing Partial differential equation (section)

=== Systems of first-order equations and characteristic surfaces ===
{{see also|First-order partial differential equation}}

The classification of partial differential equations can be extended to systems of first-order equations, where the unknown {{mvar|u}} is now a [[Euclidean vector|vector]] with {{mvar|m}} components, and the coefficient matrices {{mvar|A<sub>ν</sub>}} are {{mvar|m}} by {{mvar|m}} matrices for {{math|1=''ν'' = 1, 2, …, ''n''}}. The partial differential equation takes the form
<math display="block">Lu = \sum_{\nu=1}^{n} A_\nu \frac{\partial u}{\partial x_\nu} + B=0,</math>
where the coefficient matrices {{mvar|A<sub>ν</sub>}} and the vector {{mvar|B}} may depend upon {{mvar|x}} and {{mvar|u}}. If a [[hypersurface]] {{mvar|S}} is given in the implicit form
<math display="block">\varphi(x_1, x_2, \ldots, x_n)=0,</math>
where {{mvar|φ}} has a non-zero gradient, then {{mvar|S}} is a '''characteristic surface''' for the [[Differential_operator|operator]] {{mvar|L}} at a given point if the characteristic form vanishes:
<math display="block">Q\left(\frac{\partial\varphi}{\partial x_1}, \ldots, \frac{\partial\varphi}{\partial x_n}\right) = \det\left[\sum_{\nu=1}^n A_\nu \frac{\partial \varphi}{\partial x_\nu}\right] = 0.</math>

The geometric interpretation of this condition is as follows: if data for {{mvar|u}} are prescribed on the surface {{mvar|S}}, then it may be possible to determine the normal derivative of {{mvar|u}} on {{mvar|S}} from the differential equation. If the data on {{mvar|S}} and the differential equation determine the normal derivative of {{mvar|u}} on {{mvar|S}}, then {{mvar|S}} is non-characteristic. If the data on {{mvar|S}} and the differential equation ''do not'' determine the normal derivative of {{mvar|u}} on {{mvar|S}}, then the surface is '''characteristic''', and the differential equation restricts the data on {{mvar|S}}: the differential equation is ''internal'' to {{mvar|S}}.

# A first-order system {{math|1=''Lu'' = 0}} is ''elliptic'' if no surface is characteristic for {{mvar|L}}: the values of {{mvar|u}} on {{mvar|S}} and the differential equation always determine the normal derivative of {{mvar|u}} on {{mvar|S}}.
# A first-order system is ''hyperbolic'' at a point if there is a '''spacelike''' surface {{mvar|S}} with normal {{mvar|ξ}} at that point. This means that, given any non-trivial vector {{mvar|η}} orthogonal to {{mvar|ξ}}, and a scalar multiplier {{mvar|λ}}, the equation {{math|1=''Q''(''λξ'' + ''η'') = 0}} has {{mvar|m}} real roots {{math|''λ''<sub>1</sub>, ''λ''<sub>2</sub>, …, ''λ''<sub>''m''</sub>}}. The system is '''strictly hyperbolic''' if these roots are always distinct. The geometrical interpretation of this condition is as follows: the characteristic form {{math|1=''Q''(''ζ'') = 0}} defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has {{mvar|nm}} sheets, and the axis {{math|1=''ζ'' = ''λξ''}} runs inside these sheets: it does not intersect any of them. But when displaced from the origin by η, this axis intersects every sheet. In the elliptic case, the normal cone has no real sheets.
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