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

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

这个条件的几何解释如下: 如果数据是在表面上规定的，那么就有可能确定法向导数的微分方程。如果上面的数据和上面的微分方程确定了 on 的正常导数，那么它就是非特征的。如果上面的数据和微分方程的数据不能确定 on 的法向导数，那么表面是特征的，微分方程的数据限制在: 微分方程是内部的。

这个条件的几何解释如下: 如果关于 {{mvar|u}} 的数据是在曲面 {{mvar|S}} 上规定的，那么就有可能依据微分方程确定曲面 {{mvar|S}} 上 {{mvar|u}} 的法向导数。如果曲面 {{mvar|S}} 上的数据和上面的微分方程能确定曲面 {{mvar|S}} 上 {{mvar|u}} 的法向导数，那么它就是非特征的。如果曲面 {{mvar|S}} 上的数据和上面的微分方程能确定曲面 {{mvar|S}} 上 {{mvar|u}} 的法向导数，那么曲面是特征的，并且微分方程将数据限制在曲面 {{mvar|S}} 上：微分方程是在曲面 {{mvar|S}} 内部。

  A first-order system  0}} is elliptic if no surface is characteristic for : the values of  on  and the differential equation always determine the normal derivative of  on .

  A first-order system  0}} is elliptic if no surface is characteristic for : the values of  on  and the differential equation always determine the normal derivative of  on .

如果没有表面具有以下特征，则一阶系统0}是椭圆的: on 和微分方程的值总是决定 on 的法向导数。

如果没有表面具有以下特征，则一阶系统 {{math|''Lu'' {{=}} 0}} 是椭圆形的：{{mvar|u}}在 {{mvar|S}} 的值和微分方程总是决定 {{mvar|S}} 上 {{mvar|u}} 的法向导数。

# 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|''Q''(''λξ'' + ''η'') {{=}} 0}} has {{mvar|m}} real roots {{math|''λ''₁, ''λ''₂,… ''λ''_''m''}}. The system is '''strictly hyperbolic''' if these roots are always distinct. The geometrical interpretation of this condition is as follows: the characteristic form {{math|''Q''(''ζ'') {{=}} 0}} defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has {{mvar|m}} sheets, and the axis {{math|''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.

# 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|''Q''(''λξ'' + ''η'') {{=}} 0}} has {{mvar|m}} real roots {{math|''λ''₁, ''λ''₂,… ''λ''_''m''}}. The system is '''strictly hyperbolic''' if these roots are always distinct. The geometrical interpretation of this condition is as follows: the characteristic form {{math|''Q''(''ζ'') {{=}} 0}} defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has {{mvar|m}} sheets, and the axis {{math|''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.

  A first-order system is hyperbolic at a point if there is a spacelike surface  with normal  at that point. This means that, given any non-trivial vector  orthogonal to , and a scalar multiplier , the equation  0}} has  real roots . The system is strictly hyperbolic if these roots are always distinct. The geometrical interpretation of this condition is as follows: the characteristic form  0}} defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has  sheets, and the axis  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.

  A first-order system is hyperbolic at a point if there is a spacelike surface  with normal  at that point. This means that, given any non-trivial vector  orthogonal to , and a scalar multiplier , the equation  0}} has  real roots . The system is strictly hyperbolic if these roots are always distinct. The geometrical interpretation of this condition is as follows: the characteristic form  0}} defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has  sheets, and the axis  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.

一阶系统在某一点是双曲的，如果在该点有一个法线的类空曲面的话。这意味着，给定任意正交于的非平凡向量和一个标量乘子，方程0}有实根。如果这些根始终是不同的，则该系统是严格双曲型的。这个条件的几何解释如下: 特征形式0}定义了一个具有齐次坐标的圆锥体(法线圆锥体)。在双曲线的情况下，这个圆锥体有工作表，并且轴}在这些工作表中运行: 它不与工作表中的任何一个相交。但是当从原点偏离时，这条轴线与每一片都相交。在椭圆情况下，法锥没有实片。

如果在该点存在一个法向量为 {{mvar|ξ}} '''<font color="#ff8000">类空曲面 Spacclike Surface</font> {{mvar|S}} ，则一阶系统在某一点是双曲的。这意味着，给定任意正交于 {{mvar|ξ}} 的非平凡向量 {{mvar|η}} 和一个标量乘子 {{mvar|λ}}，方程 {{math|''Q''(''λξ'' + ''η'') {{=}} 0}} 有 {{mvar|m}} 个实根 {{math|''λ''₁, ''λ''₂,… ''λ''_''m''}}。如果这些根始终不同，则该系统是严格双曲形的。这个条件的几何解释如下: 特征形式 {{math|''Q''(''ζ'') {{=}} 0}} 定义了一个具有齐次坐标 ζ的圆锥(法线圆锥)。在双曲线的情况下，这个圆锥体有 {{mvar|m}} 层，并且轴 {{math|''ζ'' {{=}} ''λξ''}} 在这些层中运行: 它不与任何一层相交。但是当从原点偏离 η时，这条轴线与每一层都相交。在椭圆情况下，法锥没有实层。

=== Equations of mixed type ===

=== Equations of mixed type ===