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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 . |
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− | 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。 |
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| 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 . |
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− | 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。 |
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| # 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''<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|''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''<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|''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. |
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| 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. |
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− | 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''<sub>1</sub>, ''λ''<sub>2</sub>,… ''λ''<sub>''m''</sub>}}。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。 |
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| + | ==[[用户:Yuling|Yuling]]([[用户讨论:Yuling|讨论]]) sheet 这个单词也不很理解,我直译为了“层” |
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| === Equations of mixed type === | | === Equations of mixed type === |