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湖泊体积及平均水深的估算

湖泊体积及平均水深的估算. 椭球正弦曲面（ elliptic sinosoids) 是许多湖泊的湖床形状的很好的近似 . 假定湖面的边界为椭圆. 若湖的最大水深为. , 则椭球正弦曲面由. 其中 给出. 现要求湖水的总体积 V 及平均水深. 湖泊体积及平均水深的估算. 解 : 若 是湖面的椭圆区域 ,. 则湖水的总体积为. 被积函数的形状启示我们用变换. 湖泊体积及平均水深的估算. 由二重积分的变量替换公式得. 湖泊体积及平均水深的估算.

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湖泊体积及平均水深的估算

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  1. 湖泊体积及平均水深的估算 椭球正弦曲面（elliptic sinosoids)是许多湖泊的湖床形状的很好的近似. 假定湖面的边界为椭圆 若湖的最大水深为 , 则椭球正弦曲面由 其中 给出 . 现要求湖水的总体积 V及平均水深 .

  2. 湖泊体积及平均水深的估算 解:若 是湖面的椭圆区域, 则湖水的总体积为 被积函数的形状启示我们用变换

  3. 湖泊体积及平均水深的估算 由二重积分的变量替换公式得

  4. 湖泊体积及平均水深的估算 上述公式可通过测量 来估计湖水的体积(即水量). 容易证明椭圆R的面积为 ,因而湖水的平均深度为 人们对全世界107个湖泊的研究得到 的平均值为0.467.

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