偏微分计算

2023-06-22 11:41 作者:编程会一点建模不太懂 0人读过 | 我要投稿

题目选自1996年考研数学

设变换 $%5Cleft%5C%7B%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09u%3Dx-2y%5C%5C%0A%09v%3Dx%2Bay%5C%5C%0A%5Cend%7Barray%7D%20%5Cright.%20$ 可把方程 $6%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20x%5E2%7D%2B%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20x%5Cpartial%20y%7D-%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20y%5E2%7D%3D0%0A$

化为 $%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20u%5Cpartial%20v%7D%3D0$

分别计算变量 $u%2Cv$ 对 $x%2Cy$ 的偏导数

$%5Cleft%5C%7B%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09u%3Dx-2y%5C%5C%0A%09v%3Dx%2Bay%5C%5C%0A%5Cend%7Barray%7D%20%5Cright.%20%5CRightarrow%20%5Cleft%5C%7B%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20x%7D%3D1%5C%5C%0A%09%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20y%7D%3D-2%5C%5C%0A%09%5Cfrac%7B%5Cpartial%20v%7D%7B%5Cpartial%20x%7D%3D1%5C%5C%0A%09%5Cfrac%7B%5Cpartial%20v%7D%7B%5Cpartial%20y%7D%3Da%5C%5C%0A%5Cend%7Barray%7D%20%5Cright.%20$

通过链式法则将 $z$ 对 $x%2Cy$ 的一阶偏导化为 $z$ 对 $u%2Cv$ 的一阶偏导

$%5Cleft%5C%7B%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09%5Cfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20x%7D%3D%5Cfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20u%7D%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20x%7D%2B%5Cfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20v%7D%5Cfrac%7B%5Cpartial%20v%7D%7B%5Cpartial%20x%7D%3D%5Cleft(%20%5Cfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20u%7D%2B%5Cfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20v%7D%20%5Cright)%5C%5C%0A%09%5Cfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20y%7D%3D%5Cfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20u%7D%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20y%7D%2B%5Cfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20v%7D%5Cfrac%7B%5Cpartial%20v%7D%7B%5Cpartial%20y%7D%3D%5Cleft(%20-2%5Cfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20u%7D%2Ba%5Cfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20v%7D%20%5Cright)%5C%5C%0A%5Cend%7Barray%7D%20%5Cright.%20$

通过链式法则将 $z$ 对 $x%2Cy$ 的二阶偏导化为 $z$ 对 $u%2Cv$ 的二阶偏导

$%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20x%5E2%7D%3D%5Cfrac%7B%5Cpartial%20%5Cleft(%20%5Cfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20u%7D%20%5Cright)%7D%7B%5Cpartial%20x%7D%2B%5Cfrac%7B%5Cpartial%20%5Cleft(%20%5Cfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20v%7D%20%5Cright)%7D%7B%5Cpartial%20x%7D%0A$

$%3D%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20u%5E2%7D%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20x%7D%2B%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20u%5Cpartial%20v%7D%5Cfrac%7B%5Cpartial%20v%7D%7B%5Cpartial%20x%7D%2B%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20v%5Cpartial%20u%7D%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20x%7D%2B%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20v%5E2%7D%5Cfrac%7B%5Cpartial%20v%7D%7B%5Cpartial%20x%7D$

$%3D%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20u%5E2%7D%2B2%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20u%5Cpartial%20v%7D%2B%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20v%5E2%7D%0A$

$%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20x%5Cpartial%20y%7D%3D%5Cfrac%7B%5Cpartial%20%5Cleft(%20%5Cfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20u%7D%20%5Cright)%7D%7B%5Cpartial%20y%7D%2B%5Cfrac%7B%5Cpartial%20%5Cleft(%20%5Cfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20v%7D%20%5Cright)%7D%7B%5Cpartial%20y%7D%0A$

$%3D%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20u%5E2%7D%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20y%7D%2B%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20u%5Cpartial%20v%7D%5Cfrac%7B%5Cpartial%20v%7D%7B%5Cpartial%20y%7D%2B%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20v%5Cpartial%20u%7D%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20y%7D%2B%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20v%5E2%7D%5Cfrac%7B%5Cpartial%20v%7D%7B%5Cpartial%20y%7D$

$%3D-2%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20u%5E2%7D%2B%5Cleft(%20a-2%20%5Cright)%20%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20u%5Cpartial%20v%7D%2Ba%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20v%5E2%7D%0A$

$%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20y%5E2%7D%3D-2%5Cfrac%7B%5Cpartial%20%5Cleft(%20%5Cfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20u%7D%20%5Cright)%7D%7B%5Cpartial%20y%7D%2Ba%5Cfrac%7B%5Cpartial%20%5Cleft(%20%5Cfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20v%7D%20%5Cright)%7D%7B%5Cpartial%20y%7D%0A$

$%3D-2%5Cleft(%20%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20u%5E2%7D%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20y%7D%2B%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20u%5Cpartial%20v%7D%5Cfrac%7B%5Cpartial%20v%7D%7B%5Cpartial%20y%7D%20%5Cright)%20%2Ba%5Cleft(%20%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20v%5Cpartial%20u%7D%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20y%7D%2B%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20v%5E2%7D%5Cfrac%7B%5Cpartial%20v%7D%7B%5Cpartial%20y%7D%20%5Cright)%20%0A$

$%3D4%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20u%5E2%7D-4a%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20u%5Cpartial%20v%7D%2Ba%5E2%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20v%5E2%7D$

将上述偏导数带入方程

$6%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20x%5E2%7D%2B%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20x%5Cpartial%20y%7D-%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20y%5E2%7D%3D0$

$%5Cleft(%2010%2B5a%20%5Cright)%20%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20u%5Cpartial%20v%7D%2B%5Cleft(%206%2Ba-a%5E2%20%5Cright)%20%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20v%5E2%7D%3D0%0A$

要使 $%5Cfrac%7B%5Cpartial%20%5E2z%7D%7B%5Cpartial%20u%5Cpartial%20v%7D%3D0$

即 $%5Cleft%5C%7B%20%5Cbegin%7Barray%7D%7Bc%7D%0A%0910%2B5a%5Cne%200%5C%5C%0A%096%2Ba-a%5E2%3D0%5C%5C%0A%5Cend%7Barray%7D%20%5Cright.%20%5CRightarrow%20a%3D3$

本题选自1996年考研数学真题，重点考察偏微分变换与链式法则，计算量偏大，即便将此题放在现今考研数学命题中，也不失为一道好题难题。

近年来看来，考研数学的命题有往课本或者往年真题中重复命题的趋势；譬如2022年考研数学一中切比雪夫不等式和线性代数大题在00年代考研数学中出过；2021年考研数学一的数一专题的第一问也是在同济版高等数学中可找到类似的题目；2022年考研数学二中，线性代数大题瑞利商也是在同济版线性代数课后题中能找到类似的题目。

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