偏微分計算

2023-06-22 11:41 作者:編程會一點建模不太懂 0人讀過 | 我要投稿

題目選自1996年考研數(shù)學(xué)

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

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

分別計算變量 $u%2Cv$ 對 $x%2Cy$ 的偏導(dǎo)數(shù)

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

通過鏈?zhǔn)椒▌t將 $z$ 對 $x%2Cy$ 的一階偏導(dǎo)化為 $z$ 對 $u%2Cv$ 的一階偏導(dǎo)

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

通過鏈?zhǔn)椒▌t將 $z$ 對 $x%2Cy$ 的二階偏導(dǎo)化為 $z$ 對 $u%2Cv$ 的二階偏導(dǎo)

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

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

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

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

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

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

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

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

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

將上述偏導(dǎo)數(shù)帶入方程

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

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

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

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

本題選自1996年考研數(shù)學(xué)真題，重點考察偏微分變換與鏈?zhǔn)椒▌t，計算量偏大，即便將此題放在現(xiàn)今考研數(shù)學(xué)命題中，也不失為一道好題難題。

近年來看來，考研數(shù)學(xué)的命題有往課本或者往年真題中重復(fù)命題的趨勢；譬如2022年考研數(shù)學(xué)一中切比雪夫不等式和線性代數(shù)大題在00年代考研數(shù)學(xué)中出過；2021年考研數(shù)學(xué)一的數(shù)一專題的第一問也是在同濟版高等數(shù)學(xué)中可找到類似的題目；2022年考研數(shù)學(xué)二中，線性代數(shù)大題瑞利商也是在同濟版線性代數(shù)課后題中能找到類似的題目。

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