【正文】 BABABABAAB ?? ????????????????ttt dddddd矩陣 /導(dǎo)數(shù) /對(duì)時(shí)間的導(dǎo)數(shù) Friday, November 12, 2021 理論力學(xué) CAI 數(shù)學(xué)基礎(chǔ) 29 ? 例 已知 ? ?dd ? ?t A B A B? ? ?矩陣 /導(dǎo)數(shù) /對(duì)時(shí)間的導(dǎo)數(shù) ?????????? ttttc o ss ins inc o sA?????????200ttB? ? ?????????????????????????? 200c o ss i ns i nc o sddddtttttttt BAA?????????????????? ?????ttttt2001s i nc o sc o ss i nBA ??驗(yàn)證 ???????? ???? ttt tttt c o ss in s inc o sdd 2???????? ???? ttt tt s in2c o s c o ss in1? ?dd ? ?t A B A B? ? ????????? ????ttttts in2c o sc o ss in1???????? ????tttts inc o sc o ss in B?????????? t20 01Friday, November 12, 2021 理論力學(xué) CAI 數(shù)學(xué)基礎(chǔ) 30 ? 例已知 矩陣 /導(dǎo)數(shù) /對(duì)時(shí)間的導(dǎo)數(shù) ? ? ???????????????????????? 200c o ss i ns i nc o sddddtttttttt AB???????????????? ?????????????????? ?????ttttttttttt2001c o ss i ns i nc o s00s i nc o sc o ss i n2BABA ??? ?dd ? ?t AB A B A B? ??????????? ttttc o ss ins inc o sA?????????200ttB驗(yàn)證 ???????????????????????? ttttttttttttttttttttttt s i nc o s2c o ss i nc o ss i n2s i nc o sc o ss i ns i nc o sdd2222??????????????ttttttttttttts i nc o s2c o ss i nc o ss i n2s i nc o s22????????????tttts inc o sc o ss inA?????????? t20 01B???????????????????????ttttttttttttttc o s2s i ns i n2c o ss i nc o sc o ss i n22??????????????tttttttttttttts i nc o s2c o ss i nc o ss i n2s i nc o s22Friday, November 12, 2021 理論力學(xué) CAI 數(shù)學(xué)基礎(chǔ) 31 矩陣對(duì)變量的導(dǎo)數(shù) ? 標(biāo)量函數(shù)對(duì)變量的偏導(dǎo)數(shù) ? 標(biāo)量函數(shù)陣對(duì)變量的偏導(dǎo)數(shù) 矩陣 /導(dǎo)數(shù) /對(duì)變量的的偏導(dǎo)數(shù) Friday, November 12, 2021 理論力學(xué) CAI 數(shù)學(xué)基礎(chǔ) 32 ? 標(biāo)量函數(shù)對(duì)變量的偏導(dǎo)數(shù) – 多變量的標(biāo)量函數(shù) (多元函數(shù) ) 矩陣 /導(dǎo)數(shù) /對(duì)變量的偏導(dǎo)數(shù) 對(duì)于一組（ n個(gè)）變量，通常引入一 n 階列矩陣表示這組變量，即 ? ? T21 nqqq ??q如果有一個(gè)標(biāo)量 a，它是這組變量的函數(shù) (n元函數(shù) ) ，記為 )(),( 21 qaqqqaa n ?? ?21 2c o ss i n)( ???? qaa[例 ]對(duì)于二元函數(shù) 21 2c o ss in ???a? ?T21 ???q引入二階變量陣 標(biāo)量函數(shù) a可表為 Friday, November 12, 2021 理論力學(xué) CAI 數(shù)學(xué)基礎(chǔ) 33 ? 標(biāo)量函數(shù)對(duì)變量的偏導(dǎo)數(shù) – 多變量函數(shù) a 對(duì) n 階變量陣 q 的偏導(dǎo)數(shù)為一 n 階 行陣 – 其元素分別為該標(biāo)量函數(shù)對(duì)各自變量 qj 的偏導(dǎo)數(shù) 矩陣 /導(dǎo)數(shù) /對(duì)變量的偏導(dǎo)數(shù) [例 ]對(duì)于二階變量陣 的標(biāo)量函數(shù) ? ?T21 ???q 21 2c o ss i n)( ???? qaanjnqaqaqaqaaa????????????????????121d e fd e f?????????? ?qq? ?2121212s i ns i n22c osc os ?????????? ??????????? aaaq21?12?Friday, November 12, 2021 理論力學(xué) CAI 數(shù)學(xué)基礎(chǔ) 34 ? 標(biāo)量函數(shù)陣對(duì)變量的偏導(dǎo)數(shù) – 多變量的標(biāo)量函數(shù)陣 矩陣 /導(dǎo)數(shù) /對(duì)變量的偏導(dǎo)數(shù) 對(duì)于有 n個(gè)變量，通常引入一 n 階列矩陣表示這組變量，即 ? ? T21 nqqq ??q如果有一個(gè) m 階標(biāo)量函數(shù)列陣 它的元素是上述變量的函數(shù)，記為 ? ? ? ? T21T21 )()()( qqq mm ?????? ?? ?????????????????????????????????)()()(2121qqqmm ?????????Friday, November 12, 2021 理論力學(xué) CAI 數(shù)學(xué)基礎(chǔ) 35 ? 標(biāo)量函數(shù)陣對(duì)變量的偏導(dǎo)數(shù) – 多變量的標(biāo)量函數(shù)陣 矩陣 /導(dǎo)數(shù) /對(duì)變量的偏導(dǎo)數(shù) [例 ]對(duì)于二階變量陣 有如下的標(biāo)量函數(shù)陣 ? ?T21 ???q? ?? ?? ?? ? ? ? ? ?? ? T21212121212122c o s2c o s2s i n22c o s2c o s2s i n???????????????????????????????? ? T212121)()()(mmm???????????????????????????????????????????qqq?21?13?Friday, November 12, 2021 理論力學(xué) CAI 數(shù)學(xué)基礎(chǔ) 36 ? 標(biāo)量函數(shù)陣對(duì)變量的偏導(dǎo)數(shù) – m 階標(biāo)量函數(shù)列陣對(duì) n 階變量陣 q 的偏導(dǎo)數(shù)定義為一 m n 階矩陣 – 且 矩陣 /導(dǎo)數(shù) /對(duì)變量的偏導(dǎo)數(shù) nmjiq???????????????? d e fd e fqq??q1 q2 qn ?1 ?2 ?m ???????????????qqqmΦΦΦ?21????????????????????