欢迎光临散文网 会员登陆 & 注册

多元函数微分学笔记(3)

2023-06-07 13:09 作者:~Sakuno酱  | 我要投稿

今天思考了一下复合函数的二阶偏导数能不能用矩阵的方式表达,结果发现确实可以


z%3Df(y_1%2Cy_2%2C...y_n)%0A

y_i%3Dy_i(x_1%2Cx_2%2C...x_n)

%5Cbegin%7Bbmatrix%7D%0A%5Cfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20x_1%7D%20%26%20%5Cfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20x_2%7D%20%26%20..%20%26%0A%5Cfrac%7B%5Cpartial%20z%7D%7B%5Cpartial%20x_n%20%7D%20%0A%5Cend%7Bbmatrix%7D%20%3D

%5Cbegin%7Bbmatrix%7D%0Af_1'%20%26%20f_2'%20%26%20..%20%26%20f'_n%0A%5Cend%7Bbmatrix%7D%20%0A%5Cbegin%7Bbmatrix%7D%0A%5Cfrac%7B%5Cpartial%20y_1%7D%7B%5Cpartial%20x_1%7D%20%26%20%5Cfrac%7B%5Cpartial%20y_1%7D%7B%5Cpartial%20x_2%7D%20%26%20..%20%26%20%5Cfrac%7B%5Cpartial%20y_1%7D%7B%5Cpartial%20x_n%20%7D%20%5C%5C%0A%5Cfrac%7B%5Cpartial%20y_2%7D%7B%5Cpartial%20x_1%7D%20%26%20%5Cfrac%7B%5Cpartial%20y_2%7D%7B%5Cpartial%20x_2%7D%20%26%20..%20%26%20%5Cfrac%7B%5Cpartial%20y_2%7D%7B%5Cpartial%20x_n%20%7D%20%5C%5C%0A..%20%26%20..%20%26%20..%26%20..%5C%5C%0A%5Cfrac%7B%5Cpartial%20y_n%7D%7B%5Cpartial%20x_1%7D%20%26%20%5Cfrac%7B%5Cpartial%20y_n%7D%7B%5Cpartial%20x_2%7D%20%26%20..%20%26%20%5Cfrac%7B%5Cpartial%20y_n%7D%7B%5Cpartial%20x_n%20%7D%20%5C%5C%0A%5Cend%7Bbmatrix%7D


%5Cfrac%7B%5Cpartial%5E2%20z%7D%7B%5Cpartial%20x_j%20%5Cpartial%20x_i%7D%20%3D%20%5Cfrac%7B%5Cpartial(%5Csum_%7Bk%3D1%7D%5E%7Bn%7Df'_k%20%5Cfrac%7B%5Cpartial%20y_i%7D%7B%5Cpartial%20x_k%7D)%7D%7B%5Cpartial%20x_j%7D

%3D%5Csum_%7Bk%3D1%7D%5E%7Bn%7D%5Cfrac%7B%5Cpartial%20f'_k%7D%7B%5Cpartial%20x_j%7D%5Ccdot%20%5Cfrac%7B%5Cpartial%20y_i%7D%7B%5Cpartial%20x_k%7D%20%2B%5Csum_%7Bk%3D1%7D%5E%7Bn%7Df'_k%5Cfrac%7B%5Cpartial%5E2%20y_i%7D%7B%5Cpartial%20x_j%20%5Cpartial%20x_k%7D

其中

%5Cfrac%7B%5Cpartial%20f'_k%7D%7B%5Cpartial%20x_j%7D%3D%5Csum_%7Bp%3D1%7D%5E%7Bn%7Df''_%7Bkp%7D%5Cfrac%7B%5Cpartial%20y_p%7D%7B%5Cpartial%20x_j%7D

%5Cfrac%7B%5Cpartial%5E2%20z%7D%7B%5Cpartial%20x_j%20%5Cpartial%20x_i%7D%3D

%3D%5Csum_%7Bk%3D1%7D%5E%7Bn%7D%5Cfrac%7B%5Cpartial%20y_i%7D%7B%5Cpartial%20x_k%7D%20%20(%5Csum_%7Bp%3D1%7D%5E%7Bn%7Df''_%7Bkp%7D%5Cfrac%7B%5Cpartial%20y_p%7D%7B%5Cpartial%20x_j%7D)%2B%5Csum_%7Bk%3D1%7D%5E%7Bn%7Df'_k%5Cfrac%7B%5Cpartial%5E2%20y_i%7D%7B%5Cpartial%20x_j%20%5Cpartial%20x_k%7D

%3D%5Cbegin%7Bbmatrix%7D%20%5Cfrac%7B%5Cpartial%20y_i%7D%7B%5Cpartial%20x_1%7D%20%26%20%5Cfrac%7B%5Cpartial%20y_i%7D%7B%5Cpartial%20x_2%7D%20%26%20..%20%26%20%5Cfrac%7B%5Cpartial%20y_i%7D%7B%5Cpartial%20x_n%7D%20%20%20%5Cend%7Bbmatrix%7D%0A%5Cbegin%7Bbmatrix%7D%20%0Af''_%7B11%7D%20%26%20f''_%7B12%7D%20%26%20..%20%26%20f''_%7B1n%7D%20%5C%5C%0Af''_%7B21%7D%20%26%20f''_%7B22%7D%20%26%20..%20%26%20f''_%7B2n%7D%20%5C%5C%0A..%20%26%20..%20%26%20..%20%26%20..%20%5C%5C%0Af''_%7Bn1%7D%20%26%20f''_%7Bn2%7D%20%26%20..%20%26%20f''_%7Bnn%7D%20%5C%5C%0A%5Cend%7Bbmatrix%7D%0A%5Cbegin%7Bbmatrix%7D%20%5Cfrac%7B%5Cpartial%20y_1%7D%7B%5Cpartial%20x_j%7D%20%5C%5C%20%5Cfrac%7B%5Cpartial%20y_2%7D%7B%5Cpartial%20x_j%7D%20%5C%5C%20..%20%5C%5C%5Cfrac%7B%5Cpartial%20y_n%7D%7B%5Cpartial%20x_j%7D%20%20%20%5Cend%7Bbmatrix%7D%20%0A%2B%20%5Cbegin%7Bbmatrix%7D%20f'_1%20%26%20f'_2%20%26%20..%20%26%20f'_n%20%20%20%5Cend%7Bbmatrix%7D%20%0A%5Cbegin%7Bbmatrix%7D%20%0A%5Cfrac%7B%5Cpartial%5E2%20y_i%7D%7B%5Cpartial%20x_j%20%5Cpartial%20x_1%7D%20%5C%5C%0A%5Cfrac%7B%5Cpartial%5E2%20y_i%7D%7B%5Cpartial%20x_j%20%5Cpartial%20x_2%7D%20%5C%5C%20%0A..%20%5C%5C%0A%5Cfrac%7B%5Cpartial%5E2%20y_i%7D%7B%5Cpartial%20x_j%20%5Cpartial%20x_n%7D%20%5C%5C%0A%5Cend%7Bbmatrix%7D

多元函数微分学笔记(3)的评论 (共 条)

分享到微博请遵守国家法律