{"id":319,"date":"2023-07-06T06:56:27","date_gmt":"2023-07-06T06:56:27","guid":{"rendered":"https:\/\/mathority.org\/cn\/%e9%9b%85%e5%8f%af%e6%af%94%e7%9f%a9%e9%98%b5-%e9%9b%85%e5%8f%af%e6%af%94%e7%9f%a9%e9%98%b5\/"},"modified":"2023-07-06T06:56:27","modified_gmt":"2023-07-06T06:56:27","slug":"%e9%9b%85%e5%8f%af%e6%af%94%e7%9f%a9%e9%98%b5-%e9%9b%85%e5%8f%af%e6%af%94%e7%9f%a9%e9%98%b5","status":"publish","type":"post","link":"https:\/\/mathority.org\/cn\/%e9%9b%85%e5%8f%af%e6%af%94%e7%9f%a9%e9%98%b5-%e9%9b%85%e5%8f%af%e6%af%94%e7%9f%a9%e9%98%b5\/","title":{"rendered":"\u96c5\u53ef\u6bd4\u548c\u96c5\u53ef\u6bd4\u77e9\u9635"},"content":{"rendered":"<p>\u5728\u6b64\u9875\u9762\u4e0a\uff0c\u60a8\u5c06\u901a\u8fc7\u793a\u4f8b\u4e86\u89e3\u4ec0\u4e48\u662f\u96c5\u53ef\u6bd4\u77e9\u9635\u4ee5\u53ca\u5982\u4f55\u8ba1\u7b97\u5b83\u3002\u6b64\u5916\uff0c\u60a8\u8fd8\u6709\u51e0\u4e2a\u5df2\u89e3\u51b3\u7684\u96c5\u53ef\u6bd4\u77e9\u9635\u7ec3\u4e60\uff0c\u4ee5\u4fbf\u60a8\u8fdb\u884c\u7ec3\u4e60\u3002\u60a8\u8fd8\u5c06\u4e86\u89e3\u4e3a\u4ec0\u4e48\u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u884c\u5217\u5f0f\u96c5\u53ef\u6bd4\u5982\u6b64\u91cd\u8981\u3002\u6700\u540e\uff0c\u6211\u4eec\u89e3\u91ca\u8be5\u77e9\u9635\u4e0e\u5176\u4ed6\u64cd\u4f5c\u4e4b\u95f4\u7684\u5173\u7cfb\u53ca\u5176\u5e94\u7528\u3002<\/p>\n<h2 class=\"wp-block-heading\">\u4ec0\u4e48\u662f\u96c5\u53ef\u6bd4\u77e9\u9635\uff1f<\/h2>\n<p>\u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u5b9a\u4e49\u5982\u4e0b\uff1a<\/p>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"><strong>\u96c5\u53ef\u6bd4\u77e9\u9635<\/strong>\u662f\u7531\u51fd\u6570\u7684\u4e00\u9636\u504f\u5bfc\u6570\u6784\u6210\u7684\u77e9\u9635\u3002<\/p>\n<p>\u56e0\u6b64\u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u516c\u5f0f\u5982\u4e0b\uff1a <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/formule-de-la-matrice-jacobienne.webp\" alt=\"\u96c5\u53ef\u6bd4\u77e9\u9635\u516c\u5f0f\" class=\"wp-image-2796\" width=\"477\" height=\"397\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p>\u56e0\u6b64\uff0c\u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u884c\u6570\u59cb\u7ec8\u4e0e\u6807\u91cf\u51fd\u6570\u7684\u884c\u6570\u4e00\u6837\u591a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c2cded1f0c41f6c4f73404951209deec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f_1,f_2,\\ldots ,f_m)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"114\" style=\"vertical-align: -5px;\"><\/p>\n<p>\u5177\u6709\u8be5\u529f\u80fd\uff0c\u5e76\u4e14\u5217\u6570\u5c06\u5bf9\u5e94\u4e8e\u53d8\u91cf\u7684\u6570\u91cf<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-acf6e9bb52e3b1715802a12e9b93d938_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x_1, x_2, \\ldots , x_n).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"119\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p>\u53e6\u4e00\u65b9\u9762\uff0c\u8fd9\u4e2a\u77e9\u9635\u4e5f\u79f0\u4e3a<em>\u96c5\u53ef\u6bd4\u5fae\u5206<\/em>\u56fe\u6216<em>\u96c5\u53ef\u6bd4\u7ebf\u6027\u56fe<\/em>\u3002\u4e8b\u5b9e\u4e0a\uff0c\u6709\u65f6\u4e5f\u5199\u6210\u5b57\u6bcdD\u800c\u4e0d\u662f\u5b57\u6bcdJ\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a754eb7ba76edc6a0057255d4b17792c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle J_f = D_f\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"64\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p>\u51fa\u4e8e\u597d\u5947\uff0c\u96c5\u53ef\u6bd4\u77e9\u9635\u4ee5\u5361\u5c14\u00b7\u53e4\u65af\u5854\u592b\u00b7\u96c5\u53ef\u6bd4 (Carl Gustav Jacobi) \u7684\u540d\u5b57\u547d\u540d\uff0c\u5361\u5c14\u00b7\u53e4\u65af\u5854\u592b\u00b7\u96c5\u53ef\u6bd4\u662f\u4e00\u4f4d 19 \u4e16\u7eaa\u91cd\u8981\u7684\u6570\u5b66\u5bb6\u548c\u6559\u6388\uff0c\u4e3a\u6570\u5b66\u4e16\u754c\uff0c\u7279\u522b\u662f\u7ebf\u6027\u4ee3\u6570\u9886\u57df\u505a\u51fa\u4e86\u91cd\u8981\u8d21\u732e\u3002<\/p>\n<h2 class=\"wp-block-heading\">\u8ba1\u7b97\u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u793a\u4f8b<\/h2>\n<p>\u4e00\u65e6\u6211\u4eec\u4e86\u89e3\u4e86\u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u6982\u5ff5\uff0c\u6211\u4eec\u5c06\u901a\u8fc7\u793a\u4f8b\u9010\u6b65\u4e86\u89e3\u5b83\u662f\u5982\u4f55\u8ba1\u7b97\u7684\uff1a<\/p>\n<ul>\n<li>\u786e\u5b9a\u4ee5\u4e0b\u51fd\u6570\u5728\u70b9 (1,2) \u5904\u7684\u96c5\u53ef\u6bd4\u77e9\u9635\uff1a<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f0fc87451aa9fbec94159b7a916880c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x,y)= (x^4 +3y^2x \\ , \\ 5y^2-2xy+1)\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"290\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p>\u6211\u4eec\u9700\u8981\u505a\u7684\u7b2c\u4e00\u4ef6\u4e8b\u662f\u8ba1\u7b97\u51fd\u6570\u7684\u6240\u6709\u4e00\u9636\u504f\u5bfc\u6570\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dec527bbc6852cbe54551e96000f8357_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f_1}{\\partial x} = 4x^3+3y^2\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"124\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-65bf87809e33a1ea0113f875bf6d1187_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f_1}{\\partial y} = 6yx\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"79\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6c8ed9d5e68e04f45efe36891a22c088_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f_2}{\\partial x} = -2y\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"82\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-20a7a54e9162f3f423056d0638effb9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f_2}{\\partial y} = 10y-2x\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"118\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p>\u73b0\u5728\u6211\u4eec\u5e94\u7528\u96c5\u53ef\u6bd4\u77e9\u9635\u516c\u5f0f\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u8be5\u51fd\u6570\u6709\u4e24\u4e2a\u53d8\u91cf\u548c\u4e24\u4e2a\u6807\u91cf\u51fd\u6570\uff0c\u56e0\u6b64\u96c5\u53ef\u6bd4\u77e9\u9635\u5c06\u662f\u7ef4\u5ea6\u4e3a 2\u00d72 \u7684\u65b9\u9635\uff1a <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/matrice-jacobienne-22152-avec-deux-variables-et-2-fonctions-scalaires-1.webp\" alt=\"\" class=\"wp-image-2821\" width=\"504\" height=\"124\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p>\u4e00\u65e6\u6211\u4eec\u6709\u4e86\u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u8868\u8fbe\u5f0f\uff0c\u6211\u4eec\u5c31\u5728\u70b9 (1,2) \u5904\u8ba1\u7b97\u5b83\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fa6ed35890b94e3abe43b9a3f9674e36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle J_f(1,2)=\\begin{pmatrix} 4\\cdot 1^3+3\\cdot 2^2 &amp; 6\\cdot 2 \\cdot 1 \\\\[3ex] -2\\cdot 2 &amp; 10\\cdot 2-2 \\cdot 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"313\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u6700\u540e\uff0c\u6211\u4eec\u8fdb\u884c\u8fd0\u7b97\uff0c\u5f97\u5230\u89e3\uff1a <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exemple-de-calcul-de-la-matrice-jacobienne.webp\" alt=\"\u8ba1\u7b97\u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u793a\u4f8b\" class=\"wp-image-2823\" width=\"209\" height=\"71\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p>\u4e00\u65e6\u60a8\u4e86\u89e3\u4e86\u5982\u4f55\u6c42\u51fd\u6570\u7684\u96c5\u53ef\u6bd4\u77e9\u9635\uff0c\u6211\u4eec\u5c31\u4f1a\u7ed9\u60a8\u7559\u4e0b\u51e0\u4e2a\u9010\u6b65\u89e3\u51b3\u7684\u7ec3\u4e60\uff0c\u4ee5\u4fbf\u60a8\u8fdb\u884c\u7ec3\u4e60\u3002<\/p>\n<h2 class=\"wp-block-heading\">\u89e3\u51b3\u4e86\u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u95ee\u9898<\/h2>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e601<\/h3>\n<p>\u6c42\u4ee5\u4e0b 2 \u4e2a\u53d8\u91cf\u5411\u91cf\u51fd\u6570\u5728\u70b9 (0,-2) \u5904\u7684\u96c5\u53ef\u6bd4\u77e9\u9635\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-210a9fdec3d1430dd17f52f91fb8a5fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x,y)= (e^{xy}+y \\ , \\ y^2x)\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"190\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u8be5\u51fd\u6570\u6709\u4e24\u4e2a\u53d8\u91cf\u548c\u4e24\u4e2a\u6807\u91cf\u51fd\u6570\uff0c\u56e0\u6b64\u96c5\u53ef\u6bd4\u77e9\u9635\u5c06\u662f\u5927\u5c0f\u4e3a 2\u00d72 \u7684\u65b9\u9635\uff1a <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-de-la-matrice-jacobienne.webp\" alt=\"\u6c42\u89e3\u96c5\u53ef\u6bd4\u77e9\u9635\u7ec3\u4e60\" class=\"wp-image-2840\" width=\"511\" height=\"124\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\">\u4e00\u65e6\u6211\u4eec\u8ba1\u7b97\u4e86\u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u8868\u8fbe\u5f0f\uff0c\u6211\u4eec\u5c31\u5728\u70b9 (0,-2) \u5904\u5bf9\u5176\u6c42\u503c\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4f6008d8799a0a1c3a667e958d6c8818_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle J_f(0,-2)=\\begin{pmatrix}e^{0\\cdot (-2)}\\cdot (-2)\\phantom{5} &amp; \\phantom{5}e^{0\\cdot (-2)} \\cdot 0 +1 \\\\[4ex](-2)^2 &amp; 2\\cdot (-2) \\cdot 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"75\" width=\"352\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6700\u540e\uff0c\u6211\u4eec\u8fdb\u884c\u8fd0\u7b97\u5e76\u5f97\u5230\u7ed3\u679c\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5eb37dc494497a424b489235b1a55a5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{J_f(0,-2)}=\\begin{pmatrix} \\bm{-2} &amp; \\bm{1} \\\\[1.5ex] \\bm{4} &amp; \\bm{0} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"166\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e602<\/h3>\n<p>\u8ba1\u7b97\u4ee5\u4e0b\u5177\u6709 2 \u4e2a\u53d8\u91cf\u7684\u51fd\u6570\u5728\u70b9 (2,-1) \u5904\u7684\u96c5\u53ef\u6bd4\u77e9\u9635\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cf0b9acb2545e2e42f1e63a0edb5a7c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x,y)= (x^3y^2 - 5x^2y^2 \\ , \\ y^6-3y^3x+7)\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"314\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u8be5\u51fd\u6570\u6709\u4e24\u4e2a\u53d8\u91cf\u548c\u4e24\u4e2a\u6807\u91cf\u51fd\u6570\uff0c\u56e0\u6b64\u96c5\u53ef\u6bd4\u77e9\u9635\u5c06\u662f 2 \u9636\u65b9\u9635\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-48baf447fc5a448f30f13295f96cb874_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle J_f(x,y)=\\begin{pmatrix}\\cfrac{\\phantom{5}\\partial f_1}{\\partial x}\\phantom{5} &amp; \\phantom{5}\\cfrac{\\partial f_1}{\\partial y}\\phantom{5} \\\\[3ex] \\cfrac{\\partial f_2}{\\partial x} &amp; \\cfrac{\\partial f_2}{\\partial y}\\end{pmatrix} = \\begin{pmatrix} \\vphantom{\\cfrac{\\partial f_2}{\\partial x}}3x^2y^2-10xy^2&amp; 2x^3y-10x^2y \\\\[3ex] \\vphantom{\\cfrac{\\partial f_2}{\\partial x}} -3y^3 &amp; 6y^5-9y^2x \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"101\" width=\"502\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4e00\u65e6\u627e\u5230\u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u8868\u8fbe\u5f0f\uff0c\u6211\u4eec\u5c31\u5728\u70b9 (2,-1) \u5904\u5bf9\u5176\u6c42\u503c\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4f2ee2de8e72eed6956f784628353547_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle J_f(2,-1)=\\begin{pmatrix} 3\\cdot 2^2\\cdot (-1)^2-10\\cdot 2 \\cdot (-1)^2\\phantom{5} &amp; \\phantom{5}2\\cdot 2^3\\cdot (-1)-10\\cdot 2^2\\cdot (-1) \\\\[4ex] -3(-1)^3 &amp; 6\\cdot (-1)^5-9\\cdot (-1)^2\\cdot 2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"74\" width=\"573\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6700\u540e\uff0c\u6211\u4eec\u8fdb\u884c\u8fd0\u7b97\u5e76\u5f97\u5230\u7ed3\u679c\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7935318698eadf3d3af4f87e6e8f2629_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{J_f(1,2)}=\\begin{pmatrix} \\bm{-8} &amp; \\bm{24} \\\\[1.5ex] \\bm{3} &amp; \\bm{-24} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"175\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e603<\/h3>\n<p>\u786e\u5b9a\u4ee5\u4e0b\u5177\u6709 3 \u4e2a\u53d8\u91cf\u7684\u51fd\u6570\u5728\u70b9 (2,-2,2) \u5904\u7684\u96c5\u53ef\u6bd4\u77e9\u9635\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d63b0e9c8ecd37b9f100a46233ef9d48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x,y,z)= \\left(z\\tan (x^2-y^2) \\ , \\ xy\\ln \\left( \\frac{z}{2} \\right)\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"313\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u8be5\u51fd\u6570\u5177\u6709\u4e09\u4e2a\u53d8\u91cf\u548c\u4e24\u4e2a\u6807\u91cf\u51fd\u6570\uff0c\u56e0\u6b64\u96c5\u53ef\u6bd4\u77e9\u9635\u5c06\u662f\u4e00\u4e2a\u7ef4\u5ea6\u4e3a 2\u00d73 \u7684\u77e9\u5f62\u77e9\u9635\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5b327537a2e4c80c7eb38d56d94bb141_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle J_f(x,y,z)= \\begin{pmatrix}\\cfrac{\\phantom{5}\\partial f_1}{\\partial x}\\phantom{5} &amp; \\phantom{5}\\cfrac{\\partial f_1}{\\partial y}\\phantom{5} &amp; \\phantom{5}\\cfrac{\\partial f_1}{\\partial z}\\phantom{5} \\\\[3ex] \\cfrac{\\partial f_2}{\\partial x} &amp; \\cfrac{\\partial f_2}{\\partial y} &amp;\\cfrac{\\partial f_2}{\\partial z}\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"101\" width=\"297\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-de-la-matrice-jacobienne-resolu-avec-3-variables.webp\" alt=\"\u96c5\u53ef\u6bd4\u77e9\u9635\u6c42\u89e3 3 \u4e2a\u53d8\u91cf\u7684\u7ec3\u4e60\" class=\"wp-image-2870\" width=\"798\" height=\"121\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\">\u4e00\u65e6\u6211\u4eec\u6709\u4e86\u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u8868\u8fbe\u5f0f\uff0c\u6211\u4eec\u5c31\u5728\u70b9 (2,-2,2) \u5904\u5bf9\u5176\u6c42\u503c\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a62dd1b4655e9d089404028ec48fbe11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle J_f(2,-2,2)= \\begin{pmatrix} \\vphantom{\\cfrac{\\partial f_2}{\\partial x}}2\\bigl(1+\\tan^2 (2^2-(-2)^2)\\bigr) \\cdot 2\\cdot 2 &amp; 2\\bigl(1+\\tan^2 (2^2-(-2)^2)\\bigr) \\cdot (-2\\cdot (-2)) &amp; \\tan (2^2-(-2)^2)\\\\[3ex] \\vphantom{\\cfrac{\\partial f_2}{\\partial x}} \\displaystyle -2\\ln \\left( \\frac{2}{2} \\right) &amp; \\displaystyle 2\\ln \\left( \\frac{2}{2} \\right) &amp;\\displaystyle \\frac{2\\cdot (-2)}{2} \\right)\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"106\" width=\"804\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u8fdb\u884c\u8ba1\u7b97\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-05c8aaa8cca0f4cb652c95b11d2e9db1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle J_f(2,-2,2)= \\begin{pmatrix} \\vphantom{\\cfrac{\\partial f_2}{\\partial x}}2\\bigl(1+\\tan^2 (0)\\bigr) \\cdot 4 \\phantom{5} &amp; 2\\bigl(1+\\tan^2 (0)\\bigr) \\cdot 4 &amp; \\phantom{5}\\tan (0)\\\\[3ex] \\vphantom{\\cfrac{\\partial f_2}{\\partial x}} -2\\cdot 0 &amp;  2\\cdot 0 &amp;-2 \\right)\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"95\" width=\"506\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u800c\u6211\u4eec\u7ee7\u7eed\u8fd0\u7b97\uff0c\u76f4\u5230\u4e0d\u80fd\u518d\u7b80\u5316\u4e3a\u6b62\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d2b4fda9837a6287456ca469d46a2382_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{J_f(2,-2,2)=} \\begin{pmatrix}\\bm{8} &amp; \\bm{8} &amp; \\bm{0} \\\\[2ex]   \\bm{0} &amp; \\bm{0} &amp;\\bm{-2} \\right)\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"208\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e604<\/h3>\n<p>\u786e\u5b9a\u8be5\u70b9\u7684\u96c5\u53ef\u6bd4\u77e9\u9635<strong> <\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-636bdeb4752c0344a75d5969fb84917a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\pi, \\pi)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"41\" style=\"vertical-align: -5px;\"><\/p>\n<p>\u4ee5\u4e0b\u591a\u53d8\u91cf\u51fd\u6570\u7684\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a9b36f1402d71e330215cbee0169a58f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x,y)= \\left( \\frac{\\cos (x-y)}{x} \\ , \\ e^{x^2-y^2} \\ , \\ x^3\\sin (2y) \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"343\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u8be5\u51fd\u6570\u6709\u4e24\u4e2a\u53d8\u91cf\u548c\u4e09\u4e2a\u6807\u91cf\u51fd\u6570\uff0c\u56e0\u6b64\u96c5\u53ef\u6bd4\u77e9\u9635\u5c06\u662f\u4e00\u4e2a\u7ef4\u5ea6\u4e3a 3\u00d72 \u7684\u77e9\u5f62\u77e9\u9635\uff1a <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-etape-par-etape-de-jacobian-matrix-32152-1.webp\" alt=\"\u9010\u6b65\u89e3\u51b3\u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u7ec3\u4e60\" class=\"wp-image-2886\" width=\"704\" height=\"192\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\">\u4e00\u65e6\u6211\u4eec\u6709\u4e86\u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u8868\u8fbe\u5f0f\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5c06\u5176\u8ba1\u7b97\u4e3a\u70b9<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fbf56ffad010b2211cd457a72a08d870_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\pi, \\pi):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"51\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-167caa7a7d1cb34db33f7b92e21b5f78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle J_f(\\pi,\\pi)= \\begin{pmatrix} \\displaystyle \\vphantom{\\cfrac{\\partial f_3}{\\partial y}}\\frac{-\\sin(\\pi-\\pi)\\pi-\\cos(\\pi-\\pi)}{\\pi^2} &amp; \\displaystyle\\frac{\\sin (\\pi- \\pi)}{\\pi} \\\\[3ex] \\vphantom{\\cfrac{\\partial f_3}{\\partial y}}2\\pi e^{\\pi^2-\\pi^2} &amp; -2\\pi e^{\\pi^2-\\pi^2} \\\\[3ex] \\vphantom{\\cfrac{\\partial f_3}{\\partial y}} 3\\pi^2\\sin(2\\pi) &amp; \\pi^3 \\cos(2\\pi)\\cdot 2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"159\" width=\"440\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u8fdb\u884c\u4ee5\u4e0b\u64cd\u4f5c\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b05c5bfee3f874f3adec324a6bc9b43e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle J_f(\\pi,\\pi)= \\begin{pmatrix} \\displaystyle \\vphantom{\\cfrac{\\partial f_3}{\\partial y}}\\displaystyle\\frac{-0-1}{\\pi^2} &amp; \\displaystyle\\frac{0}{\\pi} \\\\[3ex] \\vphantom{\\cfrac{\\partial f_3}{\\partial y}}2\\pi e^{0} &amp; -2\\pi e^{0} \\\\[3ex] \\vphantom{\\cfrac{\\partial f_3}{\\partial y}} 3\\pi^2\\cdot 0 &amp; \\pi^3 \\cdot 1 \\cdot 2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"159\" width=\"246\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u56e0\u6b64\uff0c\u5728\u6240\u8003\u8651\u7684\u70b9\u5904\uff0c\u5411\u91cf\u51fd\u6570\u7684\u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u503c\u4e3a\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f4addee61e4664b95dbb049be217af34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{J_f(\\pi,\\pi)=} \\begin{pmatrix}\\displaystyle -\\frac{\\bm{1}}{\\bm{\\pi^2}} &amp; \\bm{0} \\\\[3ex] \\bm{2\\pi} &amp; \\bm{-2\\pi}\\\\[3ex]\\bm{0} &amp; \\bm{2\\pi^3} \\right)\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"119\" width=\"195\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e605<\/h3>\n<p>\u8ba1\u7b97\u8be5\u70b9\u7684\u96c5\u53ef\u6bd4\u77e9\u9635<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-31e57c38165a2810230c0ca075aefde9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(3,0,\\pi)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -5px;\"><\/p>\n<p>\u4ee5\u4e0b\u5177\u6709 3 \u4e2a\u53d8\u91cf\u7684\u51fd\u6570\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d7841aa7ac90f968d460391c8d1529ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x,y,z)= \\left(xe^{2y}\\cos(-z) \\ , \\ (y-2)^3\\cdot \\sin\\left(\\frac{z}{2}\\right)  \\ , \\ e^{2y}\\cdot \\ln\\left(\\frac{x}{3}\\right) \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"472\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u51fd\u6570\u7531\u4e09\u4e2a\u53d8\u91cf\u548c\u4e09\u4e2a\u6807\u91cf\u51fd\u6570\u7ec4\u6210\uff0c\u56e0\u6b64\uff0c\u96c5\u53ef\u6bd4\u77e9\u9635\u5c06\u662f\u7ef4\u5ea6\u4e3a 3\u00d73 \u7684\u65b9\u9635\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bfd9dcbb1d4961906d5b8581f70f5392_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle J_f(x,y,z)=\\begin{pmatrix}\\phantom{5}\\cfrac{\\partial f_1}{\\partial x}\\phantom{5} &amp; \\phantom{5}\\cfrac{\\partial f_1}{\\partial y}\\phantom{5} &amp; \\phantom{5}\\cfrac{\\partial f_1}{\\partial z}\\phantom{5} \\\\[3ex] \\cfrac{\\partial f_2}{\\partial x} &amp; \\cfrac{\\partial f_2}{\\partial y} &amp; \\cfrac{\\partial f_2}{\\partial z} \\\\[3ex] \\cfrac{\\partial f_3}{\\partial x} &amp; \\cfrac{\\partial f_3}{\\partial y} &amp; \\cfrac{\\partial f_3}{\\partial z}\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"161\" width=\"297\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-de-la-matrice-jacobienne-32153-avec-3-variables.webp\" alt=\"\u89e3\u51b3\u4e86\u5177\u6709 3 \u4e2a\u53d8\u91cf\u7684 3x3 \u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u7ec3\u4e60\" class=\"wp-image-2937\" width=\"629\" height=\"191\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\">\u4e00\u65e6\u6211\u4eec\u627e\u5230\u4e86\u96c5\u53ef\u6bd4\u77e9\u9635\uff0c\u6211\u4eec\u5c31\u5728\u8be5\u70b9\u5bf9\u5176\u8fdb\u884c\u8bc4\u4f30<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-696ca943dc60b9e55cbc96d72e3c0c19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(3,0,\\pi):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4f56df32b7632d1e74f014f0aab2b52a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle J_f(3,0,\\pi)= \\begin{pmatrix} \\vphantom{\\cfrac{\\partial f_2}{\\partial x}} e^{2\\cdot 0}\\cos(-\\pi) &amp; 2\\cdot 3e^{2\\cdot 0}\\cos(-\\pi) &amp; 3e^{2\\cdot 0}\\sin(-\\pi) \\\\[3ex] \\vphantom{\\cfrac{\\partial f_2}{\\partial x}} 0 &amp; \\displaystyle 3(0-2)^2\\cdot \\sin\\left(\\frac{\\pi}{2}\\right) &amp; \\displaystyle\\frac{1}{2}(0-2)^3\\cdot \\cos\\left(\\frac{\\pi}{2}\\right)\\\\[3ex] \\vphantom{\\cfrac{\\partial f_2}{\\partial x}}\\displaystyle\\frac{e^{2\\cdot 0}}{3} &amp;\\displaystyle 2e^{2\\cdot 0}\\cdot \\ln\\left(\\frac{3}{3}\\right) &amp; 0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"162\" width=\"542\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u8ba1\u7b97\u64cd\u4f5c\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5771c5e1c54eabf6df6633abd5f3e194_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle J_f(3,0,\\pi)= \\begin{pmatrix} \\vphantom{\\cfrac{\\partial f_2}{\\partial x}} 1\\cdot (-1) &amp; 6\\cdot 1\\cdot (-1) &amp; 3\\cdot 1 \\cdot 0 \\\\[3ex] \\vphantom{\\cfrac{\\partial f_2}{\\partial x}} 0 &amp; \\displaystyle 3\\cdot 4 \\cdot 1 &amp; \\displaystyle\\frac{1}{2}\\cdot (-8)\\cdot 0\\\\[3ex] \\vphantom{\\cfrac{\\partial f_2}{\\partial x}}\\displaystyle\\frac{1}{3} &amp;\\displaystyle 2\\cdot 1\\cdot  0 &amp; 0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"159\" width=\"380\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4e14\u8be5\u70b9\u7684\u96c5\u53ef\u6bd4\u77e9\u9635\u7ed3\u679c\u4e3a\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6dc1884b96ce985e1475c5cfcba2fff8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{J_f(3,0,\\pi)=} \\begin{pmatrix} \\vphantom{\\cfrac{\\partial f_2}{\\partial x}} \\bm{-1} &amp; \\bm{-6} &amp; \\phantom{-}\\bm{0} \\\\[2ex]  \\bm{0} &amp; \\bm{12} &amp; \\displaystyle \\bm{0} \\\\[2ex] \\displaystyle \\frac{\\bm{1}}{\\bm{3}} &amp;\\bm{0}&amp; \\bm{0}\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"121\" width=\"224\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-118\"><\/div>\n<\/div>\n<h2 class=\"wp-block-heading\">\u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u884c\u5217\u5f0f\uff1a\u96c5\u53ef\u6bd4<\/h2>\n<p>\u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u884c\u5217\u5f0f\u79f0\u4e3a<strong>\u96c5\u53ef\u6bd4<\/strong>\u884c\u5217\u5f0f\u6216\u96c5\u53ef\u6bd4\u884c\u5217\u5f0f\u3002\u5fc5\u987b\u8003\u8651\u5230\uff0c\u53ea\u6709\u5f53\u51fd\u6570\u4e0e\u6807\u91cf\u51fd\u6570\u5177\u6709\u76f8\u540c\u6570\u91cf\u7684\u53d8\u91cf\u65f6\uff0c\u624d\u80fd\u8ba1\u7b97\u96c5\u53ef\u6bd4\u77e9\u9635\uff0c\u56e0\u4e3a\u8fd9\u6837\u96c5\u53ef\u6bd4\u77e9\u9635\u5c06\u5177\u6709\u4e0e\u5217\u6570\u76f8\u540c\u7684\u884c\u6570\uff0c\u56e0\u6b64\u5b83\u5c06\u662f\u4e00\u4e2a\u65b9\u9635\u77e9\u9635\u3002 \u3002<\/p>\n<h3 class=\"wp-block-heading\">\u96c5\u53ef\u6bd4\u884c\u5217\u5f0f\u7684\u4f8b\u5b50<\/h3>\n<p>\u8ba9\u6211\u4eec\u770b\u4e00\u4e2a\u8ba1\u7b97\u5177\u6709\u4e24\u4e2a\u53d8\u91cf\u7684\u51fd\u6570\u7684\u96c5\u53ef\u6bd4\u884c\u5217\u5f0f\u7684\u793a\u4f8b\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a23f671a7521885bf05872fbc353e7fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x,y)= (x^2-y^2 \\ , \\ 2xy)\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"193\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p>\u6211\u4eec\u9996\u5148\u8ba1\u7b97\u51fd\u6570\u7684\u96c5\u53ef\u6bd4\u77e9\u9635\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5870e75f368ea3e554b2fa32cfa554dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle J_f(x,y)=\\begin{pmatrix}\\cfrac{\\phantom{5}\\partial f_1}{\\partial x}\\phantom{5} &amp; \\phantom{5}\\cfrac{\\partial f_1}{\\partial y}\\phantom{5} \\\\[3ex] \\cfrac{\\partial f_2}{\\partial x} &amp; \\cfrac{\\partial f_2}{\\partial y}\\end{pmatrix} = \\begin{pmatrix} \\vphantom{\\cfrac{\\partial f_2}{\\partial x}}2x \\phantom{5}&amp; -2y \\\\[2ex] \\vphantom{\\cfrac{\\partial f_2}{\\partial x}} 2y &amp; 2x \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"101\" width=\"349\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u73b0\u5728\u6211\u4eec\u6c42\u89e3 2\u00d72 \u77e9\u9635\u7684\u884c\u5217\u5f0f\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6d1ef9df1d4735e3cea235c653714439_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}\\bigl(J_f(x,y)\\bigr) =\\begin{vmatrix} 2x&amp;-2y \\\\[2ex] 2y &amp; 2x \\end{vmatrix} = \\bm{4x^2+4y^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"300\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\">\u96c5\u53ef\u6bd4\u884c\u5217\u5f0f\u548c\u51fd\u6570\u7684\u53ef\u9006\u6027<\/h2>\n<p>\u73b0\u5728\u60a8\u5df2\u7ecf\u4e86\u89e3\u4e86\u96c5\u53ef\u6bd4\u884c\u5217\u5f0f\u7684\u6982\u5ff5\uff0c\u60a8\u53ef\u80fd\u4f1a\u60f3\u2026\u2026\u8fd9\u6709\u4ec0\u4e48\u610f\u4e49\u5462\uff1f<\/p>\n<p>\u90a3\u4e48\uff0c\u96c5\u53ef\u6bd4\u884c\u5217\u5f0f\u7684\u4e3b\u8981\u7528\u9014\u662f\u786e\u5b9a\u51fd\u6570\u662f\u5426\u53ef\u4ee5\u53cd\u8f6c\u3002<strong>\u53cd\u51fd\u6570\u5b9a\u7406<\/strong>\u6307\u51fa\uff0c\u5982\u679c\u96c5\u53ef\u6bd4\u77e9\u9635\uff08\u96c5\u53ef\u6bd4\u884c\u5217\u5f0f\uff09\u7684\u884c\u5217\u5f0f\u4e0d\u4e3a 0\uff0c\u5219\u610f\u5473\u7740\u8be5\u51fd\u6570\u662f\u53ef\u9006\u7684\u3002<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d719077544284d291fe9faf0fbf0a099_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}\\bigl(J_f\\bigr) \\neq 0 \\ \\longrightarrow \\ \\exists \\ f^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"186\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u8fd9\u4e2a\u6761\u4ef6\u662f\u5fc5\u8981\u7684\uff0c\u4f46\u4e0d\u662f\u5145\u5206\u7684\uff0c\u4e5f\u5c31\u662f\u8bf4\uff0c\u5982\u679c\u884c\u5217\u5f0f\u975e\u96f6\uff0c\u6211\u4eec\u53ef\u4ee5\u65ad\u8a00\u77e9\u9635\u53ef\u4ee5\u6c42\u9006\uff0c\u4f46\u662f\uff0c\u5982\u679c\u884c\u5217\u5f0f\u4e3a0\uff0c\u6211\u4eec\u65e0\u6cd5\u77e5\u9053\u662f\u5426\u51fd\u6570\u6709\u4e00\u4e2a\u53cd\u51fd\u6570\u6216No\u3002<\/p>\n<p>\u4f8b\u5982\uff0c\u5728\u524d\u9762\u770b\u5230\u7684\u5982\u4f55\u67e5\u627e\u51fd\u6570\u7684\u96c5\u53ef\u6bd4\u884c\u5217\u5f0f\u7684\u793a\u4f8b\u4e2d\uff0c\u884c\u5217\u5f0f\u7ed9\u51fa<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f2387cef5f9f9d4963e2e311bd672bfd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4x^2+4y^2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"74\" style=\"vertical-align: -4px;\"><\/p>\n<p>\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u65ad\u8a00\u9664\u4e86\u70b9 (0,0) \u4e4b\u5916\uff0c\u8be5\u51fd\u6570\u603b\u662f\u53ef\u4ee5\u53cd\u8f6c\uff0c\u56e0\u4e3a\u8be5\u70b9\u662f\u552f\u4e00\u96c5\u53ef\u6bd4\u884c\u5217\u5f0f\u7b49\u4e8e 0 \u7684\u70b9\uff0c\u56e0\u6b64\uff0c\u6211\u4eec\u4e0d\u77e5\u9053\u53cd\u51fd\u6570\u662f\u5426\u5b58\u5728\u4e8e\u8fd9\u4e00\u70b9\u3002 <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-119\"><\/div>\n<\/div>\n<h2 class=\"wp-block-heading\">\u96c5\u53ef\u6bd4\u77e9\u9635\u4e0e\u5176\u4ed6\u8fd0\u7b97\u7684\u5173\u7cfb<\/h2>\n<p>\u96c5\u53ef\u6bd4\u77e9\u9635\u4e0e\u51fd\u6570\u7684\u68af\u5ea6\u548cHessian\u77e9\u9635\u6709\u5173\uff1a<\/p>\n<h3 class=\"wp-block-heading\">\u5761<\/h3>\n<p>\u5982\u679c\u51fd\u6570\u662f\u6807\u91cf\u51fd\u6570\uff0c\u96c5\u53ef\u6bd4\u77e9\u9635\u5c06\u662f\u884c\u77e9\u9635\uff0c\u76f8\u5f53\u4e8e<strong>\u68af\u5ea6<\/strong>\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c493f1d8b149a2ed4710288031d7be71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f: \\mathbb{R}^n \\to \\mathbb{R}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"88\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-278699b80ce58cadb5c056b945483637_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle J_f = \\nabla f = \\begin{pmatrix}\\phantom{5} \\cfrac{\\partial f}{\\partial x_1} \\phantom{5}&amp; \\cfrac{\\partial f}{\\partial x_2}&amp; \\dots &amp; \\cfrac{\\partial f}{\\partial x_n}\\phantom{5} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"304\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\">\u6d77\u68ee\u77e9\u9635<\/h3>\n<p>\u51fd\u6570\u68af\u5ea6\u7684\u96c5\u53ef\u6bd4\u77e9\u9635\u7b49\u4e8e<strong>\u6d77\u68ee\u77e9\u9635<\/strong>\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-96ab1054f3c447eedac17f9ce04b4606_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f = J(\\nabla f)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"97\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p> Hessian \u77e9\u9635\u5bf9\u4e8e\u5177\u6709\u591a\u4e2a\u53d8\u91cf\u7684\u51fd\u6570\u7684\u6c42\u5bfc\u6765\u8bf4\u662f\u4e00\u4e2a\u975e\u5e38\u91cd\u8981\u7684\u77e9\u9635\uff0c\u56e0\u4e3a\u5b83\u662f\u7531\u51fd\u6570\u7684\u4e8c\u9636\u5bfc\u6570\u5f62\u6210\u7684\u3002\u4e8b\u5b9e\u4e0a\uff0c\u53ef\u4ee5\u8bf4<a href=\"https:\/\/mathority.org\/cn\/\u7c97\u9ebb\u5e03\u77e9\u9635-\u7c97\u9ebb\u5e03-2x2-3x3\/\">Hessian\u77e9\u9635<\/a>\u662fJacobian\u77e9\u9635\u7684\u8fde\u7eed\u6027\u3002\u4f46\u8fd9\u975e\u5e38\u91cd\u8981\uff0c\u6211\u4eec\u6709\u4e00\u4e2a\u5b8c\u6574\u7684\u9875\u9762\u6765\u8be6\u7ec6\u89e3\u91ca\u5b83\u3002\u6240\u4ee5\u5982\u679c\u4f60\u60f3\u786e\u5207\u5730\u77e5\u9053\u8fd9\u4e2a\u77e9\u9635\u662f\u4ec0\u4e48\u4ee5\u53ca\u4e3a\u4ec0\u4e48\u5b83\u5982\u6b64\u7279\u522b\uff0c\u4f60\u53ef\u4ee5\u70b9\u51fb\u94fe\u63a5\u3002<\/p>\n<h2 class=\"wp-block-heading\">\u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u5e94\u7528<\/h2>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<p>\u9664\u4e86\u6211\u4eec\u6240\u770b\u5230\u7684\u96c5\u53ef\u6bd4\u884c\u5217\u5f0f\uff08\u786e\u5b9a\u51fd\u6570\u662f\u5426\u53ef\u9006\uff09\u7684\u6709\u7528\u6027\u4e4b\u5916\uff0c\u96c5\u53ef\u6bd4\u77e9\u9635\u8fd8\u6709\u5176\u4ed6\u5e94\u7528\u3002<\/p>\n<p>\u96c5\u53ef\u6bd4\u77e9\u9635\u7528\u4e8e\u8ba1\u7b97\u591a\u5143\u51fd\u6570\u7684<strong><span style=\"color:#1976d2;\">\u4e34\u754c\u70b9<\/span><\/strong>\uff0c\u7136\u540e\u901a\u8fc7Hessian\u77e9\u9635\u5c06\u5176\u5206\u7c7b\u4e3a\u6700\u5927\u503c\u3001\u6700\u5c0f\u503c\u6216\u978d\u70b9\u3002\u8981\u627e\u5230\u4e34\u754c\u70b9\uff0c\u60a8\u9700\u8981\u8ba1\u7b97\u51fd\u6570\u7684\u96c5\u53ef\u6bd4\u77e9\u9635\uff0c\u5c06\u5176\u8bbe\u7f6e\u4e3a 0\uff0c\u7136\u540e\u6c42\u89e3\u6240\u5f97\u65b9\u7a0b\u3002<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d0c053381fc5f78d85944f3f431e5537_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle J_f(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"74\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p>\u6b64\u5916\uff0c\u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u53e6\u4e00\u79cd\u5e94\u7528\u662f\u5177\u6709\u591a\u4e2a\u53d8\u91cf\u7684\u51fd\u6570\u79ef\u5206\uff0c\u5373\u4e8c\u91cd\u3001\u4e09\u91cd\u79ef\u5206\u7b49\u3002\u7531\u4e8e\u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u884c\u5217\u5f0f\u5141\u8bb8\u6839\u636e\u4ee5\u4e0b\u516c\u5f0f<span style=\"color:#1976d2;\"><strong>\u6539\u53d8\u591a\u4e2a\u79ef\u5206\u4e2d\u7684\u53d8\u91cf<\/strong><\/span>\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-32b139ab326a616a77e4b30bd6123cea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle x=T(x^*)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"77\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fc466e0a77ffd809702bfbff6981115d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\int_\\Omega  f(x)dx=\\int_{\\Omega^*} f\\bigl(T(x^*)\\bigr)\\cdot \\begin{vmatrix} \\text{det}\\bigl(JT(x^*)\\bigr)\\end{vmatrix} dx^*\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"352\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p>\u5176\u4e2d T \u662f\u5c06\u539f\u59cb\u53d8\u91cf\u4e0e\u65b0\u53d8\u91cf\u76f8\u5173\u8054\u7684\u53d8\u91cf\u53d8\u5316\u51fd\u6570\u3002<\/p>\n<p>\u6700\u540e\uff0c\u96c5\u53ef\u6bd4\u77e9\u9635\u8fd8\u7528\u4e8e\u5bf9\u4efb\u610f\u51fd\u6570\u8fdb\u884c<span style=\"color:#1976d2;\"><strong>\u7ebf\u6027\u903c\u8fd1<\/strong><\/span><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p>\u56f4\u7ed5\u4e00\u4e2a\u70b9<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3bf85f1087e9fbed3a319341134ac1a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"p\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p>\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-65ba36b611b690e470a1f4c464200fbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x) \\approx f(p) + J_f(p)(x-p)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"208\" style=\"vertical-align: -6px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5728\u6b64\u9875\u9762\u4e0a\uff0c\u60a8\u5c06\u901a\u8fc7\u793a\u4f8b\u4e86\u89e3\u4ec0\u4e48\u662f\u96c5\u53ef\u6bd4\u77e9\u9635\u4ee5\u53ca\u5982\u4f55\u8ba1\u7b97\u5b83\u3002\u6b64\u5916\uff0c\u60a8\u8fd8\u6709\u51e0\u4e2a\u5df2\u89e3\u51b3\u7684\u96c5\u53ef\u6bd4\u77e9\u9635\u7ec3\u4e60\uff0c\u4ee5\u4fbf\u60a8\u8fdb\u884c &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" 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