{"id":329,"date":"2023-07-06T06:56:27","date_gmt":"2023-07-06T06:56:27","guid":{"rendered":"https:\/\/mathority.org\/id\/matriks-jacobian-jacobian\/"},"modified":"2023-07-06T06:56:27","modified_gmt":"2023-07-06T06:56:27","slug":"matriks-jacobian-jacobian","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/matriks-jacobian-jacobian\/","title":{"rendered":"Matriks jacobian dan jacobian"},"content":{"rendered":"<p>Di halaman ini Anda akan menemukan apa itu matriks Jacobian dan cara menghitungnya menggunakan contoh. Selain itu, Anda memiliki beberapa latihan matriks Jacobian yang telah diselesaikan sehingga Anda dapat berlatih. Anda juga akan melihat mengapa determinan matriks Jacobian, Jacobian, sangat penting. Terakhir, kami menjelaskan hubungan yang dipertahankan matriks ini dengan operasi lain dan aplikasi yang dimilikinya.<\/p>\n<h2 class=\"wp-block-heading\"> Apa yang dimaksud dengan matriks Jacobian?<\/h2>\n<p> Pengertian matriks Jacobian adalah sebagai berikut:<\/p>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> <strong>Matriks Jacobian<\/strong> adalah matriks yang dibentuk oleh turunan parsial suatu fungsi orde pertama.<\/p>\n<p> Oleh karena itu, rumus matriks Jacobian adalah sebagai berikut: <\/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=\"Rumus matriks Jacobian\" class=\"wp-image-2796\" width=\"477\" height=\"397\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Oleh karena itu, matriks Jacobian akan selalu memiliki baris sebanyak fungsi skalar<\/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> memiliki fungsi, dan jumlah kolom akan sesuai dengan jumlah variabel<\/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> Di sisi lain, matriks ini juga dikenal sebagai peta <em>diferensial Jacobian<\/em> atau <em>peta linier Jacobian<\/em> . Bahkan terkadang juga ditulis dengan huruf D bukan huruf J:<\/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> Sebagai rasa ingin tahu, matriks Jacobian dinamai Carl Gustav Jacobi, seorang matematikawan dan profesor penting abad ke-19 yang memberikan kontribusi penting bagi dunia matematika, khususnya di bidang aljabar linier.<\/p>\n<h2 class=\"wp-block-heading\"> Contoh penghitungan matriks Jacobian<\/h2>\n<p> Setelah kita melihat konsep matriks Jacobian, kita akan melihat langkah demi langkah cara menghitungnya menggunakan contoh:<\/p>\n<ul>\n<li> Tentukan matriks Jacobian pada titik (1,2) dari fungsi berikut:<\/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> Hal pertama yang perlu kita lakukan adalah menghitung semua turunan parsial orde pertama dari fungsi tersebut: <\/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> Sekarang kita terapkan rumus matriks Jacobian. Dalam hal ini fungsi tersebut memiliki dua variabel dan dua fungsi skalar, sehingga matriks Jacobian akan menjadi matriks persegi berdimensi 2\u00d72: <\/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> Setelah kita mendapatkan ekspresi untuk matriks Jacobian, kita mengevaluasinya pada titik (1,2):<\/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> Dan akhirnya, kami melakukan operasi dan mendapatkan solusinya: <\/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=\"contoh penghitungan matriks Jacobian\" class=\"wp-image-2823\" width=\"209\" height=\"71\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Setelah Anda melihat cara mencari matriks Jacobian suatu fungsi, kami memberikan beberapa latihan yang diselesaikan selangkah demi selangkah sehingga Anda dapat berlatih.<\/p>\n<h2 class=\"wp-block-heading\"> Memecahkan masalah matriks Jacobian<\/h2>\n<h3 class=\"wp-block-heading\"> Latihan 1<\/h3>\n<p> Carilah matriks Jacobian di titik (0,-2) fungsi vektor berikut pada 2 variabel: <\/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>lihat solusi<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Fungsi tersebut memiliki dua variabel dan dua fungsi skalar, sehingga matriks Jacobian akan menjadi matriks persegi berukuran 2\u00d72: <\/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=\"menyelesaikan latihan matriks Jacobian\" class=\"wp-image-2840\" width=\"511\" height=\"124\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Setelah kita menghitung ekspresi matriks Jacobian, kita mengevaluasinya pada titik (0,-2):<\/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\"> Dan akhirnya, kami melakukan operasi dan mendapatkan hasilnya: <\/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\">Latihan 2<\/h3>\n<p> Hitung matriks Jacobian di titik (2,-1) fungsi berikut dengan 2 variabel: <\/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>lihat solusi<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Dalam hal ini fungsi tersebut memiliki dua variabel dan dua fungsi skalar, sehingga matriks Jacobian akan menjadi matriks persegi berorde 2:<\/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\"> Setelah kita menemukan ekspresi matriks Jacobian, kita evaluasinya pada titik (2,-1):<\/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\"> Dan akhirnya, kami melakukan operasi dan mendapatkan hasilnya: <\/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\">Latihan 3<\/h3>\n<p> Tentukan matriks Jacobian di titik (2,-2,2) fungsi berikut dengan 3 variabel: <\/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>lihat solusi<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Dalam hal ini fungsi tersebut memiliki tiga variabel dan dua fungsi skalar, sehingga matriks Jacobian akan menjadi matriks persegi panjang berdimensi 2\u00d73: <\/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=\"Matriks Jacobian menyelesaikan latihan 3 variabel\" class=\"wp-image-2870\" width=\"798\" height=\"121\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Setelah kita mendapatkan ekspresi matriks Jacobian, kita evaluasinya pada titik (2,-2,2):<\/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\"> Kami melakukan perhitungan:<\/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\"> Dan kami terus beroperasi hingga tidak dapat disederhanakan lagi: <\/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\"> Latihan 4<\/h3>\n<p> Tentukan matriks Jacobian pada titik tersebut<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>dari fungsi multivariabel berikut: <\/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>lihat solusi<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Dalam hal ini fungsi tersebut memiliki dua variabel dan tiga fungsi skalar, sehingga matriks Jacobian akan menjadi matriks persegi panjang berdimensi 3\u00d72: <\/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=\"latihan diselesaikan selangkah demi selangkah dari matriks Jacobian\" class=\"wp-image-2886\" width=\"704\" height=\"192\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Setelah kita mendapatkan ekspresi untuk matriks Jacobian, kita mengevaluasinya secara langsung <\/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\"> Kami melakukan operasi:<\/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\"> Dengan demikian matriks Jacobian dari fungsi vektor pada titik yang dipertimbangkan bernilai: <\/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\">Latihan 5<\/h3>\n<p> Hitung matriks Jacobian pada titik tersebut<\/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> dari fungsi berikut dengan 3 variabel: <\/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>lihat solusi<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Dalam hal ini fungsinya terdiri dari tiga variabel dan tiga fungsi skalar, sehingga matriks Jacobian akan berupa matriks persegi berdimensi 3\u00d73: <\/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=\"Latihan penyelesaian matriks Jacobian 3x3 dengan 3 variabel\" class=\"wp-image-2937\" width=\"629\" height=\"191\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Setelah kami menemukan matriks Jacobian, kami mengevaluasinya pada saat itu juga <\/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\"> Kami menghitung operasi:<\/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\"> Dan hasil matriks Jacobian pada titik tersebut adalah: <\/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\"> Penentu matriks Jacobian: Jacobian<\/h2>\n<p> Penentu matriks Jacobian disebut determinan <strong>Jacobian<\/strong> atau Jacobian. Harus diingat bahwa Jacobian hanya dapat dihitung jika fungsi tersebut memiliki jumlah variabel yang sama dengan fungsi skalar, karena matriks Jacobian akan memiliki jumlah baris yang sama dengan kolom dan oleh karena itu akan berbentuk persegi. matriks. .<\/p>\n<h3 class=\"wp-block-heading\"> Contoh Jacobian<\/h3>\n<p> Mari kita lihat contoh penghitungan determinan Jacobian suatu fungsi dengan dua variabel:<\/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> Pertama-tama kita menghitung matriks Jacobian dari fungsi tersebut:<\/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\"> Dan sekarang kita selesaikan determinan matriks 2\u00d72:<\/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\"> Jacobian dan invertibilitas suatu fungsi<\/h2>\n<p> Sekarang setelah Anda melihat konsep Jacobian, Anda mungkin berpikir&#8230; apa gunanya?<\/p>\n<p> Nah, kegunaan utama dari Jacobian adalah untuk menentukan apakah suatu fungsi dapat dibalik. <strong>Teorema fungsi invers<\/strong> menyatakan bahwa jika determinan matriks Jacobian (Jacobian) berbeda dengan 0, berarti fungsi tersebut dapat dibalik.<\/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> Perlu diperhatikan bahwa kondisi ini perlu tetapi tidak cukup, yaitu jika determinannya bukan nol maka kita dapat menyatakan bahwa matriksnya dapat dibalik, namun jika determinannya 0 maka kita tidak dapat mengetahui apakah matriks tersebut. fungsi memiliki invers atau No.<\/p>\n<p> Misalnya, dalam contoh yang terlihat sebelumnya tentang cara mencari Jacobian suatu fungsi, determinannya diberikan<\/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> . Dalam hal ini kita dapat menyatakan bahwa fungsi tersebut selalu dapat dibalik kecuali pada titik (0,0), karena titik ini adalah satu-satunya titik yang determinan Jacobiannya sama dengan nol dan, oleh karena itu, kita tidak mengetahui apakah invers fungsinya ada pada titik ini. <\/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\"> Hubungan matriks Jacobian dengan operasi lain<\/h2>\n<p> Matriks Jacobian berhubungan dengan gradien dan matriks Hessian suatu fungsi:<\/p>\n<h3 class=\"wp-block-heading\"> Lereng<\/h3>\n<p> Jika fungsinya merupakan fungsi skalar, maka matriks Jacobian akan berupa matriks baris yang ekuivalen dengan <strong>gradien<\/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-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\"> Matriks Goni<\/h3>\n<p> Matriks Jacobian dari gradien suatu fungsi sama dengan <strong>matriks Hessian<\/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-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> Matriks Hessian merupakan matriks yang sangat penting untuk menurunkan fungsi dengan lebih dari satu variabel, karena matriks tersebut dibentuk oleh turunan kedua dari fungsi tersebut. Bahkan, bisa dikatakan <a href=\"https:\/\/mathority.org\/id\/matriks-goni-goni-2x2-3x3\/\">matriks Hessian<\/a> merupakan kesinambungan dari matriks Jacobian. Namun sangat penting bagi kita untuk memiliki satu halaman penuh yang menjelaskannya secara detail. Jadi jika Anda ingin mengetahui secara pasti apa itu matriks dan mengapa begitu istimewa, Anda bisa mengklik link tersebut.<\/p>\n<h2 class=\"wp-block-heading\"> Penerapan matriks Jacobian<\/h2>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<p> Selain kegunaan Jacobian yang telah kita lihat, yang menentukan apakah suatu fungsi dapat dibalik, matriks Jacobian memiliki penerapan lain.<\/p>\n<p> Matriks Jacobian digunakan untuk menghitung <strong><span style=\"color:#1976d2;\">titik kritis<\/span><\/strong> suatu fungsi multivariat, yang kemudian diklasifikasikan menjadi titik maxima, minima atau saddle melalui matriks Hessian. Untuk menemukan titik kritis, Anda perlu menghitung matriks Jacobian dari fungsi tersebut, menetapkannya sama dengan 0, dan menyelesaikan persamaan yang dihasilkan.<\/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> Selain itu, penerapan lain dari matriks Jacobian terdapat pada integrasi fungsi dengan lebih dari satu variabel, yaitu pada integral rangkap, integral rangkap tiga, dan sebagainya. Karena determinan matriks Jacobian memungkinkan <span style=\"color:#1976d2;\"><strong>perubahan variabel dalam integral berganda<\/strong><\/span> sesuai dengan rumus berikut:<\/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> Dimana T adalah fungsi perubahan variabel yang menghubungkan variabel awal dengan variabel baru.<\/p>\n<p> Terakhir, matriks Jacobian juga digunakan untuk membuat <span style=\"color:#1976d2;\"><strong>pendekatan linier<\/strong><\/span> terhadap fungsi apa pun<\/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> sekitar suatu titik<\/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> :<\/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>Di halaman ini Anda akan menemukan apa itu matriks Jacobian dan cara menghitungnya menggunakan contoh. Selain itu, Anda memiliki beberapa latihan matriks Jacobian yang telah diselesaikan sehingga Anda dapat berlatih. Anda juga akan melihat mengapa determinan matriks Jacobian, Jacobian, sangat penting. Terakhir, kami menjelaskan hubungan yang dipertahankan matriks ini dengan operasi lain dan aplikasi yang &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/matriks-jacobian-jacobian\/\"> <span class=\"screen-reader-text\">Matriks jacobian dan jacobian<\/span> Selengkapnya &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[52],"tags":[],"class_list":["post-329","post","type-post","status-publish","format-standard","hentry","category-lukisan"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Matriks Jacobian dan Jacobian - Mathority<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/id\/matriks-jacobian-jacobian\/\" \/>\n<meta property=\"og:locale\" content=\"id_ID\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Matriks Jacobian dan Jacobian - Mathority\" \/>\n<meta property=\"og:description\" content=\"Di halaman ini Anda akan menemukan apa itu matriks Jacobian dan cara menghitungnya menggunakan contoh. Selain itu, Anda memiliki beberapa latihan matriks Jacobian yang telah diselesaikan sehingga Anda dapat berlatih. Anda juga akan melihat mengapa determinan matriks Jacobian, Jacobian, sangat penting. Terakhir, kami menjelaskan hubungan yang dipertahankan matriks ini dengan operasi lain dan aplikasi yang &hellip; Matriks jacobian dan jacobian Selengkapnya &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/id\/matriks-jacobian-jacobian\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-06T06:56:27+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/formule-de-la-matrice-jacobienne.webp\" \/>\n<meta name=\"author\" content=\"Tim Mathority\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Ditulis oleh\" \/>\n\t<meta name=\"twitter:data1\" content=\"Tim Mathority\" \/>\n\t<meta name=\"twitter:label2\" content=\"Estimasi waktu membaca\" \/>\n\t<meta name=\"twitter:data2\" content=\"5 menit\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/id\/matriks-jacobian-jacobian\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/id\/matriks-jacobian-jacobian\/\"},\"author\":{\"name\":\"Tim Mathority\",\"@id\":\"https:\/\/mathority.org\/id\/#\/schema\/person\/ea4523caf53a07e2ebf32e306a925b38\"},\"headline\":\"Matriks jacobian dan jacobian\",\"datePublished\":\"2023-07-06T06:56:27+00:00\",\"dateModified\":\"2023-07-06T06:56:27+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/id\/matriks-jacobian-jacobian\/\"},\"wordCount\":1021,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/id\/#organization\"},\"articleSection\":[\"Lukisan\"],\"inLanguage\":\"id\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/id\/matriks-jacobian-jacobian\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/id\/matriks-jacobian-jacobian\/\",\"url\":\"https:\/\/mathority.org\/id\/matriks-jacobian-jacobian\/\",\"name\":\"Matriks Jacobian dan Jacobian - 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Selain itu, Anda memiliki beberapa latihan matriks Jacobian yang telah diselesaikan sehingga Anda dapat berlatih. Anda juga akan melihat mengapa determinan matriks Jacobian, Jacobian, sangat penting. Terakhir, kami menjelaskan hubungan yang dipertahankan matriks ini dengan operasi lain dan aplikasi yang &hellip; Matriks jacobian dan jacobian Selengkapnya &raquo;","og_url":"https:\/\/mathority.org\/id\/matriks-jacobian-jacobian\/","article_published_time":"2023-07-06T06:56:27+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/formule-de-la-matrice-jacobienne.webp"}],"author":"Tim Mathority","twitter_card":"summary_large_image","twitter_misc":{"Ditulis oleh":"Tim Mathority","Estimasi waktu membaca":"5 menit"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/id\/matriks-jacobian-jacobian\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/id\/matriks-jacobian-jacobian\/"},"author":{"name":"Tim Mathority","@id":"https:\/\/mathority.org\/id\/#\/schema\/person\/ea4523caf53a07e2ebf32e306a925b38"},"headline":"Matriks jacobian dan jacobian","datePublished":"2023-07-06T06:56:27+00:00","dateModified":"2023-07-06T06:56:27+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/id\/matriks-jacobian-jacobian\/"},"wordCount":1021,"commentCount":0,"publisher":{"@id":"https:\/\/mathority.org\/id\/#organization"},"articleSection":["Lukisan"],"inLanguage":"id","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathority.org\/id\/matriks-jacobian-jacobian\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/id\/matriks-jacobian-jacobian\/","url":"https:\/\/mathority.org\/id\/matriks-jacobian-jacobian\/","name":"Matriks Jacobian dan Jacobian - Mathority","isPartOf":{"@id":"https:\/\/mathority.org\/id\/#website"},"datePublished":"2023-07-06T06:56:27+00:00","dateModified":"2023-07-06T06:56:27+00:00","breadcrumb":{"@id":"https:\/\/mathority.org\/id\/matriks-jacobian-jacobian\/#breadcrumb"},"inLanguage":"id","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathority.org\/id\/matriks-jacobian-jacobian\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathority.org\/id\/matriks-jacobian-jacobian\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathority.org\/id\/"},{"@type":"ListItem","position":2,"name":"Matriks jacobian dan jacobian"}]},{"@type":"WebSite","@id":"https:\/\/mathority.org\/id\/#website","url":"https:\/\/mathority.org\/id\/","name":"Mathority","description":"Di mana rasa ingin tahu bertemu dengan perhitungan!","publisher":{"@id":"https:\/\/mathority.org\/id\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathority.org\/id\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"id"},{"@type":"Organization","@id":"https:\/\/mathority.org\/id\/#organization","name":"Mathority","url":"https:\/\/mathority.org\/id\/","logo":{"@type":"ImageObject","inLanguage":"id","@id":"https:\/\/mathority.org\/id\/#\/schema\/logo\/image\/","url":"https:\/\/mathority.org\/id\/wp-content\/uploads\/2023\/09\/mathority-logo.png","contentUrl":"https:\/\/mathority.org\/id\/wp-content\/uploads\/2023\/09\/mathority-logo.png","width":703,"height":151,"caption":"Mathority"},"image":{"@id":"https:\/\/mathority.org\/id\/#\/schema\/logo\/image\/"}},{"@type":"Person","@id":"https:\/\/mathority.org\/id\/#\/schema\/person\/ea4523caf53a07e2ebf32e306a925b38","name":"Tim Mathority","image":{"@type":"ImageObject","inLanguage":"id","@id":"https:\/\/mathority.org\/id\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","caption":"Tim Mathority"},"sameAs":["http:\/\/mathority.org\/id"]}]}},"yoast_meta":{"yoast_wpseo_title":"","yoast_wpseo_metadesc":"","yoast_wpseo_canonical":""},"_links":{"self":[{"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/posts\/329","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/comments?post=329"}],"version-history":[{"count":0,"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/posts\/329\/revisions"}],"wp:attachment":[{"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/media?parent=329"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/categories?post=329"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/tags?post=329"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}