{"id":433,"date":"2017-08-17T08:43:20","date_gmt":"2017-08-17T08:43:20","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=433"},"modified":"2017-12-17T13:43:15","modified_gmt":"2017-12-17T13:43:15","slug":"liuxingjianyingshedejacobiyuyingshedu","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=433","title":{"rendered":"\u6d41\u5f62\u95f4\u6620\u5c04\u7684Jacobian\u4e0e\u6620\u5c04\u5ea6"},"content":{"rendered":"

\u5047\u8bbe$\\phi\\mathpunct{:}(M,g)\\to(N,h)$\u662f\u4e24\u4e2a$n$\u7ef4\u9ece\u66fc\u6d41\u5f62\u95f4\u7684\u5149\u6ed1\u6620\u7167\u3002\u6211\u4eec\u7684\u76ee\u6807\u662f\u6765\u5b9a\u4e49\u8be5\u6620\u7167\u7684Jacobian\u3002\u8fd9\u5bf9\u5b9a\u4e49\u6620\u7167\u7684\u5ea6<\/span>\u662f\u975e\u5e38\u91cd\u8981\u7684\u3002<\/p>\n

\u8be5\u5b9a\u4e49\u6700\u91cd\u8981\u7684\u662f\u4e0e\u6d41\u5f62\u4e0a\u7684\u5750\u6807\u9009\u53d6\u65e0\u5173\u3002\u4e3a\u6b64\u5047\u8bbe$M$\u4e0a\u6709\u4e24\u5957\u5750\u6807\u7cfb\u4e14\u5176\u8f6c\u5316\u51fd\u6570\u4e3a$\\Phi\\mathpunct{:}(x^1,x^2,\\ldots,x^n)\\to(y^1,y^2,\\ldots,y^n)$. \u7c7b\u4f3c\u5730\uff0c$N$\u4e0a\u4e5f\u6709\u4e24\u5957\u5750\u6807\u7cfb\u4e14\u5176\u8f6c\u5316\u51fd\u6570\u4e3a$\\Psi\\mathpunct{:}(u^1,u^2,\\ldots,u^n)\\to(v^1,v^2,\\ldots,v^n)$. \u8fd9\u6837\uff0c \u6211\u4eec\u77e5\u9053\u5ea6\u91cf\u7684\u5c40\u90e8\u8868\u793a\u4e3a\uff1a
\n\\begin{align*}
\n\\tilde g_{kl}&=g\\left(\\frac{\\partial}{\\partial y^k},\\frac{\\partial}{\\partial y^l}\\right),\\quad\\tilde h_{\\alpha\\beta}=h\\left(\\frac{\\partial}{\\partial v^\\alpha},\\frac{\\partial}{\\partial v^\\beta}\\right),\\\\
\ng_{ij}&=g\\left(\\frac{\\partial}{\\partial x^i},\\frac{\\partial}{\\partial x^j}\\right)
\n=g\\left(\\frac{\\partial y^k}{\\partial x^i}\\frac{\\partial}{\\partial y^k},\\frac{\\partial y^l}{\\partial x^j}\\frac{\\partial}{\\partial y^l}\\right)=\\frac{\\partial y^k}{\\partial x^i}\\frac{\\partial y^l}{\\partial x^j}\\tilde g_{kl},\\\\
\nh_{\\alpha\\beta}&=\\frac{\\partial v^\\gamma}{\\partial u^\\alpha}\\frac{\\partial v^\\delta}{\\partial u^\\beta}\\tilde h_{\\gamma\\delta}.
\n\\end{align*}<\/p>\n

\u7279\u522b\u5730\uff0c\u82e5\u8bb0
\n$$
\nP_{ij}=\\frac{\\partial y^j}{\\partial x^i},\\quad Q_{\\alpha\\beta}=\\frac{\\partial v^\\beta}{\\partial u^\\alpha},\\quad
\nA_{i\\alpha}=\\frac{\\partial u^\\alpha}{\\partial x^i},\\quad
\n\\tilde A_{j\\beta}=\\frac{\\partial v^\\beta}{\\partial y^j}.
\n$$
\n\u5219
\n$$
\ng=P\\tilde gP^T,\\quad h=Q\\tilde h Q^T.
\n$$<\/p>\n

\u7531\u4e8e$v=\\tilde\\phi(y)=\\tilde\\phi\\circ\\Phi(x)\\mathpunct{:}=\\Psi\\circ\\phi(x)=\\Psi(u)$, \u6211\u4eec\u77e5\u9053
\n$$
\n\\frac{\\partial v^\\alpha}{\\partial y^k}\\frac{\\partial y^k}{\\partial x^i}=\\frac{\\partial (\\tilde\\phi\\circ\\Phi)^\\alpha}{\\partial x^i}=\\frac{\\partial (\\Psi\\circ\\phi)^\\alpha}{\\partial x^i}=\\frac{\\partial v^\\alpha}{\\partial u^\\beta}\\frac{\\partial u^\\beta}{\\partial x^i},
\n$$
\n\u5373
\n$$
\nP_{ik}\\tilde A_{k\\alpha}=A_{i\\beta}Q_{\\beta\\alpha}\\implies P\\tilde A=AQ.
\n$$
\n

\u5b9a\u4e49 1<\/span>.<\/span> \u6211\u4eec\u79f0
\n$$
\nJ(\\phi)\\mathpunct{:}=\\sqrt{\\det(g^{-1}h)}\\det A=\\sqrt{\\det h\/\\det g}\\det A
\n$$
\n\u4e3a\u6620\u5c04$\\phi$\u7684Jacobian<\/span>.
\n<\/div>
\n\u5bb9\u6613\u9a8c\u8bc1
\n\\begin{align*}
\nJ(\\phi)&=\\sqrt{\\det h\/\\det g}\\det A\\\\
\n&=\\det Q\/\\det P\\cdot\\sqrt{\\det \\tilde h\/\\det\\tilde g}\\cdot \\det P\\det\\tilde A\/\\det Q\\\\
\n&=\\sqrt{\\det(\\tilde g^{-1}\\tilde h)}\\det \\tilde A=J(\\tilde\\phi).
\n\\end{align*}
\n\u4e0b\u9762\uff0c \u6211\u4eec\u5c06\u7ed9\u51fa\u4e0a\u9762\u5b9a\u4e49Jacobian\u7684\u4e00\u4e2a\u6bd4\u8f83\u51e0\u4f55\u89e3\u91ca\u3002<\/p>\n

\u56de\u5fc6, \u9ece\u66fc\u6d41\u5f62\u7684\u4f53\u79ef\u5143\u5b9a\u4e49\u4e3a:
\n$$
\ndv_h=\\sqrt{\\det h}du^1\\wedge\\cdots\\wedge du^n.
\n$$
\n\u6ce8\u610f\u5230$dv^\\alpha=\\frac{\\partial v^\\alpha}{\\partial u^\\beta}du^\\beta$, \u6545
\n\\begin{align*}
\ndv^1\\wedge\\cdots\\wedge dv^n&=(Q_{\\beta_1 1}du^{\\beta_1})\\wedge(Q_{\\beta_2 2}du^{\\beta_2})\\wedge\\cdots\\wedge(Q_{\\beta_n n}du^{\\beta_n})\\\\
\n&=\\det Q du^1\\wedge\\cdots\\wedge du^n.
\n\\end{align*}
\n\u5229\u7528\u4e0a\u9762\u7684\u8ba1\u7b97\u77e5\u9053
\n\\begin{align*}
\ndv_h&=\\sqrt{\\det h}du^1\\wedge\\cdots\\wedge du^n\\\\
\n&=\\det Q\\sqrt{\\det\\tilde h}du^1\\wedge\\cdots\\wedge du^n\\\\
\n&=\\sqrt{\\det\\tilde h}dv^1\\wedge\\cdots\\wedge dv^n=dv_{\\tilde h}.
\n\\end{align*}
\n\u5373\u5b83\u4e0e\u5750\u6807\u7cfb\u7684\u9009\u62e9\u662f\u65e0\u5173\u7684.<\/p>\n

\u73b0\u5728, \u6ce8\u610f\u5230$\\phi^* du^\\alpha=\\frac{\\partial u^\\alpha}{\\partial x^i}dx^i=A_{i\\alpha}dx^i$. \u6211\u4eec\u6709
\n\\begin{align*}
\n\\phi^*(dv_h)&=\\sqrt{\\det h}\\circ\\phi\\cdot (\\phi^*du^1)\\wedge\\cdots\\wedge(\\phi^*du^n)\\\\
\n&=\\det A\\sqrt{\\det h}dx^1\\wedge\\cdots\\wedge dx^n\\\\
\n&=\\det A\\sqrt{\\det h\/\\det g}\\sqrt{\\det g}dx^1\\wedge\\cdots\\wedge dx^n\\\\
\n&=J(\\phi)dv_g.
\n\\end{align*}
\n\u8fd9\u6b63\u662f\u5728\u901a\u5e38\u7684\u9ece\u66fc\u79ef\u5206\u7684\u6362\u5143\u6cd5\u4e2d\u7684\u9762\u79ef\u5143\u7684\u53d8\u6362\u516c\u5f0f.<\/p>\n

\u6ce8\u610f\u5230, $dv_g$, $dv_h$\u90fd\u4e0e\u5750\u6807\u7cfb\u7684\u9009\u62e9\u65e0\u5173, \u6545$J(\\phi)$\u4e5f\u4e0d\u4f9d\u8d56\u4e8e\u5750\u6807\u7cfb\u7684\u9009\u62e9. \u5b83\u662f\u4e00\u4e2a\u51e0\u4f55\u91cf, \u5c40\u90e8\u5730, \u5b83\u523b\u753b\u4e86\u6620\u7167 $\\phi$ \u7684\u50cf\u96c6\u7684\u9762\u79ef\u5fae\u5143\u4e0e\u539f\u50cf\u96c6\u7684\u9762\u79ef\u5fae\u5143\u4e4b\u6bd4.\u00a0 \u79ef\u5206\u540e\u6211\u4eec\u5f97\u5230\u4e00\u4e2a\u6574\u4f53\u7684\u4e0d\u53d8\u91cf. \u5373\u5982\u4e0b\u5b9a\u4e49\u7684
\n

\u5b9a\u4e49 2<\/span> (Brouwer\u5ea6<\/span>).<\/span> \u6211\u4eec\u79f0
\n$$
\n\\mathrm{deg}(\\phi)\\mathpunct{:}=\\frac{\\int_M\\phi^*(dv_h)}{\\int_{N}dv_h}
\n=\\frac{\\int_M J(\\phi)dv_g}{\\int_N dv_h}
\n$$
\n\u4e3a\u6620\u7167$\\phi$\u7684Brouwer\u5ea6<\/span>\u3002
\n<\/div><\/p>\n","protected":false},"excerpt":{"rendered":"

\u5047\u8bbe$\\phi\\mathpunct{:}(M,g)\\to(N,h)$\u662f\u4e24\u4e2a$n$\u7ef4\u9ece\u66fc\u6d41\u5f62\u95f4\u7684\u5149\u6ed1\u6620\u7167\u3002\u6211\u4eec\u7684… \u7ee7\u7eed\u9605\u8bfb\u6d41\u5f62\u95f4\u6620\u5c04\u7684Jacobian\u4e0e\u6620\u5c04\u5ea6<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[118,89],"class_list":["post-433","post","type-post","status-publish","format-standard","hentry","category-math","tag-jacobian","tag-yingshedu","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/433","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=433"}],"version-history":[{"count":7,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/433\/revisions"}],"predecessor-version":[{"id":568,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/433\/revisions\/568"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=433"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=433"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=433"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}