\u5047\u8bbe$A,F$\u5206\u522b\u4e3a\u6d41\u5f62$M$\u4e0a\u76841\u5f62\u5f0f\u548c2\u5f62\u5f0f, \u5728\u5c40\u90e8\u5750\u6807\u4e0b, \u5176\u5206\u91cf\u6ee1\u8db3\u65b9\u7a0b
\n\\[
\n x^kF_{kj}=\\partial_r(rA_j),
\n\\]
\n\u5176\u4e2d$x=(r,\\psi)$\u8868\u793a\u6781\u5750\u6807.<\/p>\n
\u4e00\u4e2a\u81ea\u7136\u7684\u95ee\u9898\u662f, \u5982\u679c\u4ee4$y=\\tau x$, \u90a3\u4e48\u4e0a\u8ff0\u65b9\u7a0b\u53d8\u4e3a\u4ec0\u4e48?<\/p>\n
\n\u6211\u4eec\u6ce8\u610f\u5230, $y=\\tau x$\u672c\u8d28\u4e0a\u5e94\u8be5\u770b\u4f5c\u6d41\u5f62\u4e0a\u7684\u4e00\u4e2a\u5750\u6807\u53d8\u6362$\\phi: x\\to y=\\tau x$. \u9996\u5148\u6765\u5206\u6790\u5750\u6807\u53d8\u6362\u4e0b, \u6d41\u5f62\u4e0a\u5404\u79cd\u5bf9\u8c61\u7684\u53d8\u6362\u89c4\u5f8b.<\/p>\n
\u4e3a\u6b64, \u5047\u8bbe$M$\u4e3a\u5149\u6ed1\u6d41\u5f62, $p\\in M$\u662f\u4e00\u4e2a\u56fa\u5b9a\u70b9, $U\\ni p$\u4e3a$M$\u4e0a$p$\u70b9\u7684\u4e00\u4e2a\u5f00\u90bb\u57df. \u800c\u4e14\u6709\u5750\u6807\u51fd\u6570$(U,\\psi_1;x^i)$, $(U,\\psi_2;y^j)$, \u5176\u4e2d 2. \u5411\u91cf\u573a<\/a><\/span>\n\n\u8003\u5bdf\u6d41\u5f62$M$\u4e0a\u7684\u5149\u6ed1\u5411\u91cf\u573a$X\\in \\mathfrak{X}(M)$, \u5219\u5728\u5c40\u90e8\u5750\u6807\u4e0b \u4ee3\u4eba, \u5e76\u6bd4\u8f83\u6211\u4eec\u5f97\u5230 3. 1\u5f62\u5f0f<\/a><\/span>\n\n\u5047\u8bbe$\\omega$\u4e3a$M$\u4e0a\u76841\u5f62\u5f0f, \u5728\u5c40\u90e8\u5750\u6807\u4e0b, \u7531\u4e8e \u6839\u636e\u5916\u79ef\u7684\u57fa\u672c\u6027\u8d28, \u5bb9\u6613\u77e5\u9053\u4f53\u79ef\u5143
\n\\[
\n \\psi_1:U\\to \\psi_1(U)\\subset\\mathbb{R}^n,\\quad \\psi_2:U\\to \\psi_2(U)\\subset\\mathbb{R}^n,
\n\\]
\n\u4e3a\u540c\u80da\u6620\u5c04.
\n1. \u51fd\u6570<\/a><\/span>\n\n\u6700\u7b80\u5355\u7684\u662f\u6d41\u5f62\u4e0a\u7684\u51fd\u6570\u53d8\u6362\u89c4\u5f8b. \u5047\u8bbe$f\\in C^\\infty(M)$\u4e3a\u5149\u6ed1\u51fd\u6570, \u5219
\n\\[
\n f(p)=f(\\psi_1^{-1}(x))=f(\\psi_2^{-1}(y)).
\n\\]
\n\u6ce8\u610f\u5230,
\n\\[
\n \\psi_1^{-1}(x)=\\psi_2^{-1}\\circ \\psi_2\\circ\\psi_1^{-1}(x),
\n\\]
\n\u6545, \u82e5\u5047\u8bbe$\\phi=\\psi_2\\circ \\psi_1^{-1}$, \u5219\u5f97\u5230
\n\\[
\n \\psi_1^{-1}(x)=\\psi_2^{-1}\\circ \\phi(x).
\n\\]
\n\u6362\u8a00\u4e4b,
\n\\[
\n f(p)=f(\\psi_2^{-1}(y))=f(\\psi_1^{-1}(x))=f(\\psi_2^{-1}(\\phi(x))).
\n\\]
\n\u53ef\u89c1, \u51fd\u6570\u7684\u53d8\u6362\u89c4\u5f8b\u5c31\u662f\u76f4\u63a5\u53d8\u91cf\u66ff\u6362$x$\u4e3a$\\phi(x)$.<\/p>\n
\n\\[
\n X(p)=X^i(\\psi_1^{-1}(x))\\left. \\frac{\\partial}{\\partial x^i} \\right\\rvert_{\\psi_1^{-1}(x)}
\n =\\tilde{X}^j(\\psi_2^{-1}(y))\\left. \\frac{\\partial}{\\partial y^j} \\right\\rvert_{\\psi_2^{-1}(y)}.
\n\\]
\n\u6ce8\u610f\u5230
\n\\[
\n Xf(p)=X^i(\\psi_1^{-1}(x))\\left. \\frac{\\partial}{\\partial x^i} \\right\\rvert_{\\psi_1^{-1}(x)} f
\n =X^i(\\psi_1^{-1}(x))\\left. \\frac{\\partial}{\\partial x^i} (f\\circ \\psi_1^{-1})\\right\\rvert _{x},
\n\\]
\n\u7c7b\u4f3c\u5730,
\n\\[
\n Xf(p)=\\tilde{X}^j(\\psi_2^{-1}(y))\\left. \\frac{\\partial}{\\partial y^j} (f\\circ \\psi_2^{-1})\\right\\rvert _{y}.
\n\\]
\n\u4f46\u662f, \u6211\u4eec\u5df2\u7ecf\u5f97\u5230
\n\\[
\n f\\circ \\psi_1^{-1}(x)=f\\circ\\psi_2^{-1}\\circ \\phi(x).
\n\\]
\n\u56e0\u6b64, \u4e24\u8fb9\u5bf9$x^i$\u6c42\u5bfc, \u5e76\u6ce8\u610f\u5230$y=\\phi(x)$, \u5f97\u5230
\n\\[
\n \\frac{\\partial}{\\partial x^i}\\left( f\\circ\\psi^{-1}_1 \\right)
\n =\\frac{\\partial \\phi^j}{\\partial x^i} \\cdot
\n \\left. \\frac{\\partial }{\\partial y^j} \\left( f\\circ \\psi_2^{-2} \\right)\\right\\rvert_{\\phi(x)} .
\n\\]
\n(\u4e8b\u5b9e\u4e0a, \u6211\u4eec\u8fd8\u5f97\u5230\u4e86
\n\\[
\n \\left. \\frac{\\partial}{ \\partial x^i} \\right\\rvert_{\\psi_1^{-1}(x)}=\\left. \\frac{\\partial\\phi^j}{\\partial x^i} \\right\\rvert_{x}\\left. \\frac{\\partial}{\\partial y^j} \\right\\rvert_{\\psi_2^{-1}(y)}.
\n\\]
\n\u8fd9\u5728\u540e\u9762\u7684\u63a8\u5bfc\u4e2d\u6709\u7528.)<\/p>\n
\n\\[
\n \\tilde{X}^j\\left( \\psi_2^{-1}(\\phi(x)) \\right)=\\tilde{X}^j\\left( \\psi_2^{-1}(y) \\right)=X^i(\\psi_1^{-1}(x))\\frac{\\partial\\phi^j}{\\partial x^i}.
\n\\]
\n\u53ef\u89c1, \u5bf9\u5411\u91cf\u573a\u800c\u8a00, \u5176\u5206\u91cf\u5728\u5750\u6807\u53d8\u6362\u4e0b, \u4e0d\u4ec5\u8981\u50cf\u51fd\u6570\u4e00\u6837\u66ff\u6362\u5206\u91cf\u91cc\u9762\u7684\u53d8\u91cf$x$\u4e3a$\\phi(x)$, \u800c\u4e14\u8fd8\u8981\u989d\u5916\u7684\u4e58\u4ee5Jacobi\u77e9\u9635$\\left(\\frac{\\partial\\phi^j}{\\partial x^i}\\right)$\u7684\u9006.<\/p>\n
\n\\[
\n \\omega(p)=w_i(\\phi_1^{-1}(x))dx^i=\\tilde{w}_j(\\psi_2^{-1}(y))dy^j.
\n\\]
\n\u7531\u4e8e
\n\\[
\n w_i(\\psi_1^{-1}(x))=\\langle \\omega(p),\\left. \\frac{\\partial}{\\partial x^i} \\right\\rvert_p \\rangle=\\langle \\tilde{w}_k(\\psi_2^{-1}(y))dy^k,\\left. \\frac{\\partial\\phi^j}{\\partial x^i} \\right\\rvert_x \\cdot\\frac{\\partial}{\\partial y^j} \\rangle=\\tilde{w}_j(\\psi_2^{-1}(\\phi(x)))\\frac{\\partial \\phi^j}{\\partial x^i}.
\n\\]
\n\u7531\u6b64\u53ef\u89c1, \u5bf91\u5f62\u5f0f\u800c\u8a00, \u5176\u5206\u91cf\u5728\u5750\u6807\u53d8\u6362\u4e0b, \u4e0d\u4ec5\u8981\u50cf\u51fd\u6570\u4e00\u6837\u66ff\u6362\u5206\u91cf\u91cc\u9762\u7684\u53d8\u91cf$x$\u4e3a$\\phi(x)$, \u800c\u4e14\u8fd8\u8981\u989d\u5916\u7684\u4e58\u4ee5Jacobi\u77e9\u9635$\\left( \\frac{\\partial\\phi^j}{\\partial x^i} \\right)$.
\n4. \u6c42\u5bfc\u6570\u3001\u5fae\u5206\u4e0e\u79ef\u5206<\/a><\/span>\n\n\u524d\u9762\u5df2\u7ecf\u77e5\u9053, \u5173\u4e8e\u6c42\u5bfc\u6570\u6ee1\u8db3\u5982\u4e0b\u6cd5\u5219
\n\\[
\n \\left. \\frac{\\partial}{ \\partial x^i} \\right\\rvert_{\\psi_1^{-1}(x)}=\\left. \\frac{\\partial\\phi^j}{\\partial x^i} \\right\\rvert_{x}\\left. \\frac{\\partial}{\\partial y^j} \\right\\rvert_{\\psi_2^{-1}(y)}.
\n\\]
\n\u5373, \u5728\u5750\u6807\u53d8\u6362\u4e0b, \u5bfc\u6570\u7684\u66ff\u6362\u89c4\u5219\u4e3a\u4e58\u4ee5Jacobi\u77e9\u9635$\\left( \\frac{\\partial\\phi^j}{\\partial x^i} \\right)$.<\/p>\n
\n\\[
\n dy^j=\\frac{\\partial \\phi^j}{\\partial x^i}dx^i,
\n\\]
\n\u53ef\u89c1\u5fae\u5206\u7684\u66ff\u6362\u89c4\u5219\u4e3a\u4e58\u4ee5Jacobi\u77e9\u9635\u7684\u9006\u77e9\u9635.<\/p>\n
\n\\[
\n dv_g=\\sqrt{(g_{ij}(x))}dx^1\\wedge\\cdots\\wedge dx^n
\n\\]
\n\u662f\u4fdd\u6301\u4e0d\u53d8\u7684. \u4e8b\u5b9e\u4e0a, \u82e5\u5c06Jacobi\u77e9\u9635$J=\\left( \\frac{\\partial \\phi^j}{\\partial x^i} \\right)$\u7684\u5143\u7d20\u8bb0\u4e3a$a_{ij}$, \u5176\u9006\u5143\u7d20\u8bb0\u4e3a$b_{ij}$. \u5219$g_{ij}$\u4f5c\u4e3a\u4e3a2\u5f62\u5f0f\u7684\u7cfb\u6570\u53d8\u4e3a$a_{ik}a_{jl}g_{kl}$. \u6545$\\sqrt{\\det (g_{ij}(x))}=\\sqrt{\\det (g_{kl}(y))}\\det J$. \u800c
\n\\[
\n dy^j=a_{ij}dx^i,\\quad
\n dy^1\\wedge\\cdots\\wedge dy^n=\\det Jdx^1\\wedge \\cdots \\wedge dx^n.
\n\\]
\n\u7531\u6b64\u53ef\u89c1$dv_g$\u662f\u4e0d\u53d8\u7684.
\n5. \u56de\u5230\u539f\u59cb\u95ee\u9898<\/a><\/span>\n\n\u6ce8\u610f\u5230$F_{kj}$\u662f\u4e8c\u5f62\u5f0f, \u6545\u5728\u5750\u6807\u53d8\u6362$\\phi:x\\to y=\\tau x$\u4e0b, \u7b49\u5f0f\u5de6\u8fb9\u53d8\u4e3a
\n\\[
\n \\tau x^k\\tau^2F_{kj}(\\tau x).
\n\\]
\n\u7b49\u5f0f\u53f3\u8fb9\u91c7\u7528\u4e86\u6781\u5750\u6807, \u5176\u8bf1\u5bfc\u53d8\u6362\u4e3a$\\varphi:(r,\\psi)\\to(\\tau r,\\psi)$, \u6545\u53d8\u4e3a
\n\\[
\n \\tau\\partial_\\rho(\\tau r\\cdot \\tau A_j(\\tau r,\\psi)),
\n\\]
\n\u53ef\u89c1\u7b49\u5f0f\u53d8\u4e3a(\u4ee4$\\rho=\\tau r$),
\n\\[
\n x^kF_{kj}(\\tau x)=\\partial_\\rho\\left( r A_j(\\tau r, \\psi) \\right).
\n\\]
\n\u7531\u4e8e\u4e0a\u8ff0\u7b49\u5f0f\u6052\u6210\u7acb, \u7279\u522b\u5730, \u56fa\u5b9a$r$, \u6211\u4eec\u5c06\u5176\u89c6\u4e3a$(\\tau,\\psi)$\u7684\u51fd\u6570, \u5f97\u5230$r\\frac{\\partial}{\\partial\\rho}=\\frac{\\partial}{\\partial\\tau}$, \u4ece\u800c
\n\\[
\n x^kF_{kj}(\\tau x)=\\partial_\\tau(A_j(\\tau r,\\psi).
\n\\]<\/p>\n","protected":false},"excerpt":{"rendered":"