\u5047\u8bbe$(M,g)$\u662f\u4e00\u4e2a\u9ece\u66fc\u6d41\u5f62, \u5bf9\u4efb\u610f\u7684$S\\in \\Gamma(T_s^rM)$\u662f\u4e00\u4e2a$r$\u9636\u53cd\u53d8, $s$\u9636\u534f\u53d8\u5f20\u91cf\u573a. \u7279\u522b\u5730, \u5bf9$S\\in\\Gamma(TM)$, $S=S^i\\ppt{x^i}$, \u6709<\/p>\n $$ \u518d\u6b21, \u7279\u522b\u5730, \u5047\u8bbe$S=\\mathrm{grad} f$\u4e3a\u4e00\u4e2a\u5149\u6ed1\u51fd\u6570\u7684\u68af\u5ea6\u573a, \u5219\u6309\u7167\u68af\u5ea6<\/strong>\u7684\u5b9a\u4e49$Xf=g(X,\\mathrm{grad}f),\\quad\\forall X\\in\\Gamma(TM),$ $$\\begin{align*} $$\\Delta f:=-\\div(\\mathrm{grad}f).$$<\/p>\n<\/div>\n \u5047\u8bbe$h\\in C^\\infty_0(M)$, \u5373\u4e3a$M$\u4e0a\u5177\u6709\u7d27\u652f\u96c6\u7684\u5149\u6ed1\u51fd\u6570. \u5219\u7531\u6563\u5ea6\u5b9a\u7406<\/strong>\u6709:<\/p>\n $$ \u4e00\u822c\u5f20\u91cf\u573a\u7684\u6563\u5ea6 \u5047\u8bbe$(M,g)$\u662f\u4e00\u4e2a\u9ece\u66fc\u6d41\u5f62, \u5bf9\u4efb\u610f\u7684$S\\in \\Ga …<\/p>\n
\n$\\newcommand{\\ppt}[1]{\\frac{\\pt}{\\pt{#1}}}$
\n\u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49$S$\u7684\u6563\u5ea6<\/strong>\u5982\u4e0b:
\n$$
\n\\div(S)=\\tr(X\\to\\nabla_X S),
\n$$
\n\u5176\u4e2d$X\\in\\Gamma(TM)$. \u5b83\u662f$\\Gamma(T^r_sM)\\to\\Gamma(T^{r-1}_sM)$\u7684\u7ebf\u6027\u6620\u5c04.
\n
\n\u5728\u5c40\u90e8\u5750\u6807\u7cfb\u4e0b, \u6211\u4eec\u5047\u8bbe$$S=S_{j_1j_2\\ldots j_s}^{i_1i_2\\ldots i_r}\\ppt{x^{i_1}}\\otimes \\ppt{x^{i_2}}\\otimes\\cdots\\otimes \\ppt{x^{i_r}}\\otimes \\rd x^{j_1}\\otimes\\rd x^{j_2}\\cdots\\otimes\\rd x^{j_s},$$
\n\u5219
\n$$\\begin{align*}
\n[\\div(S)]_{j_1j_2\\ldots j_s}^{i_1i_2\\ldots i_{r-1}}&
\n=[\\div(S)]\\bigg(\\rd x^{i_1},\\rd x^{i_2},\\cdots,\\rd x^{i_{r-1}},\\ppt{x^{j_1}},\\ppt{x^{j_2}},\\cdots,\\ppt{x^{j_s}}\\bigg)\\\\
\n&=\\tr\\left(X\\mapsto(\\nabla_X S)\\bigg(\\rd x^{i_1},\\cdots,\\rd x^{i_{r-1}},\\ppt{x^{j_1}},\\cdots,\\ppt{x^{j_s}}\\bigg)\\right)\\\\
\n&=g^{ij}\\bigg\\langle\\ppt{x^{i}},
\n\\left(\\nabla_{\\ppt{x^j}}S\\right)
\n\\bigg(\\rd x^{i_1},\\cdots,\\rd x^{i_{r-1}},\\ppt{x^{j_1}},\\cdots,\\ppt{x^{j_s}}\\bigg)\\bigg\\rangle\\\\
\n&=g^{ij}\\bigg\\langle
\n\\ppt{x^{i}},S^{i_1i_2\\ldots i_{r-1}i_r}_{j_1j_2\\ldots j_s;j}\\ppt{x^{i_r}}
\n\\bigg\\rangle\\\\
\n&=S^{i_1i_2\\ldots i_{r-1}j}_{j_1j_2\\ldots j_s;j}.
\n\\end{align*}$$<\/p>\n\u5207\u5411\u91cf\u573a\u7684\u6563\u5ea6<\/h4>\n
\n\\div(S)=S^i_{;i}=\\frac{\\pt S^i}{\\pt x^i}+S^k\\Gamma_{ki}^i,
\n$$
\n\u8fd9\u91cc$\\Gamma_{ki}^i$\u4e3a\u5ea6\u91cf$g$\u51b3\u5b9a\u7684Christoffel, \u5b83\u7528$g$\u53ef\u4ee5\u8868\u793a\u4e3a
\n$$
\n\\Gamma_{ki}^i=\\frac{1}{2}g^{il}\\left\\{
\n\\pt_kg_{li}+\\pt_{i}g_{kl}-\\pt_l g_{ki}
\n\\right\\}=\\frac{1}{2}g^{il}\\pt_kg_{li}.
\n$$
\n\u82e5\u8bb0$G=\\det(g_{ij})$, \u5219\u6309\u7167\u884c\u5217\u5f0f\u7684\u6c42\u5bfc\u6cd5\u5219, \u6709
\n$$
\n\\pt_k G=(\\pt_k g_{li})\\cdot(g^{il}G),
\n$$
\n\u8fd9\u53ef\u4ee5\u6539\u5199\u6210
\n$$
\n\\pt_k\\log G=g^{il}\\pt_k g_{li}.
\n$$
\n\u56e0\u6b64
\n$$
\n\\Gamma_{ki}^i=\\frac{1}{2}\\pt_k\\log G=\\pt_k\\log\\sqrt{G}=\\frac{1}{\\sqrt G}\\pt_k\\sqrt{G}.
\n$$<\/p>\n\u51fd\u6570\u7684\u68af\u5ea6\u573a\u7684\u6563\u5ea6<\/h4>\n
\n\u77e5$\\mathrm{grad}f=g^{ij}\\pt_i f\\ppt{x^j}:=f^j\\pt_j$, \u8fd9\u6837<\/p>\n
\n\\div S&=\\div(\\mathrm{grad} f)
\n=f^j_{;j}\\\\
\n&=\\pt_j f^j+f^j\\frac{1}{\\sqrt G}\\pt_j\\sqrt{G}\\\\
\n&=\\frac{1}{\\sqrt G}\\pt_j(f^j\\sqrt{G})\\\\
\n&=\\frac{1}{\\sqrt G}\\pt_j\\bigg(\\sqrt{G}g^{ij}\\ppt{x^i} f\\bigg).
\n\\end{align*}$$<\/p>\n\u5206\u90e8\u79ef\u5206\u516c\u5f0f<\/h5>\n
\n\\int_M\\Delta f h \\;\\rd v_g=-\\int_M\\div(\\mathrm{grad}f)h\\rd v_g
\n=\\int_M\\langle\\mathrm{grad}f,\\mathrm{grad}h\\rangle\\rd v_g,
\n$$
\n\u8fd9\u662f\u56e0\u4e3a, $\\div (hX)=h \\div X+Xh,\\forall X\\in\\Gamma(TM)$.<\/p>\n","protected":false},"excerpt":{"rendered":"