\u5728\u65e2\u5750\u6807\u4e0b\uff0c\u6b27\u6c0f\u5ea6\u91cf\u53ef\u8868\u793a\u4e3a
\n\\[
\nds^2=dr^2+r^2d\\theta^2.
\n\\]
\n\u53ef\u89c1\uff0c$e_1=\\partial_r$, $e_2=r^{-1}\\partial\\theta$\u662f\u4e00\u4e2a\u5e7a\u6b63\u57fa, \u5176\u5bf9\u5076\u57fa\u4e3a$\\omega^1=dr$, $\\omega^2=rd\\theta$\u3002\u6211\u4eec\u77e5\u9053\uff0c\u5c40\u90e8\u5e93\u4f26\u89c4\u8303\u7684\u6761\u4ef6\u662f
\n\\[
\nd^*A=0,\\quad A=A_rdr+A_\\theta d\\theta=A_r\\omega^1+rA_\\theta\\omega^2.
\n\\]
\n\u76f4\u63a5\u8ba1\u7b97\u53ef\u5f97
\n\\begin{align*}
\n-d^*A&=\\left( \\nabla_{e_i}A \\right)(e_i)=\\left( \\nabla_{e_i}(A_j\\omega^j) \\right)(e_i)\\\\
\n&=\\left( \\nabla_{e_i}A_j\\omega^j+A_i\\nabla_{e_i}\\omega^j \\right)(e_i)\\\\
\n&=\\nabla_{e_i}A_i-A_j\\omega^j(\\nabla_{e_i}e_i).
\n\\end{align*}
\n\u53c8\uff0c\u6211\u4eec\u77e5\u9053\uff0c\u5728\u6781\u5750\u6807\u4e0b\u76f4\u63a5\u8ba1\u7b97\u5f97\u5230\u514b\u91cc\u65af\u6258\u5f17\u7b26\u53f7$\\Gamma_{ij}^k$,
\n\\[
\n\\Gamma_{ij}^k=\\frac{1}{2}g^{kl}\\left( \\partial_ig_{lj}+\\partial_jg_{il}-\\partial_lg_{ij} \\right)=
\n\\begin{cases}
\n-r,&i=j=\\theta\\,,k=r\\\\
\nr^{-1},&i=\\theta\\,,j=r\\,,k=\\theta\\\\
\nr^{-1},&i=r\\,,j=\\theta\\,,k=\\theta\\\\
\n0,&\\text{\u5176\u4ed6}
\n\\end{cases}
\n\\]
\n\u8fd9\u6837\uff0c\u6211\u4eec\u5f97\u5230
\n\\begin{align*}
\n\\nabla_{\\partial_r}\\partial_r&=\\Gamma_{rr}^r\\partial_r+\\Gamma_{rr}^\\theta\\partial_\\theta=0\\\\
\n\\nabla_{\\partial_\\theta}\\partial_\\theta&=\\Gamma_{\\theta\\theta}^r\\partial_r+\\Gamma_{\\theta\\theta}^\\theta\\partial_\\theta
\n=-r\\partial_r\\\\
\n-d^*A&=\\nabla_{e_i}A_i-A_j\\omega^j(\\nabla_{e_i}e_i)\\\\
\n&=\\partial_rA_r+r^{-1}\\partial_\\theta(rA_\\theta)
\n-A_rdr(-r\\partial_r)-(rA_\\theta)(r^{-1}d\\theta\\left( -r\\partial_r \\right))\\\\
\n&=\\partial_r+A_r\\partial_\\theta A_\\theta+rA_r.
\n\\end{align*}
\n\u6700\u7ec8\uff0c\u6211\u4eec\u5f97\u5230\u5e93\u4f26\u89c4\u8303\u5728\u6781\u5750\u6807\u4e0b
\n\\[
\n0=-d^*A=\\partial_rA_r+\\partial_\\theta A_\\theta+rA_r.
\n\\]<\/p>\n","protected":false},"excerpt":{"rendered":"