{"id":3389,"date":"2024-06-23T22:33:23","date_gmt":"2024-06-23T14:33:23","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=3389"},"modified":"2025-02-26T10:56:48","modified_gmt":"2025-02-26T02:56:48","slug":"games001_einstein_summation_convention_levi_civita_symbol","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2024\/06\/23\/games001_einstein_summation_convention_levi_civita_symbol\/","title":{"rendered":"[Games 001] \u7231\u56e0\u65af\u5766\u6c42\u548c\u7ea6\u5b9a\u4e0eLevi-Civita\u7b26\u53f7"},"content":{"rendered":"<p><script type=\"text\/javascript\" async src=\"https:\/\/www.caiqinyi.cn\/wp-content\/MathJax\/MathJax.js?config=TeX-AMS_CHTML\">\n<\/script><br \/>\n<script type=\"text\/x-mathjax-config\">\n    MathJax.Hub.Config({\n        tex2jax: {inlineMath: [['$','$']]},\n        TeX: {equationNumbers: {autoNumber: [\"AMS\"], useLabelIds: true}},\n        \"HTML-CSS\": {linebreaks: {automatic: true}},\n        SVG: {linebreaks: {automatic: true}}\n    });\n<\/script><\/p>\n<p>\u524d\u6bb5\u65f6\u95f42024\u5e74\u963f\u91cc\u5df4\u5df4\u5168\u7403\u6570\u5b66\u7ade\u8d5b\u521d\u8d5b\u7684\u6210\u7ee9\u51fa\u4e86, \u610f\u6599\u4e4b\u5185\u5730\u6ca1\u6709\u8fdb\u51b3\u8d5b, \u8fd8\u662f\u6709\u4e00\u4e9b\u5931\u671b\u53ed(\uff1b\u2032\u2312`) \u4f46\u7ec8\u5f52\u8fd8\u662f\u81ea\u8eab\u5b9e\u529b\u4e0d\u6d4e, \u53ea\u80fd\u6765\u5e74\u6709\u7a7a\u518d\u6218\u53ed~ \u6700\u8fd1\u5728\u5b66\u4e60\u77e2\u91cf\u5206\u6790, \u67e5\u9605\u4e86\u5404\u79cd\u8d44\u6599\u540e, \u603b\u7b97\u638c\u63e1\u4e86\u7231\u56e0\u65af\u5766\u6c42\u548c\u7ea6\u5b9a\u4e0eLevi-Civita\u7b26\u53f7\u8fd9\u4e24\u4e2a\u5f3a\u6709\u529b\u7684\u77e2\u91cf\u5206\u6790\u5de5\u5177, \u7279\u6b64\u8bb0\u5f55.<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/medium.com\/%E9%87%8F%E5%8C%96%E4%BA%A4%E6%98%93%E7%9A%84%E8%B5%B7%E9%BB%9E-%E9%82%81%E5%90%91%E9%87%8F%E5%8C%96%E4%BA%A4%E6%98%93%E7%85%89%E9%87%91%E8%A1%93%E5%B8%AB%E4%B9%8B%E8%B7%AF\/%E5%B7%A5%E6%95%B8%E7%AD%86%E8%A8%98-%E6%84%9B%E5%9B%A0%E6%96%AF%E5%9D%A6%E6%B1%82%E5%92%8C%E7%B4%84%E5%AE%9A-einstein-summation-convention-ae5b786de439\">[\u5de5\u6578\u7b46\u8a18] \u611b\u56e0\u65af\u5766\u6c42\u548c\u7d04\u5b9a\uff08Einstein summation convention\uff09<\/a><br \/>\n2. <a href=\"https:\/\/math.stackexchange.com\/questions\/496060\/gradient-of-a-dot-product\">Gradient of a dot product<\/a><br \/>\n3. <a href=\"https:\/\/dsl.nju.edu.cn\/litao\/teach\/Ch0.pdf\">\u9884\u5907\u77e5\u8bc6\u2014\u77e2\u91cf\u573a\u8bba\u590d\u4e60<\/a><br \/>\n4. <a href=\"https:\/\/math.stackexchange.com\/questions\/809584\/how-to-prove-that-nabla-a-cdot-b-a-cdot-nabla-bb-cdot-nabla-aa-times\">how-to-prove-that-nabla-a-cdot-b-a-cdot-nabla-bb-cdot-nabla-aa-times<\/a><br \/>\n5. <a href=\"https:\/\/icourse.club\/uploads\/files\/9c928d95d613077c2676f00932fa821eb69b28b2.pdf\">\u77e2\u91cf\u5206\u6790\u7b80\u4ecb<\/a><\/p>\n<p><strong>1. \u57fa\u672c\u5b9a\u4e49<\/strong><\/p>\n<p>$$\\delta_{ij} = \\left\\{\\begin{matrix}<br \/>\n1, &#038; i = j, \\\\<br \/>\n0, &#038; i \\ne j,<br \/>\n\\end{matrix}\\right. \\\\ \\epsilon_{ijk} = \\left\\{\\begin{matrix}<br \/>\n1, &#038; \\text{ if } (i, j, k) \\in \\{ (1, 2, 3), (3, 1, 2), (2, 3, 1), \\} \\\\<br \/>\n-1, &#038; \\text{ if } (i, j, k) \\in \\{ (3, 2, 1), (1, 3, 2), (2, 1, 3), \\} \\\\<br \/>\n0, &#038; \\text{ if } i = j\\ or\\ j = k\\ or\\ j = i.<br \/>\n\\end{matrix}\\right.$$<br \/>\n<strong>2. \u5e38\u89c1\u8868\u8fbe\u5f0f\u53ca\u5176\u5bf9\u5e94\u7684\u7231\u56e0\u65af\u5766\u6c42\u548c\u7ea6\u5b9a\u4e0eLevi-Civita\u7b26\u53f7<\/strong><\/p>\n<p>$$\\begin{matrix}<br \/>\n\\nabla \\textbf{v} &#038; | &#038; \\frac{\\partial}{\\partial x_i} \\textbf{e}_i v_i \\textbf{e}_i = \\frac{\\partial}{\\partial x_i} v_i \\\\<br \/>\n\\nabla \\cdot \\textbf{v} &#038; | &#038; \\frac{\\partial}{\\partial x_j} \\textbf{e}_j v_i \\textbf{e}_i \\\\<br \/>\n\\nabla \\times \\textbf{v} &#038; | &#038; \\epsilon_{ijk} \\frac{\\partial}{\\partial x_i} \\textbf{e}_i \\textbf{e}_j v_k \\textbf{e}_k<br \/>\n\\end{matrix}$$<br \/>\n<strong>3. \u76f8\u5173\u4f8b\u9898<\/strong><\/p>\n<p><strong>\u4f8b3.1<\/strong> $$\\nabla(\\textbf{u}\\cdot \\textbf{v}) = (\\nabla\\textbf{v})\\cdot\\textbf{u}+(\\nabla\\textbf{u})\\cdot\\textbf{v}.$$<strong>\u8bc1:<\/strong> \u56e0\u4e3a$\\nabla \\textbf{u} = (\\frac{\\partial u_j}{x_i})$, $\\nabla \\textbf{v} = (\\frac{\\partial v_j}{x_i})$, \u6545<br \/>\n$$\\begin{equation}<br \/>\n   \\begin{aligned}<br \/>\n       &#038;(\\nabla\\textbf{v})\\cdot\\textbf{u}+(\\nabla\\textbf{u})\\cdot\\textbf{v} \\\\ =&#038; ((\\frac{\\partial v_j}{x_i} u_j + \\frac{\\partial u_j}{x_i} v_j) e_i) \\\\ =&#038; (\\frac{\\partial (u_j v_j)}{x_i} e_i) \\\\ =&#038; \\nabla(\\textbf{u} \\cdot \\textbf{v}).<br \/>\n   \\end{aligned}<br \/>\n\\end{equation}$$<br \/>\n<strong>\u4f8b3.2<\/strong> $$(\\nabla \\times \\textbf{v})\\times\\textbf{a}=[\\textbf{v}\\nabla-\\nabla\\textbf{v}]\\cdot\\textbf{a}.$$<strong>\u8bc1:<\/strong> $$\\begin{equation}<br \/>\n   \\begin{aligned}<br \/>\n      &#038;(\\nabla \\times \\textbf{v})\\times\\textbf{a} \\\\ =&#038; \\epsilon_{ijk} (\\nabla \\times \\textbf{v})_i \\textbf{e}_j a_k \\\\ =&#038; \\epsilon_{ijk} \\epsilon_{ipq} (\\frac{\\partial}{\\partial x_q} v_p) \\textbf{e}_j a_k \\\\ =&#038; (\\delta_{jp} \\delta_{kq} &#8211; \\delta_{jq} \\delta_{kp}) (\\frac{\\partial}{\\partial x_q} v_p) \\textbf{e}_j a_k \\\\ =&#038; (\\frac{\\partial}{\\partial x_k} v_j) \\textbf{e}_j a_k &#8211; (\\frac{\\partial}{\\partial x_j} v_k) \\textbf{e}_j a_k \\\\ =&#038; a_k (\\frac{\\partial}{\\partial x_k} v_j e_j) &#8211; (\\textbf{e}_j \\frac{\\partial}{\\partial x_j} v_k) a_k \\\\ =&#038; a_k \\textbf{e}_k (\\textbf{e}_k \\frac{\\partial}{\\partial x_k} v_j e_j) &#8211; (\\textbf{e}_j \\frac{\\partial}{\\partial x_j} v_k \\textbf{e}_k) a_k \\textbf{e}_k \\\\ =&#038; (a_k \\textbf{e}_k \\cdot \\textbf{e}_k \\frac{\\partial}{\\partial x_k}) v_j e_j &#8211; (\\nabla \\textbf{v}) \\cdot \\textbf{a} \\\\ =&#038; (\\textbf{a} \\cdot \\nabla) \\textbf{v} &#8211; (\\nabla \\textbf{v}) \\cdot \\textbf{a}.<br \/>\n   \\end{aligned}<br \/>\n\\end{equation}$$<br \/>\n<strong>\u4f8b3.3<\/strong> $$\\nabla(\\textbf{u}\\cdot\\textbf{v})=\\textbf{u}\\times(\\nabla\\times\\textbf{v})+\\textbf{v}\\times(\\nabla\\times\\textbf{u})+\\textbf{u}\\cdot(\\nabla\\textbf{v})+\\textbf{v}\\cdot(\\nabla\\textbf{u}).$$<strong>\u8bc1:<\/strong> $$\\begin{equation}<br \/>\n   \\begin{aligned}<br \/>\n      Rhs =&#038; \\textbf{u}\\times(\\nabla\\times\\textbf{v})+\\textbf{v}\\times(\\nabla\\times\\textbf{u})+\\textbf{u}\\cdot(\\nabla\\textbf{v})+\\textbf{v}\\cdot(\\nabla\\textbf{u}) \\\\ =&#038; (\\nabla \\textbf{v}) \\cdot \\textbf{u} &#8211; (\\textbf{u} \\cdot \\nabla) \\textbf{v} + (\\nabla \\textbf{u}) \\cdot \\textbf{v} &#8211; (\\textbf{v} \\cdot \\nabla) \\textbf{u} + \\textbf{u} \\cdot (\\nabla \\textbf{v}) + \\textbf{v} \\cdot (\\nabla \\textbf{u}) \\\\ =&#038; (\\nabla \\textbf{v}) \\cdot \\textbf{u} + (\\nabla \\textbf{u}) \\cdot \\textbf{v} \\\\ =&#038; \\nabla(\\textbf{u}\\cdot\\textbf{v}).<br \/>\n   \\end{aligned}<br \/>\n\\end{equation}$$<br \/>\n<strong>\u4f8b3.4<\/strong> $$\\nabla\\times(\\textbf{u}\\times\\textbf{v}) = \\textbf{v}\\cdot(\\nabla\\textbf{u})-\\textbf{v}(\\nabla\\cdot\\textbf{u})+\\textbf{u}(\\nabla\\cdot\\textbf{v})-\\textbf{u}\\cdot(\\nabla\\textbf{v}).$$<strong>\u8bc1:<\/strong> $$\\begin{equation}<br \/>\n   \\begin{aligned}<br \/>\n      Lhs =&#038; \\nabla\\times(\\textbf{u}\\times\\textbf{v}) \\\\ =&#038; \\epsilon_{ijk} \\frac{\\partial}{\\partial x_i} \\textbf{e}_j (\\textbf{u} \\times \\textbf{v})_k \\\\ =&#038; \\epsilon_{ijk} \\frac{\\partial}{\\partial x_i} \\textbf{e}_j \\epsilon_{kpq} (u_p v_q) \\\\ =&#038; (\\delta_{ip} \\delta_{jq} &#8211; \\delta_{iq} \\delta_{jp}) \\frac{\\partial}{\\partial x_i} \\textbf{e}_j (u_p v_q) \\\\ =&#038; \\frac{\\partial}{\\partial x_i} \\textbf{e}_j (u_i v_j) &#8211; \\frac{\\partial}{\\partial x_i} \\textbf{e}_j (u_j v_i) \\\\ =&#038; v_j \\textbf{e}_j \\frac{\\partial}{\\partial x_i} u_i + u_i (\\frac{\\partial}{\\partial x_i} v_j \\textbf{e}_j) &#8211; v_i (\\frac{\\partial}{\\partial x_i} u_j \\textbf{e}_j) &#8211; u_j \\textbf{e}_j \\frac{\\partial}{\\partial x_i} v_i \\\\ =&#038; \\textbf{v} \\cdot (\\nabla \\textbf{u}) + \\textbf{u} (\\nabla \\cdot \\textbf{v}) &#8211; \\textbf{v} (\\nabla \\cdot \\textbf{u}) &#8211; \\textbf{u} \\cdot (\\nabla \\textbf{v}).<br \/>\n   \\end{aligned}<br \/>\n\\end{equation}$$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u524d\u6bb5\u65f6\u95f42024\u5e74\u963f\u91cc\u5df4\u5df4\u5168\u7403\u6570\u5b66\u7ade\u8d5b\u521d\u8d5b\u7684\u6210\u7ee9\u51fa\u4e86, \u610f\u6599\u4e4b\u5185\u5730\u6ca1\u6709\u8fdb\u51b3\u8d5b, \u8fd8\u662f\u6709\u4e00\u4e9b\u5931\u671b\u53ed(\uff1b\u2032\u2312`)  &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2024\/06\/23\/games001_einstein_summation_convention_levi_civita_symbol\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">[Games 001] \u7231\u56e0\u65af\u5766\u6c42\u548c\u7ea6\u5b9a\u4e0eLevi-Civita\u7b26\u53f7<\/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":[23],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/3389"}],"collection":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/comments?post=3389"}],"version-history":[{"count":23,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/3389\/revisions"}],"predecessor-version":[{"id":3597,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/3389\/revisions\/3597"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=3389"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=3389"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=3389"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}