\u6628\u5929\u5728\u505a\u7535\u78c1\u5b66\u9898\u76ee\u7684\u65f6\u5019\u9047\u5230\u51e0\u4e2a\u624b\u7b97\u6bd4\u8f83\u9ebb\u70e6\u7684\u5f0f\u5b50\uff0c\u4e8e\u662f\u6539\u7528Mathematica\u8f6f\u4ef6\u5b8c\u6210\u4e86\u8ba1\u7b97\u3002<\/p>\n
\u5728\u8fd9\u91cc\u603b\u7ed3\u4e24\u4e2a\u89e3\u51b3\u65b9\u6cd5\uff1a<\/p>\n
1\u3001\u505a\u8fd1\u4f3c\u8ba1\u7b97\uff0c\u7565\u53bb\u9ad8\u9636\u5c0f\u91cf<\/p>\n
\u6709\u4e00\u9053\u9898\u8981\u8ba1\u7b97 \\(r \\gg l\\) \u65f6\u7684\u7535\u52bf\u5206\u5e03\uff0c\u539f\u59cb\u7684\u5f0f\u5b50\u662f\uff08\u53bb\u6389\u4e86\u5e38\u6570\u90e8\u5206\uff09<\/p>\n
$$\\frac{1}{{\\sqrt {{l^2} + 2lr\\cos (\\theta ) + {r^2}} }} + \\frac{1}{{\\sqrt {{l^2} – 2lr\\cos (\\theta ) + {r^2}} }} – \\frac{2}{r}$$<\/p>\n
\u5728Mathematica\u4e2d\u53ef\u4ee5\u7528Series\u51fd\u6570\u8ba1\u7b97\u8fd9\u4e2a\u51fd\u6570\u5728 \\(l=0\\) \u5904\u7684\u6cf0\u52d2\u5c55\u5f00\uff0c\u5c55\u5f00\u5230\u7b2c\u4e8c\u9879\u5c31\u53ef\u4ee5\u4e86\u3002<\/p>\n
Mathematica\u8f93\u5165\u5982\u4e0b\uff1a<\/p>\n
\r\nAssuming[r>0,Simplify[Series[-2\/r+1\/Sqrt[r^2+l^2-2r l Cos[\\[Theta]]]+1\/Sqrt[r^2+l^2+2 r l Cos[\\[Theta]]],{l,0,2}]]]\r\n<\/pre>\n\u5f97\u5230\u7684\u7ed3\u679c\u662f\uff1a<\/p>\n
$$\\frac{{\\left( {1 + 3\\cos \\left[ {2 \\theta} \\right]} \\right){l^2}}}{{2{r^3}}} + O{\\left[ l \\right]^3}$$<\/p>\n
\u7b2c\u4e00\u9879\u5c31\u662f\u6240\u6c42\u3002<\/p>\n
2\u3001\u5728\u6781\u5750\u6807\u7cfb\u4e2d\u6c42\u68af\u5ea6<\/p>\n
\u6839\u636e\u521a\u624d\u7b97\u51fa\u6765\u7684\u7535\u52bf\u6c42\u7535\u573a\u5f3a\u5ea6\u77e2\u91cf\uff0c\u9700\u8981\u5728\u6781\u5750\u6807\u7cfb\u4e2d\u6c42\u68af\u5ea6\u3002<\/p>\n
\u7528Mathematica\u7684Grad\u51fd\u6570\uff0c\u8f93\u5165\u5982\u4e0b\uff1a<\/p>\n
\r\nAssuming[r>0,Simplify[-Grad[((1+3 Cos[2 \\[Theta]]) l^2)\/(2 r^3),{r,\\[Theta],\\[Phi]},"Spherical"]]]\r\n<\/pre>\n\u5f97\u5230\u7684\u7ed3\u679c\u662f\uff1a<\/p>\n
$$\\left\\{ {\\frac{{3{l^2}\\left( {1 + 3\\cos \\left[ {2\\theta } \\right]} \\right)}}{{2{r^4}}},\\frac{{3{l^2}\\sin \\left[ {2\\theta } \\right]}}{{{r^4}}},0} \\right\\}$$<\/p>\n
\u8fd9\u91cc\u7528\u7684\u662f\u7403\u9762\u5750\u6807\u7cfb\uff0c\u6240\u4ee5\u7ed3\u679c\u7684\u7b2c\u4e09\u4e2a\u5206\u91cf\u662f0\uff0c\u53ef\u4ee5\u7565\u53bb\u3002\u524d\u4e24\u4e2a\u5206\u91cf\u5206\u522b\u4ee3\u8868\\(r\\)\u65b9\u5411\u548c\\(\\theta\\)\u65b9\u5411\u7684\u7ed3\u679c\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
\u6628\u5929\u5728\u505a\u7535\u78c1\u5b66\u9898\u76ee\u7684\u65f6\u5019\u9047\u5230\u51e0\u4e2a\u624b\u7b97\u6bd4\u8f83\u9ebb\u70e6\u7684\u5f0f\u5b50\uff0c\u4e8e\u662f\u6539\u7528Mathematica\u8f6f\u4ef6\u5b8c\u6210\u4e86\u8ba1\u7b97\u3002 \u5728\u8fd9\u91cc\u603b\u7ed3 … \u7ee7\u7eed\u9605\u8bfb\u201c\u7528Mathematica\u8ba1\u7b97\u7269\u7406\u9898\u201d<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":"","_links_to":"","_links_to_target":""},"categories":[3,4],"tags":[],"class_list":["post-167","post","type-post","status-publish","format-standard","hentry","category-math","category-computer"],"_links":{"self":[{"href":"https:\/\/sqrt-1.me\/index.php?rest_route=\/wp\/v2\/posts\/167","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sqrt-1.me\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sqrt-1.me\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sqrt-1.me\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sqrt-1.me\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=167"}],"version-history":[{"count":16,"href":"https:\/\/sqrt-1.me\/index.php?rest_route=\/wp\/v2\/posts\/167\/revisions"}],"predecessor-version":[{"id":341,"href":"https:\/\/sqrt-1.me\/index.php?rest_route=\/wp\/v2\/posts\/167\/revisions\/341"}],"wp:attachment":[{"href":"https:\/\/sqrt-1.me\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=167"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sqrt-1.me\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=167"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sqrt-1.me\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=167"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}