{"id":166,"date":"2011-12-02T22:15:24","date_gmt":"2011-12-02T13:15:24","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=166"},"modified":"2021-03-16T16:15:37","modified_gmt":"2021-03-16T07:15:37","slug":"tkr200904","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkr200904\/","title":{"rendered":"\u6771\u5927\u7406\u7cfb2009\uff1a\u7b2c4\u554f"},"content":{"rendered":"
\n

\\(a\\) \u3092\u6b63\u306e\u5b9f\u6570\u3068\u3057, \u7a7a\u9593\u5185\u306e \\(2\\) \u3064\u306e\u5186\u677f\r\n\\[\\begin{align}\r\nD _ 1 & = \\big\\{ (x, y, z) \\big| x^2+y^2 \\leqq 1 , z=a \\big\\} , \\\\\r\nD _ 2 & = \\big\\{ (x, y, z) \\big| x^2+y^2 \\leqq 1 , z=-a \\big\\}\r\n\\end{align}\\]\r\n\u3092\u8003\u3048\u308b.\r\n\\(D _ 1\\) \u3092 \\(y\\) \u8ef8\u306e\u307e\u308f\u308a\u306b \\(180^{\\circ}\\) \u56de\u8ee2\u3057\u3066 \\(D _ 2\\) \u306b\u91cd\u306d\u308b.\r\n\u305f\u3060\u3057\u56de\u8ee2\u306f \\(z\\) \u8ef8\u306e\u6b63\u306e\u90e8\u5206\u3092 \\(x\\) \u8ef8\u306e\u6b63\u306e\u65b9\u5411\u306b\u50be\u3051\u308b\u5411\u304d\u3068\u3059\u308b.\r\n\u3053\u306e\u56de\u8ee2\u306e\u9593\u306b \\(D _ 1\\) \u304c\u901a\u308b\u90e8\u5206\u3092 \\(E\\) \u3068\u3059\u308b.\r\n\\(E\\) \u306e\u4f53\u7a4d\u3092 \\(V(a)\\) \u3068\u3057, \\(E\\) \u3068 \\(\\big\\{ (x, y, z) \\big| x \\geqq 0 \\big\\}\\) \u3068\u306e\u5171\u901a\u90e8\u5206\u306e\u4f53\u7a4d\u3092 \\(W(a)\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(W(a)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(\\displaystyle\\lim _ {a \\rightarrow \\infty} V(a)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

    (1)<\/strong><\/p>\r\n

    \u5bfe\u79f0\u6027\u304b\u3089 \\(y \\geqq 0\\) \u306e\u90e8\u5206\u306b\u3064\u3044\u3066\u8003\u3048\u308b.<\/p>\r\n\"\"\r\n

    \\(y=k \\quad ( 0 \\leqq k \\leqq 1 )\\) \u306b\u304a\u3051\u308b\u65ad\u9762\u7a4d\u3092 \\(S(k)\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nS(k) & = \\dfrac{\\pi}{2} \\left\\{ (1+a^2-k^2) -a^2 \\right\\} \\\\\r\n& = \\dfrac{\\pi (1-k^2)}{2}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nW(a) & = 2 \\displaystyle\\int _ 0^1 S(y) \\, dy \\\\\r\n& = \\pi \\left[ y -\\dfrac{y^3}{3} \\right] _ 0^1 = \\underline{\\dfrac{2 \\pi}{3}}\r\n\\end{align}\\]\r\n

    (2)<\/strong><\/p>\r\n\"\"\r\n

    \\(E\\) \u306e\u3046\u3061 \\(x \\lt 0\\) \u306e\u9818\u57df\u306b\u542b\u307e\u308c\u308b\u90e8\u5206\u306e\u4f53\u7a4d\u3092 \\(U(a)\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n0 \\lt U(a) \\lt 2 \\cdot 2 \\cdot 1 \\left( \\sqrt{1+a^2} -a \\right) = \\dfrac{4}{\\sqrt{1+a^2} +a}\r\n\\]\r\n\\(a \\rightarrow \\infty\\) \u306e\u3068\u304d, \\(( \\text{\u53f3\u8fba} ) \\rightarrow 0\\) \u306a\u306e\u3067, \u306f\u3055\u307f\u3046\u3061\u306e\u539f\u7406\u3088\u308a\r\n\\[\r\n\\displaystyle\\lim _ {a \\rightarrow \\infty} U(a) =0\r\n\\]\r\n\u3088\u3063\u3066, \\(V(a) = W(a) +U(a)\\) \u306a\u306e\u3067\r\n\\[\r\n\\displaystyle\\lim _ {a \\rightarrow \\infty} V(a) =\\underline{\\dfrac{2 \\pi}{3}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(a\\) \u3092\u6b63\u306e\u5b9f\u6570\u3068\u3057, \u7a7a\u9593\u5185\u306e \\(2\\) \u3064\u306e\u5186\u677f \\[\\begin{align} D _ 1 & = \\big\\{ (x, y, z) \\big| x^2+y^2 \\leqq 1 , z=a \\big\\} , … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[18],"tags":[139,15],"class_list":["post-166","post","type-post","status-publish","format-standard","hentry","category-tokyo_r_2009","tag-tokyo_r","tag-15"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/166","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=166"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/166\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=166"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=166"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=166"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}