{"id":1868,"date":"2021-09-12T09:05:41","date_gmt":"2021-09-12T00:05:41","guid":{"rendered":"https:\/\/www.roundown.net\/nyushi\/?p=1868"},"modified":"2021-09-12T09:08:50","modified_gmt":"2021-09-12T00:08:50","slug":"iks201602","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/iks201602\/","title":{"rendered":"\u533b\u79d1\u6b6f\u79d1\u59272016\uff1a\u7b2c2\u554f"},"content":{"rendered":"<hr \/>\n<p>\\(xyz\\) \u7a7a\u9593\u306b\u304a\u3044\u3066\u9023\u7acb\u4e0d\u7b49\u5f0f\r\n\\[\r\n|x| \\leqq 1 , \\quad |y| \\leqq 1 , \\quad |z| \\leqq 1\r\n\\]\r\n\u306e\u8868\u3059\u9818\u57df\u3092 \\(Q\\) \u3068\u3057, \u6b63\u306e\u5b9f\u6570 \\(r\\) \u306b\u5bfe\u3057\u3066 \\(x^2 +y^2 +z^2 \\leqq r^2\\) \u306e\u8868\u3059\u9818\u57df\u3092 \\(S\\) \u3068\u3059\u308b. \u307e\u305f, \\(Q\\) \u3068 \\(S\\) \u306e\u3044\u305a\u308c\u304b\u4e00\u65b9\u306e\u307f\u306b\u542b\u307e\u308c\u308b\u70b9\u5168\u4f53\u304c\u306a\u3059\u9818\u57df\u3092 \\(R\\) \u3068\u3057, \\(R\\) \u306e\u4f53\u7a4d\u3092 \\(V(r)\\) \u3068\u3059\u308b. \u3055\u3089\u306b<\/p>\r\n<ul>\r\n<li><p>\\(x \\geqq 1\\) \u306e\u8868\u3059\u9818\u57df\u3068 \\(S\\) \u306e\u5171\u901a\u90e8\u5206\u3092 \\(S _ x\\)<\/p><\/li>\r\n<li><p>\\(y \\geqq 1\\) \u306e\u8868\u3059\u9818\u57df\u3068 \\(S\\) \u306e\u5171\u901a\u90e8\u5206\u3092 \\(S _ y\\)<\/p><\/li>\r\n<li><p>\\(z \\geqq 1\\) \u306e\u8868\u3059\u9818\u57df\u3068 \\(S\\) \u306e\u5171\u901a\u90e8\u5206\u3092 \\(S _ z\\)<\/p><\/li>\r\n<\/ul>\r\n<p>\u3068\u3057,<\/p>\r\n<ul>\r\n<li><p>\\(S _ x \\neq \\emptyset\\) \u3092\u6e80\u305f\u3059 \\(r\\) \u306e\u6700\u5c0f\u5024\u3092 \\(r _ 1\\)<\/p><\/li>\r\n<li><p>\\(S _ x \\cap S _ y \\neq \\emptyset\\) \u3092\u6e80\u305f\u3059 \\(r\\) \u306e\u6700\u5c0f\u5024\u3092 \\(r _ 2\\)<\/p><\/li>\r\n<li><p>\\(S _ x \\cap S _ y \\cap S _ z \\neq \\emptyset\\) \u3092\u6e80\u305f\u3059 \\(r\\) \u306e\u6700\u5c0f\u5024\u3092 \\(r _ 3\\)<\/p><\/li>\r\n<\/ul>\r\n<p>\u3068\u3059\u308b. \u305f\u3060\u3057, \\(\\emptyset\\) \u306f\u7a7a\u96c6\u5408\u3092\u8868\u3059. \u3053\u306e\u3068\u304d\u4ee5\u4e0b\u306e\u5404\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\\(r = \\dfrac{\\sqrt{10}}{3}\\) \u306e\u3068\u304d, \\(R\\) \u306e \\(xy\\) \u5e73\u9762\u306b\u3088\u308b\u65ad\u9762\u3092\u56f3\u793a\u305b\u3088.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\\(r _ 1 , r _ 2 , r _ 3\\) \u304a\u3088\u3073 \\(V( r _ 1 ) , V( r _ 3 )\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(3)<\/strong>\u3000\\(r \\geqq r _ 1\\) \u306e\u3068\u304d, \\(S _ x\\) \u306e\u4f53\u7a4d\u3092 \\(r\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n<li><p><strong>(4)<\/strong>\u3000\\(0 \\lt r \\leqq r _ 2\\) \u306b\u304a\u3044\u3066, \\(V(r)\\) \u304c\u6700\u5c0f\u3068\u306a\u308b \\(r\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n<hr \/>\r\n<!--more-->\r\n<h4>\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n<p><strong>(1)<\/strong><\/p>\r\n<p>\\(\\sqrt{\\left( \\dfrac{\\sqrt{10}}{3} \\right)^2 -1 } = \\dfrac{1}{3}\\) \u306a\u306e\u3067, \\(R\\) \u306e\u65ad\u9762\u306f\u4e0b\u56f3\u659c\u7dda\u90e8.<\/p>\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/ika20160201.svg\" alt=\"\" class=\"aligncenter size-full\" \/>\r\n<p><strong>(2)<\/strong><\/p>\r\n<p>\\(r_1\\) \u306f, \\(xy\\) \u5e73\u9762\u306b\u3088\u308b\u65ad\u9762\u4e0a\u3067, \\(S\\) \u304c \\(Q\\) \u306b\u5185\u63a5\u3057\u3066\u3044\u308b\u3068\u304d\u306a\u306e\u3067\r\n\\[\r\nr_1 = \\underline{1} \\ .\r\n\\]\r\n\\(r_2\\) \u306f, \\(xy\\) \u5e73\u9762\u306b\u3088\u308b\u65ad\u9762\u4e0a\u3067, \\(S\\) \u304c \\(Q\\) \u306b\u5916\u63a5\u3057\u3066\u3044\u308b\u3068\u304d\u306a\u306e\u3067\r\n\\[\r\nr_2 = \\underline{\\sqrt{2}} \\ .\r\n\\]\r\n\\(r_3\\) \u306f, \u5e73\u9762 \\(z=y\\) \u306b\u3088\u308b\u65ad\u9762\u4e0a\u3067, \\(S\\) \u304c \\(Q\\) \u306b\u5916\u63a5\u3057\u3066\u3044\u308b\u3068\u304d\u306a\u306e\u3067\r\n\\[\r\nr_3 = \\underline{\\sqrt{3}} \\ .\r\n\\]\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/ika20160202.svg\" alt=\"\" class=\"aligncenter size-full\" \/>\r\n<p>\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\nV( r_1 ) = 2^3 -\\dfrac{4}{3} \\pi \\cdot 1^3 = \\underline{8 -\\dfrac{4}{3} \\pi} \\ .\r\n\\]\r\n\u307e\u305f\r\n\\[\r\nV( r_3 ) = \\dfrac{4}{3} \\pi \\cdot \\left( \\sqrt{3} \\right)^3 -2^3 = \\underline{4 \\sqrt{3} \\pi -8} \\ .\r\n\\]\r\n<p><strong>(3)<\/strong><\/p>\r\n<p>\u6c42\u3081\u308b\u4f53\u7a4d\u3092 \\(W\\) \u3068\u304a\u304f.<br \/>\r\n\u4e0b\u56f3\u306e\u659c\u7dda\u90e8\u3092 \\(x\\) \u8ef8\u306e\u5468\u308a\u306b \\(1\\) \u56de\u8ee2\u3055\u305b\u305f\u7acb\u4f53\u306e\u4f53\u7a4d\u3092\u6c42\u3081\u308c\u3070\u3088\u3044\u306e\u3067<\/p>\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/ika20160203.svg\" alt=\"\" class=\"aligncenter size-full\" \/>\r\n<p>\\[\\begin{align}\r\nW & = \\pi \\displaystyle\\int_{1}^{r} \\left( r^2 -x^2 \\right) \\, dx \\\\\r\n& = \\pi \\left[ r^2 x -\\dfrac{x^3}{3} \\right]_{1}^{r} \\\\\r\n& = \\pi \\left\\{ \\dfrac{2 r^3}{3} -\\left( r^2 -\\dfrac{1}{3} \\right) \\right\\} \\\\\r\n& = \\underline{\\dfrac{\\pi}{3} ( 2 r^3 -3 r^2 +1 )} \\ .\r\n\\end{align}\\]\r\n<p><strong>(4)<\/strong><\/p>\r\n<p>\\(0 \\lt r \\leqq r_1\\) \u306b\u304a\u3044\u3066\u306f, \\(V(r)\\) \u306f\u5358\u8abf\u6e1b\u5c11\u3059\u308b\u306e\u3067, \\(r_1 \\lt r \\leqq r_2\\) \u306b\u304a\u3044\u3066\u6700\u5c0f\u3068\u306a\u308b\u5834\u5408\u3092\u8003\u3048\u308c\u3070\u3088\u3044.<br \/>\r\n\\(Q\\) \u306e\u5185\u90e8\u3067, \\(S\\) \u306e\u5916\u90e8\u306b\u3042\u305f\u308b\u90e8\u5206\u306e\u4f53\u7a4d\u3092 \\(U\\) \u3068\u304a\u3051\u3070\r\n\\[\r\nU = 2^3 -\\left( \\dfrac{4}{3} \\pi r^3 -6 W \\right) \\ .\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nV(r) & = 6W +U \\\\\r\n& = 8 -\\dfrac{4}{3} \\pi r^3 +12W \\\\\r\n& = 8 +4 \\pi \\underline{\\left( \\dfrac{5}{3} r^3 -3 r^2 +1 \\right)} _ {[1]} \\ .\r\n\\end{align}\\]\r\n\u4e0b\u7dda\u90e8 [1] \u3092 \\(f(r)\\) \u3068\u304a\u3044\u3066, \u6700\u5c0f\u3068\u306a\u308b\u5834\u5408\u3092\u8003\u3048\u308c\u3070\u3088\u3044.\r\n\\[\r\nf'(r) = 5r^2 -6r = r ( 5r -6 ) \\ .\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(r)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u901a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccc} r & (1) & \\cdots & \\dfrac{6}{5} & \\cdots & \\sqrt{2} \\\\ \\hline f'(r) & & - & 0 & + & \\\\ \\hline f(r) & & \\searrow & \\text{\u6700\u5c0f} & \\nearrow & \\end{array}\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u5024\u306f\r\n\\[\r\nr = \\underline{\\dfrac{6}{5}} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xyz\\) \u7a7a\u9593\u306b\u304a\u3044\u3066\u9023\u7acb\u4e0d\u7b49\u5f0f \\[ |x| \\leqq 1 , \\quad |y| \\leqq 1 , \\quad |z| \\leqq 1 \\] \u306e\u8868\u3059\u9818\u57df\u3092 \\(Q\\) \u3068\u3057, \u6b63\u306e\u5b9f\u6570 \\(r\\) \u306b\u5bfe\u3057\u3066 &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/iks201602\/\">\u7d9a\u304d\u3092\u8aad\u3080 <span class=\"meta-nav\">&rarr;<\/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":[154],"tags":[145,162],"class_list":["post-1868","post","type-post","status-publish","format-standard","hentry","category-ikashika_2016","tag-ikashika","tag-162"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1868","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=1868"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1868\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1868"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1868"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1868"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}