{"id":5675,"date":"2014-09-01T06:05:19","date_gmt":"2014-08-31T21:05:19","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5675"},"modified":"2014-08-08T19:46:35","modified_gmt":"2014-08-08T10:46:35","slug":"properties-of-line-integrals","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5675","title":{"rendered":"PROPERTIES OF LINE INTEGRALS"},"content":{"rendered":"<div class=\"theContentWrap-ccc\"><blockquote>\n<p>Line integrals have properties which are analogous to those of ordinary integrals. For example: <\/p>\n<ol>\n<li><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+%5Cint_C+%5BP%28x%2C+y%29dx+%2B+Q%28x%2C+y%29dy%5D+%3D+%5Cint_C+P%28x%2C+y%29dx+%2B+%5Cint_C+Q%28x%2C+y%29dy&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle \\int_C [P(x, y)dx + Q(x, y)dy] = \\int_C P(x, y)dx + \\int_C Q(x, y)dy' title='\\displaystyle \\int_C [P(x, y)dx + Q(x, y)dy] = \\int_C P(x, y)dx + \\int_C Q(x, y)dy' class='latex' \/><\/li>\n<li><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%28a_1%2C+b_1%29%7D%5E%7B%28a_2%2C+b_2%29%7D+%5BPdx+%2B+Qdy%5D+%3D+-+%5Cint_%7B%28a_2%2C+b_2%29%7D%5E%7B%28a_1%2C+b_1%29%7D%5BPdx+%2B+Qdy%5D&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle \\int_{(a_1, b_1)}^{(a_2, b_2)} [Pdx + Qdy] = - \\int_{(a_2, b_2)}^{(a_1, b_1)}[Pdx + Qdy]' title='\\displaystyle \\int_{(a_1, b_1)}^{(a_2, b_2)} [Pdx + Qdy] = - \\int_{(a_2, b_2)}^{(a_1, b_1)}[Pdx + Qdy]' class='latex' \/><\/li>\n<p>Thus reversal of the path of integration changes the sign of the line integral. <\/p>\n<li><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%28a_1%2C+b_1%29%7D%5E%7B%28a_2%2C+b_2%29%7D+%5BPdx+%2B+Qdy%5D+%3D+%5Cint_%7B%28a_1%2C+b_1%29%7D%5E%7B%28a_3%2C+b_3%29%7D+%5BPdx+%2B+Qdy%5D+%2B+%5Cint_%7B%28a_3%2C+b_3%29%7D%5E%7B%28a_2%2C+b_2%29%7D+%5BPdx+%2B+Qdy%5D+&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle \\int_{(a_1, b_1)}^{(a_2, b_2)} [Pdx + Qdy] = \\int_{(a_1, b_1)}^{(a_3, b_3)} [Pdx + Qdy] + \\int_{(a_3, b_3)}^{(a_2, b_2)} [Pdx + Qdy] ' title='\\displaystyle \\int_{(a_1, b_1)}^{(a_2, b_2)} [Pdx + Qdy] = \\int_{(a_1, b_1)}^{(a_3, b_3)} [Pdx + Qdy] + \\int_{(a_3, b_3)}^{(a_2, b_2)} [Pdx + Qdy] ' class='latex' \/><\/li>\n<p>where <img src='https:\/\/s0.wp.com\/latex.php?latex=%28a_3%2C+b_3%29&#038;bg=T&#038;fg=000000&#038;s=0' alt='(a_3, b_3)' title='(a_3, b_3)' class='latex' \/> is another point on <em>C<\/em>. <\/p>\n<\/ol>\n<p>Similar properties hold for line integrals in space. <\/p>\n<\/blockquote>\n<p><iframe src=\"\/\/rcm-fe.amazon-adsystem.com\/e\/cm?lt1=_blank&#038;bc1=000000&#038;IS2=1&#038;bg1=FFFFFF&#038;fc1=000000&#038;lc1=0000FF&#038;t=fujiitoshiki-22&#038;o=9&#038;p=8&#038;l=as4&#038;m=amazon&#038;f=ifr&#038;ref=ss_til&#038;asins=0071635408\" style=\"width:120px;height:240px;\" scrolling=\"no\" marginwidth=\"0\" marginheight=\"0\" frameborder=\"0\"><\/iframe><\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>Line integrals have properties which are analogous to those of ordinary integrals. For example: Thus reversal  &hellip; <a href=\"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5675\" class=\"more-link\"><span class=\"screen-reader-text\">&#8220;PROPERTIES OF LINE INTEGRALS&#8221; \u306e<\/span>\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":1,"featured_media":6040,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_crdt_document":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[9],"tags":[],"class_list":["post-5675","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/5675","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5675"}],"version-history":[{"count":2,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/5675\/revisions"}],"predecessor-version":[{"id":5677,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/5675\/revisions\/5677"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/media\/6040"}],"wp:attachment":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5675"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5675"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5675"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}