{"id":5742,"date":"2014-09-22T06:05:12","date_gmt":"2014-09-21T21:05:12","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5742"},"modified":"2014-08-08T19:43:57","modified_gmt":"2014-08-08T10:43:57","slug":"conditions-for-a-line-integral-to-be-independent-of-the-path","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5742","title":{"rendered":"CONDITIONS FOR A LINE INTEGRAL TO BE INDEPENDENT OF THE PATH"},"content":{"rendered":"<div class=\"theContentWrap-ccc\"><blockquote>\n<ul>\n<li><strong><em>Theorem 6-1. <\/em><\/strong><\/li>\n<p>A necessary and sufficient condition for <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+%5Cint_C+%5BPdx+%2B+Qdy%5D&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle \\int_C [Pdx + Qdy]' title='\\displaystyle \\int_C [Pdx + Qdy]' class='latex' \/> to be independent of the path <img src='https:\/\/s0.wp.com\/latex.php?latex=C&#038;bg=T&#038;fg=000000&#038;s=0' alt='C' title='C' class='latex' \/> joining any two given points in a region <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Ccal+R&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\cal R' title='\\cal R' class='latex' \/> is that in <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Ccal+R&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\cal R' title='\\cal R' class='latex' \/> <\/p>\n<p><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cpartial+P%2F%5Cpartial+y+%3D+%5Cpartial+Q%2F%5Cpartial+x%5Ccdots+%2823%29&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\partial P\/\\partial y = \\partial Q\/\\partial x\\cdots (23)' title='\\partial P\/\\partial y = \\partial Q\/\\partial x\\cdots (23)' class='latex' \/><\/p>\n<p>where it is supposed that these partial derivatives are continuous in <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Ccal+R&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\cal R' title='\\cal R' class='latex' \/>. <\/p>\n<\/ul>\n<p>The condition (23) is also the condition that <img src='https:\/\/s0.wp.com\/latex.php?latex=Pdx+%2B+Qdy&#038;bg=T&#038;fg=000000&#038;s=0' alt='Pdx + Qdy' title='Pdx + Qdy' class='latex' \/> is an exact differential, i.e. that there exists a function <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cphi%28x%2C+y%29&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\phi(x, y)' title='\\phi(x, y)' class='latex' \/> such that <img src='https:\/\/s0.wp.com\/latex.php?latex=Pdx+%2B+Qdy+%3D+d%5Cphi&#038;bg=T&#038;fg=000000&#038;s=0' alt='Pdx + Qdy = d\\phi' title='Pdx + Qdy = d\\phi' class='latex' \/>. In such case if the end points of curve <img src='https:\/\/s0.wp.com\/latex.php?latex=C&#038;bg=T&#038;fg=000000&#038;s=0' alt='C' title='C' class='latex' \/> are <img src='https:\/\/s0.wp.com\/latex.php?latex=%28x_1%2C+y_1%29&#038;bg=T&#038;fg=000000&#038;s=0' alt='(x_1, y_1)' title='(x_1, y_1)' class='latex' \/> and <img src='https:\/\/s0.wp.com\/latex.php?latex=%28x_2%2C+y_2%29&#038;bg=T&#038;fg=000000&#038;s=0' alt='(x_2, y_2)' title='(x_2, y_2)' class='latex' \/>, the value of the line integral is given by<\/p>\n<p><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%28x_1%2C+y_1%29%7D%5E%7B%28x_2%2C+y_2%29%7D%5BPdx+%2B+Qdy%5D+%3D+%5Cint_%7B%28x_1%2C+y_1%29%7D%5E%7B%28x_2%2C+y_2%29%7D+d%5Cphi+%3D+%5Cphi%28x_2%2C+y_2%29+-+%5Cphi%28x_1%2C+y_1%29+%5Ccdots%2824%29&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle \\int_{(x_1, y_1)}^{(x_2, y_2)}[Pdx + Qdy] = \\int_{(x_1, y_1)}^{(x_2, y_2)} d\\phi = \\phi(x_2, y_2) - \\phi(x_1, y_1) \\cdots(24)' title='\\displaystyle \\int_{(x_1, y_1)}^{(x_2, y_2)}[Pdx + Qdy] = \\int_{(x_1, y_1)}^{(x_2, y_2)} d\\phi = \\phi(x_2, y_2) - \\phi(x_1, y_1) \\cdots(24)' class='latex' \/><\/p>\n<p>In particular if (23) holds and <img src='https:\/\/s0.wp.com\/latex.php?latex=C&#038;bg=T&#038;fg=000000&#038;s=0' alt='C' title='C' class='latex' \/> is closed, we have <img src='https:\/\/s0.wp.com\/latex.php?latex=x_1+%3D+x_2%2C%5C+y_1+%3D+y_2&#038;bg=T&#038;fg=000000&#038;s=0' alt='x_1 = x_2,\\ y_1 = y_2' title='x_1 = x_2,\\ y_1 = y_2' class='latex' \/> and <\/p>\n<p><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+%5Coint_C+%5BPdx+%2B+Qdy%5D+%3D+0%5Ccdots%2825%29&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle \\oint_C [Pdx + Qdy] = 0\\cdots(25)' title='\\displaystyle \\oint_C [Pdx + Qdy] = 0\\cdots(25)' class='latex' \/><\/p>\n<p>The results in Theorem 6-1 can be extended to line integrals in space. Thus we have<\/p>\n<ul>\n<li><em><strong>Theorem 6-2. <\/strong><\/em><\/li>\n<p>A necessary and sufficient condition for <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+%5Cint_C+%5BA_1dx+%2B+A_2dy+%2B+A_3dz%5D&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle \\int_C [A_1dx + A_2dy + A_3dz]' title='\\displaystyle \\int_C [A_1dx + A_2dy + A_3dz]' class='latex' \/> to be independent of the path <img src='https:\/\/s0.wp.com\/latex.php?latex=C&#038;bg=T&#038;fg=000000&#038;s=0' alt='C' title='C' class='latex' \/> joining any two given points in a region <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Ccal+R&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\cal R' title='\\cal R' class='latex' \/> is that in <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Ccal+R&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\cal R' title='\\cal R' class='latex' \/><\/p>\n<p><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7B%5Cpartial+A_1%7D%7B%5Cpartial+y%7D+%3D+%5Cfrac%7B%5Cpartial+A_2%7D%7B%5Cpartial+x%7D%2C%5C+%5Cfrac%7B%5Cpartial+A_3%7D%7B%5Cpartial+x%7D+%3D+%5Cfrac%7B%5Cpartial+A_1%7D%7B%5Cpartial+z%7D%2C%5C+%5Cfrac%7B%5Cpartial+A_2%7D%7B%5Cpartial+z%7D+%3D+%5Cfrac%7B%5Cpartial+A_3%7D%7B%5Cpartial+y%7D+%5Ccdots%2826%29&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle \\frac{\\partial A_1}{\\partial y} = \\frac{\\partial A_2}{\\partial x},\\ \\frac{\\partial A_3}{\\partial x} = \\frac{\\partial A_1}{\\partial z},\\ \\frac{\\partial A_2}{\\partial z} = \\frac{\\partial A_3}{\\partial y} \\cdots(26)' title='\\displaystyle \\frac{\\partial A_1}{\\partial y} = \\frac{\\partial A_2}{\\partial x},\\ \\frac{\\partial A_3}{\\partial x} = \\frac{\\partial A_1}{\\partial z},\\ \\frac{\\partial A_2}{\\partial z} = \\frac{\\partial A_3}{\\partial y} \\cdots(26)' class='latex' \/><\/p>\n<p>where it is supposed that these partial derivatives are continuous in <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Ccal+R&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\cal R' title='\\cal R' class='latex' \/>. <\/p>\n<\/ul>\n<p>The results can be expressed concisely in terms of vectors. If <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cbold%7BA%7D+%3D+A_1%5Cbold%7Bi%7D+%2B+A_2%5Cbold%7Bj%7D+%2B+A_3%5Cbold%7Bk%7D+&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\bold{A} = A_1\\bold{i} + A_2\\bold{j} + A_3\\bold{k} ' title='\\bold{A} = A_1\\bold{i} + A_2\\bold{j} + A_3\\bold{k} ' class='latex' \/>, the line integral can be written <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+%5Cint_C+%5Cbold%7BA%7D%5Ccdot+d%5Cbold%7Br%7D&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle \\int_C \\bold{A}\\cdot d\\bold{r}' title='\\displaystyle \\int_C \\bold{A}\\cdot d\\bold{r}' class='latex' \/> and condition (26) is equivalent to the condition <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cnabla+%5Ctimes+%5Cbold%7BA%7D+%3D+0&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\nabla \\times \\bold{A} = 0' title='\\nabla \\times \\bold{A} = 0' class='latex' \/>. If <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cbold%7BA%7D&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\bold{A}' title='\\bold{A}' class='latex' \/> represents a force field <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cbold%7BF%7D&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\bold{F}' title='\\bold{F}' class='latex' \/> which acts on an object, the result is equivalent to the statement that the work done in moving the object from one point to another is independent of the path joining the two points if and only if <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cnabla+%5Ctimes+%5Cbold%7BA%7D+%3D+0&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\nabla \\times \\bold{A} = 0' title='\\nabla \\times \\bold{A} = 0' class='latex' \/>. Such a force field is often called <em>conservative<\/em>. <\/p>\n<p>The condition (26) [or the equivalent condition <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cnabla%5Ctimes%5Cbold%7BA%7D%3D0&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\nabla\\times\\bold{A}=0' title='\\nabla\\times\\bold{A}=0' class='latex' \/>] is also the condition that <img src='https:\/\/s0.wp.com\/latex.php?latex=A_1dx+%2B+A_2dy+%2B+A_3dz&#038;bg=T&#038;fg=000000&#038;s=0' alt='A_1dx + A_2dy + A_3dz' title='A_1dx + A_2dy + A_3dz' class='latex' \/> [or <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cbold%7BA%7D%5Ccdot%5Cbold%7Br%7D&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\bold{A}\\cdot\\bold{r}' title='\\bold{A}\\cdot\\bold{r}' class='latex' \/>] is an exact differential, i.e. that there exists a function <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cphi%28x%2C+y%2C+z%29&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\phi(x, y, z)' title='\\phi(x, y, z)' class='latex' \/> such that <img src='https:\/\/s0.wp.com\/latex.php?latex=A_1dx+%2B+A_2dy+%2B+A_3dz+%3Dd%5Cphi&#038;bg=T&#038;fg=000000&#038;s=0' alt='A_1dx + A_2dy + A_3dz =d\\phi' title='A_1dx + A_2dy + A_3dz =d\\phi' class='latex' \/>. In such case if the endpoints of curve <img src='https:\/\/s0.wp.com\/latex.php?latex=C&#038;bg=T&#038;fg=000000&#038;s=0' alt='C' title='C' class='latex' \/> are <img src='https:\/\/s0.wp.com\/latex.php?latex=%28x_1%2C+y_1%2C+z_1%29&#038;bg=T&#038;fg=000000&#038;s=0' alt='(x_1, y_1, z_1)' title='(x_1, y_1, z_1)' class='latex' \/> and <img src='https:\/\/s0.wp.com\/latex.php?latex=%28x_2%2C+y_2%2C+z_2%29&#038;bg=T&#038;fg=000000&#038;s=0' alt='(x_2, y_2, z_2)' title='(x_2, y_2, z_2)' class='latex' \/>, the value of the line integral is given by<\/p>\n<p><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%28x_1%2C+y_1%2C+z_1%29%7D%5E%7B%28x_2%2C+y_2%2C+z_2%29%7D%5Cbold%7BA%7D%5Ccdot%5Cbold%7Br%7D+%3D+%5Cint_%7B%28x_1%2C+y_1%2C+z_1%29%7D%5E%7B%28x_2%2C+y_2%2C+z_2%29%7Dd%5Cphi+%3D+%5Cphi%28x_2%2C+y_2%2C+z_2%29-+%5Cphi%28x_1%2C+y_1%2C+z_1%29%5Ccdots%2827%29&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle \\int_{(x_1, y_1, z_1)}^{(x_2, y_2, z_2)}\\bold{A}\\cdot\\bold{r} = \\int_{(x_1, y_1, z_1)}^{(x_2, y_2, z_2)}d\\phi = \\phi(x_2, y_2, z_2)- \\phi(x_1, y_1, z_1)\\cdots(27)' title='\\displaystyle \\int_{(x_1, y_1, z_1)}^{(x_2, y_2, z_2)}\\bold{A}\\cdot\\bold{r} = \\int_{(x_1, y_1, z_1)}^{(x_2, y_2, z_2)}d\\phi = \\phi(x_2, y_2, z_2)- \\phi(x_1, y_1, z_1)\\cdots(27)' class='latex' \/><\/p>\n<p>In particular if <img src='https:\/\/s0.wp.com\/latex.php?latex=C&#038;bg=T&#038;fg=000000&#038;s=0' alt='C' title='C' class='latex' \/> is closed and <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cnabla%5Ctimes%5Cbold%7BA%7D+%3D+0&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\nabla\\times\\bold{A} = 0' title='\\nabla\\times\\bold{A} = 0' class='latex' \/>, we have <\/p>\n<p><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+%5Coint_C+%5Cbold%7BA%7D%5Ccdot+d%5Cbold%7Br%7D+%3D+0+%5Ccdots%2828%29&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle \\oint_C \\bold{A}\\cdot d\\bold{r} = 0 \\cdots(28)' title='\\displaystyle \\oint_C \\bold{A}\\cdot d\\bold{r} = 0 \\cdots(28)' class='latex' \/>\n<\/p><\/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>Theorem 6-1. A necessary and sufficient condition for to be independent of the path joining any two given poin &hellip; <a href=\"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5742\" class=\"more-link\"><span class=\"screen-reader-text\">&#8220;CONDITIONS FOR A LINE INTEGRAL TO BE INDEPENDENT OF THE PATH&#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":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[9],"tags":[101,100,98,97,99],"class_list":["post-5742","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics","tag-conservative","tag-force-field","tag-independent-of-the-path","tag-necessary-and-sufficient-condition","tag-partial-derivatives"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 4.9.9 - aioseo.com -->\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<meta name=\"author\" content=\"admin\"\/>\n\t<meta name=\"keywords\" content=\"necessary and sufficient condition,independent of the path,partial 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