{"id":5651,"date":"2014-08-11T06:05:16","date_gmt":"2014-08-10T21:05:16","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5651"},"modified":"2014-08-08T19:49:04","modified_gmt":"2014-08-08T10:49:04","slug":"line-integrals","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5651","title":{"rendered":"LINE INTEGRALS"},"content":{"rendered":"
\"Fig.<\/a>
Fig. 6-2<\/figcaption><\/figure>\n

Let C<\/em> be a curve in the xy<\/em> plane which connects points A (a_1, b_1) and B (a_2, b_2), (see Fig. 6-2). Let P(x, y) and Q(x, y) be single-valued functions defined at all points of C<\/em>. Subdivide C<\/em> into n<\/em> parts by choosing n – 1<\/em> points on it given by (x_1, y_1),\\ (x_2, y_2),\\ \\dots,\\ (x_{n-1}, y_{n-1}). Call \\Delta x_k = x_k - x_{k-1} and \\Delta y_k = y_k - y_{k-1},\\ k = 1,\\ 2,\\ \\dots\\ n and suppose that points (\\xi_k, \\eta_k) are chosen so that they are situated on C<\/em> between points (x_{k-1}, y_{k-1}) and (x_k, y_k). Form the sum<\/p>\n

\\displaystyle \\sum_{k=1}^{n}\\{P(\\xi_k, \\eta_k)\\Delta x_k + Q(\\xi_k, \\eta_k)\\Delta y_k\\}\\cdots(13)<\/p>\n

The limit of this sum as n\\rightarrow\\infty in such a way that all quantities \\Delta x_k,\\ \\Delta y approaches zero, if such limit exists, is called a line integral<\/em> along C<\/em> and is denoted by<\/p>\n

\\displaystyle \\int_C \\left[ P(x, y)dx + Q(x, y)dy \\right] or \\displaystyle \\int_{(a_1, b_1)}^{(a_2, b_2)}\\left[ Pdx + Qdy \\right]\\cdots(14)<\/p>\n

The limit does exist if P<\/em> and Q<\/em> are continuous (or piecewise continuous) at all points of C<\/em>. The value of the integral depends in general on P<\/em>, Q<\/em>, the particular curve C<\/em>, and on the limits (a_1, b_1) and (a_2, b_2). <\/p>\n

In an exactly analogous manner one may define a line integral along a curve C<\/em> in three dimensional space as <\/p>\n

\\displaystyle \\lim\\limits_{n \\rightarrow\\infty}\\sum_{k=1}^{n}\\left\\{ A_1(\\xi_k, \\eta_k, \\zeta_k)\\Delta x_k + A_2(\\xi_k, \\eta_k, \\zeta_k)\\Delta y_k + A_3(\\xi_k, \\eta_k, \\zeta_k)\\Delta z_k \\right\\} \\\\ = \\int_C \\left[ A_1dx + A_2dy + A_3dz \\right] \\cdots(15)<\/p>\n

where A_1, A_2 and A_3 are functions of x, y and z. <\/p>\n

Other types of line integrals, depending on particular curves, can be defined. For example, if \\Delta s_k denotes the arc length along curve C<\/em> in the above figure between points (x_k, y_k) and (x_{k+1}, y_{k+1}), then<\/p>\n

\\displaystyle \\lim\\limits_{n \\rightarrow \\infty} \\sum_{k=1}^{n} U(\\xi_k, \\eta_k)\\Delta s_k = \\int_C U(x, y)ds\\cdots(16)<\/p>\n

is called the line integral of U(x, y) along curve C<\/em>. Extensions to three (or higher) dimensions are possible. <\/p>\n<\/blockquote>\n