{"id":5770,"date":"2014-09-29T06:05:37","date_gmt":"2014-09-28T21:05:37","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5770"},"modified":"2014-08-08T19:42:50","modified_gmt":"2014-08-08T10:42:50","slug":"surface-integrals","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5770","title":{"rendered":"SURFACE INTEGRALS"},"content":{"rendered":"
\"Fig.<\/a>
Fig. 6-3<\/figcaption><\/figure>\n

Let S be a two-sided surface having projection \\cal R on the xy plane as in the adjoining Fig. 6-3. Assume that an equation for S is z = f(x, y), where f is single-valued and continuous for all x and y in \\cal R. Divide \\cal R into n subregions of area \\Delta A_p,\\ p = 1,\\ 2,\\ \\dots,\\ n, and erect a vertical column on each of these subregions to intersect S in an area \\Delta S_p. <\/p>\n

Let \\phi (x, y, z) be single-valued and continuous at all points of S. Form the sum<\/p>\n

\\displaystyle \\sum_{p=1}^{n}\\phi(\\xi_p, \\eta_p, \\zeta_p)\\Delta S_p \\cdots(29)<\/p>\n

where (\\xi_p, \\eta_p, \\zeta_p) is some point of \\Delta S_p. If the limit of this sum as n \\rightarrow \\infty in such a way that each \\Delta S_p \\rightarrow 0 exists, the resulting limit is called the surface integral<\/em> of \\phi(x, y, z) over S and is designated by <\/p>\n

\\displaystyle \\underset{S}{\\iint}\\phi(x, y, z)dS\\cdots(30)<\/p>\n

Since \\Delta S_p = |\\sec\\gamma_p|\\Delta A_p approximately, where \\gamma_p is the angle between the normal line to S and the positive z axis, the limit of the sum (29) can be written <\/p>\n

\\displaystyle \\underset{\\cal R}{\\iint}\\phi(x, y, z)|\\sec\\gamma|dA\\cdots(31)<\/p>\n

The quantity |\\sec\\gamma| is given by <\/p>\n

\\displaystyle |\\sec\\gamma| = \\frac{1}{|\\bold{n}_p\\cdot\\bold{k}|} = \\sqrt{1 + \\left( \\frac{\\partial z}{\\partial x} \\right)^2 + \\left( \\frac{\\partial z}{\\partial y} \\right)^2}\\cdots(32)<\/p>\n

Then assuming that x = f(x, y) has continuous (or sectionally continuous) derivatives in \\cal R, (31) can be written in rectangular form as <\/p>\n

\\displaystyle \\underset{\\cal R}{\\iint}\\phi(x, y, z)\\sqrt{1 + \\left( \\frac{\\partial z}{\\partial x} \\right)^2 + \\left( \\frac{\\partial z}{\\partial y} \\right)^2}dxdy \\cdots(33)<\/p>\n

In case the equation for S is given as F(x, y, z) = 0, (33) can also be written <\/p>\n

\\displaystyle \\underset{S}{\\iint}\\phi(x, y, z)\\frac{\\sqrt{(F_x)^2 + (F_y)^2 + (F_z)^2}}{|F_z|}dxdy\\cdots(34)<\/p>\n

The results (33) or (34) can be used to evaluate (30). <\/p>\n

In the above we have assumed that S is such that any line parallel to the z axis intersects S in only one point. In case S is not of this type, we can usually subdivide S into surfaces S_1,\\ S_2,\\ \\dots which are of this type. Then the surface integral over S is defined as the sum of the surface integrals over S_1,\\ S_2,\\ \\dots<\/p>\n

The results stated hold when S is projected on to a region \\cal R of the xy plane. In some cases it is better to project S on to the yz or xz planes. For such cases (30) can be evaluated by appropriately modifying (33) and (34). <\/p>\n<\/blockquote>\n