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integration are infinite, when the functions tend to zero at infinite distances in a sufficiently high order (see FUNCTION). The method of contour integration involves the introduction of complex variables (see FUNCTION! § Complex Variables). A few results are added

~ F x“"' 'rr

(1~>J 0 (I>l1>0),

f°° ta-1 xbarbitrarily

chosen length. e may devise a rule for increasing the number of marked points indefinitely and decreasing the lengths of all the edges of the polyhedral indefinitely. If the sum of the areas of the faces tends to a limit, this limit is the area of the surface. If we multiply the value of a function fat a point of the surface by the measure of the area of the corresponding face of the polyhedron, sum for all the faces, and pass to a limit as before, the result is a surface fdS.

The extension to the case of an open surface bounded Li by an edge presents no difficulty. A line integral taken I fe 'Is along a curve is defined in a similar way, and is written U ep ° ffds

where ds is the element of arc of the curve (§ 33). The direction cosines of the tangent of a curve are dx/ds, dy/ds, dz/ds, and line integrals usually present themselves in the form uds vds 'wds sor /t(u x z y w z). I(eye., ey, ,, ., .., + .,

In like manner surface integrals usually present themselves in the form

integral, and is written


where l, m, n are the direction cosines of the normal to the surface drawn in a specified sense.

The area of a bounded portion of the plane of (x, y) may be expressed either as



the former integral being a line integral taken round the boundary of the portion, and the latter a surface integral taken over the area within this boundary: In forming the line integral the boundary is OF 3.5

supposed to be described in the positive sense, so that the included area is on the left hand.

5311. We have two theorems of transformation connect- Theorems ing volume integrals with surface integrals and surface ofGreen integrals with line integrals. The first theorem, called and “ Green's theorem, " is expressed by the equation Stokes. (QS 61/ 6§

ff (@+5+@ dxdydz=ff<1s+m»+»f>dS.

(ii) 4TTdx=1r(cota1r-cotb1r), (0 <a or b<I) J o - I


x'1"1 log x 7l'2

(ll1~)J 3 -Trfdx-Sinzar, (a> I),

rm 2

iv.) x'1.cos 2x.e“" dx = - LC lv7I', 0 4


“I-t2 dr 1r

v. —4 ' =l t -,

(>, 0I+x4lOgx og M18

(, »)"°°sinmx, I I I

“- V 0 § °f;dx=§ 1-#5 »

ua F" -

(v11.) I log(1 -2acosx+a.2)dx ==oor21rlog a according as a <or> I, I o

fx, ' .

(viii.) § dx= 51.-,

J 0 x

I cos ax

ix.) + d =1 1, -1 -ab,

(.» 0 x”+b2 V 2” e


- b.

(Xb) eosax Zcos »cdx=;7r (Bu (iw 8v (Qu

s(udx+vdy-I-wdz) -ffl! ay 62 -l-m az ax) where the integral on the left is taken round the curve s in the chosen sense. When the axes are left-handed, we may either reverse the sense of Z, m, n and maintain the formula, or retain the sense of Z, m, n and change the sign of the right-hand member of the equation. For the validity of the theorems of Green and Stokes it is in general necessary that the functions involved should satisfy certain conditions of continuity. For example, in Green's theorem the differential coefficients 6E/Bx, 617/By, 65”/62 must be continuous within S. Further, there are restrictions upon the nature of the curves or surfaces involved. For example, Green's theorem, as here stated, applies only to simply-connected regions of space. The correction for multiply-connected regions is important in several ph 'sical 3


54. The process of changing the variables in a multiple integral, such as a surface or volume integral, is divisible into two " is necessary in the first place to determine the differential S

tages. It

element expressed by the product of the differentials of the change °f first set of variables in terms of the differentials of the variables second set of variables. It is necessary in the second place 1; in I to determine the limits of integration which must be em- I i: pi ployed when the integral in terms of the new variables is H eg" evaluated as a repeated integral. The first part of the problem is solved at once by the introduction of the Jacobian. If the variables of one set are denoted by xl, xg, . . ., x, ,, and those of the other set by ul, 142, . . ., un, we have the relation 6(x1 xg


dxdx...dx =—L-l-dud¢.., d

2 " 0(141, u2, ...,11, .) 1 12 M"