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Handbook of mathematics for engineers and scienteists part 53

The circle passing through the vertices of a triangle is called the circumcircle of the triangle. The center O[ of the circumcircle. called the circumcemer, is the point where the perpendicular bisectors of the sides of the triangle meet (Fig. 3.3a). | 332 Integrals 7.4.2-4. Necessary and sufficient conditions for a vector field to be potential. Let U be a simply connected domain in R3 i.e. a domain in which any closed contour can be deformed to a point without leaving U and let a x y z be a vector field in U. Then the following four assertions are equivalent to each other 1 the vector field a is potential 2 curl a 0 3 the circulation of a around any closed contour C G U is zero or equivalently C a dr 0 4 the integral a dr is independent of the shape of AB U it depends on the starting and the finishing point only . 7.4.3. Surface Integral of the First Kind 7.4.3-1. Definition of the surface integral of the first kind. Let a function f x y z be defined on a smooth surface D. Let us break up this surface into n elements cells that do not have common internal points and let us denote this partition by Dn. The diameter X Dn of a partition Dn is the largest of the diameters of the cells see Paragraph 7.3.4-1 . Let us select in each cell an arbitrary point Xi yi Zi i 1 2 . n and make up an integral sum n Sn f Xi yi Zi ASi where ASi is the area of the ith element. If there exists a finite limit of the sums sn as X Dn 0 that depends on neither the partition Dn nor the selection of the points xi yi zi then it is called the surface integral of the first kind of the function f x y z and is denoted f x y z dS. 7.4.3-2. Computation of the surface integral of the first kind. 1 . If a surface D is defined by an equation z z x y with x y g D1 then d f x y z dS D f x y z x y 1 z x 2 z y 2 dx dy. 2 . If a surface D is defined by a vector equation r r x y z x u v i y u v j z u v k where u v g D2 then UD f x z dS d f x v y z u v In u v dudu. where n u v ru X rv is the unit normal to the surface D the subscripts u and v denote the respective partial derivatives. 7.4. Line and Surface Integrals 333 7.4.3-3. Applications of the surface integral of the first kind. 1 . Area of a surface D SD jj dS. 2 . Mass of a material surface D with a