tailieunhanh - Handbook of mathematics for engineers and scienteists part 52

A straight line passing through the midpoint of a segment and perpendicular to it is called the perpendicular bisector of the segment. The circle passing through the vertices of a triangle is called the circumcircle of the triangle. The center O[ of the circumcircle. | . Double and Triple Integrals 325 2. Additivity. If a domain U is split into two subdomains U1 and U2 that do not have common internal points and if a function f x y z is integrable in either subdomain then f x y z dxdydz f x y z dxdydz f x y z dxdydz. J J Ju JJJu1 J J JU2 3. Estimation theorem. If m f x y z M in a domain U then mV J J J f x y z dx dy dz MV where V is the volume of U. 4. Mean value theorem. If f x y z is continuous in U then there exists at least one internal point x y z G U such that JJJ f x y z dx dy dz f x y z V. The number f r y z is called the mean value of the function f in the domain U. 5. Integration of inequalities. If x y z f x y z g x y z in a domain U then yyy x y z dxdydz JJJ f x y z dxdydz JJJ g x y z dxdydz. 6. Absolute value theorem JJJ f x y z dxdydz IJJ f x y z dxdydz. . Computation of the Triple Integral. Some Applications. Iterated Integrals and Asymptotic Formulas . Use of iterated integrals. 1 . Consider a three-dimensional body U bounded by a surface z g x y from above and a surface z h x y from below with a domain D being the projection of it onto the x y plane. In other words the domain U is defined as x y e D h x y z g x y . Then rr r tc i-g J J J f x y z dxdydz J J dx dy J f x y z dz. 2 . If under the same conditions as in Item 1 the domain D of the x y plane is defined as a x b y1 x y y2 x then f x y z dxdydz Îb dx Î dy f x y z dz. U Ja Jyi x Jh x y 326 Integrals . Change of variables in the triple integral. 1 . Let x x u v w y y u v w and z z u v w be continuously differentiable functions that map one to one a domain Q of the u v w space into a domain U of the x y z space and let a function f x y z be continuous in U. Then yyy f x y z dxdy dz JJJ f x u v w y u v w z u v w J u v w dudv dw where J u v w is the Jacobian of the mapping of Q into U J u v w d x y z d u v w dx dx dx du dv dw dy dy dy du dv dw dz dz dz du dv dw The expression in the middle is a very common notation for a Jacobian. The