Handbook of mathematics for engineers and scienteists part 30 pot

... Press, 1995 and 2003; A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, 1998; A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, ... V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, 2002; and A. D. Polyanin and V. F. Zaitsev, Hand...

Ngày tải lên: 02/07/2014, 13:20

... stands for the union of sets A and B ∩ intersection (Boolean multiplication); A ∩B stands for the intersection (com- mon part) of sets A and B ⊂ inclusion; A ⊂ B means that the set A is part of ... of numbers a and b is denoted by a + b and has the property a + b = b + a – minus sign; the difference of numbers a and b is denoted by a – b ⋅ multiplication sign; the p...

Ngày tải lên: 02/07/2014, 13:20

... 2πn]. For x = π 2 + 2πn, it attains its maximal value (y = 1), and for x =– π 2 + 2πn it attains its minimal value (y =–1). The graph of the function y =sinx is called the sinusoid or sine curve and ... The graph of the function y =cotx isgiveninFig.2 .9. O 1 x y π π 1 yx= tan π 2 π 2 Figure 2.8. The graph of the function y =tanx. O 1 x y π π 1 yx= cot π 2 π 2 3π 2 Figure 2 .9....

Ngày tải lên: 02/07/2014, 13:20

... r is the radius of the upper base and h is the altitude of the frustum of a cone. 3.2.3-3. Sphere. Spherical parts. Torus. 1 ◦ .Thesphere of radius R centered at O is the set of points in space ... h), (3.2.3.10) where R and h are the radius and the height of the spherical cap. 3 ◦ . A portion of a ball bounded by the curved surface of a spherical cap and the conical s...

Ngày tải lên: 02/07/2014, 13:20

... r 1 (M 0 )andr 2 (M 0 ) are the lengths of the focal radii of M 0 . The tangent at an arbitrary point M 0 (x 0 , y 0 ) of an ellipse forms acute angles ϕ 1 and ϕ 2 with the focal radii of the point of ... y) of an ellipse with the foci F 1 (–c, 0)andF 2 (c, 0) are called the left and right focal radii of this point. We denote the lengths of the left and right focal radii...

Ngày tải lên: 02/07/2014, 13:20

... is the locus of points for which the ratio of distances to the focus and the directrix is equal to 1: r |x + p/2| = 1.(4.4.4.6) 4.4.4-4. Equation of tangent and optical property of parabola. The ... ϕ ,(4.4.3.16) where upper and lower signs correspond to right and left parts of a hyperbola, respectively. 4.4.4. Parabola 4.4.4-1. Definition and canonical equation of para...

Ngày tải lên: 02/07/2014, 13:20

... 0. 4.6.1-12. Equation of plane passing through line of intersection of planes. The planes passing through the line of intersection of the planes A 1 x +B 1 y + C 1 z + D 1 = 0 and A 2 x + B 2 y + ... cosines of the angles α, β ,and formed by this straight line (the direction of the vector R 0 ) with the coordinate axes OX, OY ,andOZ. These cosines can be expressed via the coor...

Ngày tải lên: 02/07/2014, 13:20

... a nn         = n  k=1 a ik A i k = n  k=1 a kj A k j . This formula is also called the ith row expansion of the determinant of A andalsothejth column expansion of the determinant of A. Example 1. Let us find the third-order determinant of the matrix A ... [a i a j a B]. Note that if A and B are square matrices and the number of rows in C is equal to the number of ro...

Ngày tải lên: 02/07/2014, 13:20

... to both spaces L 1 and L 2 .Such elements form a subspace of V. The sum of subspaces L 1 and L 2 of one and the same linear space V is, by definition, the set of all elements of V that can be represented ... of matrices A and B consist of eigenvalues λ j and μ k , respec- tively. Then the spectrum of the Kronecker product A ⊗ B is the set of all products λ j μ k . T...

Ngày tải lên: 02/07/2014, 13:20

... called addition and c is called the sum of a and b. 2. Multiplicative form: c = ab; the corresponding composition law is called multiplication and c is called the product of a and b. A composition ... group and let  G be a set with a composition law. A mapping f : G →  G is called a homomorphism if f(ab)=f(a)f(b)foralla, b G; and the subset of  G consisting of all eleme...

Ngày tải lên: 02/07/2014, 13:20

... Baltimore, Maryland, 1996. Hazewinkel, M. (Editor), Handbook of Algebra, Vol. 1, North Holland, Amsterdam, 1996. Hazewinkel, M. (Editor), Handbook of Algebra, Vol. 2, North Holland, Amsterdam, ... transformations,forwhichdetP =+1. 2. Improper orthogonal transformations,forwhichdetP =–1. The set of proper orthogonal transformations forms a group called the special orthogonal group o...

Ngày tải lên: 02/07/2014, 13:20

... point of its domain. 6.1.5-3. Points of discontinuity of a function. A point a is called a point of discontinuity of the first kind for a function f(x)ifthereexist finite one-sided limits f(a+0)andf(a–0), ... small for x → a. Functions f (x)andg(x) are said to be of the same order for x → a, and one writes f(x)=O  g(x)  if lim x→a f(x) g(x) = K, 0 < |K| < ∞.* A funct...

Ngày tải lên: 02/07/2014, 13:20

... expressions of the form 0/0 and ∞ /∞. THEOREM 1. Let f(x) and g(x) be two functions defined in a neighborhood of a point a , vanishing at this point, f(a)=g(a)=0 , and having the derivatives f  (a) and g  (a) , ... Calculus for Functions of a Single Variable 6.2.1. Derivative and Differential, Their Geometrical and Physical Meaning 6.2.1-1. Definition of derivative and...

Ngày tải lên: 02/07/2014, 13:20

... integrals of the form 3 ◦ from Paragraph 7.1.4-1. 7.1.5. Integration of Exponential and Trigonometric Functions 7.1.5-1. Integration of exponential and hyperbolic functions. 1. Integrals of the form  R(e px , ... of a proper fraction. 1 ◦ . To integrate a proper fraction, one should first rewrite the integrand (7.1.3.1) in the form of a sum of partial fractions. Below are th...

Ngày tải lên: 02/07/2014, 13:20

... S(t) is the boundary of the domain U(t), n is the unit normal to S(t), and v is the velocity of motion of the points of S(t). 7.3.5-4. Some geometric and physical applications of the triple integral. 1. ... neither the partition L n nor the selection of the points (x i , y i , z i ), then it is called the line integral of the first kind of the function f(x, y, z) over the...

Ngày tải lên: 02/07/2014, 13:20