Handbook of mathematics for engineers and scienteists part 178 pot
Handbook of mathematics for engineers and scienteists part 178 pot
... (1977) and Polyanin and Zaitsev (2003).24. yyx= y + f (x).Abel equation of the second kind in the canonical form. Many solvable equations of thisform can be found in the books by Zaitsev and ... (1994) and Polyanin and Zaitsev(2003).25. yyx= f(x)y + g(x).Abel equation of the second kind. Many solvable equations of this form can be found in thebooks by Zaitsev and Polyanin (1994) and ... Polyanin and Zaitsev (2003).26. yyx= f(x)y2+ g(x)y + h(x).Abel equation of the second kind. Many solvable equations of this form can be found in thebooks by Zaitsev and Polyanin (1994) and...
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Handbook of mathematics for engineers and scienteists part 4 potx
... 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, ... compendium of mathematicaldefinitions, formulas, and theorems intended for researchers, university teachers, engineers, and students of various backgrounds in mathematics. The absence of proofs and ... V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis,2002; and A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial DifferentialEquation,...
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Handbook of mathematics for engineers and scienteists part 5 potx
... 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 the set B⊆ nonstrict inclusion; A ⊆B means that the set A is part of the set ... of numbers a and b is denoted by a + b and has the propertya + b = b + a– minus sign; the difference of numbers a and b is denoted by a – b⋅ multiplication sign; the product of numbersa and ... also appreciate the help of Vladimir Nazaikinskii and GrigoriiYosifian for translating several chapters of this book and are thankful to Kirill Kazakov and Mikhail Mikhin for their assistance in...
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Handbook of mathematics for engineers and scienteists part 9 pot
... 2πn]. For x =π2+ 2πn, it attains its maximal value (y = 1), and for x =–π2+ 2πn it attains its minimal value (y =–1). The graph of the function y =sinx iscalled the sinusoid or sine curve and ... on everysegment [–π + 2πn, 2πn] and is a decreasing function on every segment [2πn, π + 2πn],n = 0,1, 2, For x = 2πn it attains its maximal value (y = 1), and for x = π + 2πnit attains its ... 2.6.O1ππxyπ21yx= sinπ2Figure 2.6. The graph of the function y =sinx.2.2.2-2. Cosine: y =cosx.This function is defined for all x and its range is y [–1, 1]. The cosine is a bounded,even,...
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Handbook of mathematics for engineers and scienteists part 15 pot
... 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 conicalsurface whose base is the base of the ... called a spherical zone, and the plane circular surfaces are the bases of a spherical segment. The radius R of theball, the radii a and b of the bases, and the height h of a spherical segment...
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Handbook of mathematics for engineers and scienteists part 19 potx
... r1(M0)andr2(M0) are the lengths of the focal radii of M0.The tangent at an arbitrary point M0(x0, y0) of an ellipse forms acute angles ϕ1 and ϕ2with the focal radii of the point of ... y) of an ellipse with the foci F1(–c, 0)andF2(c, 0)are called the left and right focal radii of this point. We denote the lengths of the left and right focal radii by r1= |F1M0| and ... a2.(4.4.1.2)The area of the disk bounded by a circle of radius a is given by the formula S = πa2.The length of this circle is L = 2πa. The area of the figure bounded by the circle and thechord passing...
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Handbook of mathematics for engineers and scienteists part 20 pot
... is the locus of points for which theratio 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. Parabola4.4.4-1. Definition and canonical equation of parabola.A curve on the ... parameter of the hyperbola.The point O(0, 0) is called the center of the hyperbola. The points A1(–a, 0)andA2(a, 0) of intersection of the hyperbola with the real axis are called the vertices of...
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Handbook of mathematics for engineers and scienteists part 24 pot
... 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 A1x +B1y + C1z + D1= 0 and A2x + B2y + ... cosines of the angles α, β ,and formed by this straight line (the direction of the vector R0) with the coordinate axes OX, OY ,andOZ. These cosines can be expressedvia the coordinates of the ... equations of the form (4.6.2.4) define a straight line if and only if the coefficientsA1, B1,andC1in one of them are not proportional to the respective coefficients A2, B2,andC2in the...
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Handbook of mathematics for engineers and scienteists part 30 pot
... ann=nk=1aikAik=nk=1akjAkj.This formula is also called the ith row expansion of the determinant of A andalsothejthcolumn expansion of the determinant of A.Example 1. Let us find the third-order determinant of the matrixA ... [aiajaB].Note that if A and B are square matrices and the number of rows in C is equal to thenumber of rows in A, and the number of rows in D is equal to the number of rows in B,then(A ⊗ B)(C ... [Cαβ]=[γAαγBγβ].4. For a matrix A ≡ [aij]ofsizem× n regarded as a block matrix A ≡ [Aαβ]ofsizeM × N,the transpose has the form AT=[ATβα].5. For a matrix A ≡ [aij]ofsizem× n regarded...
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Handbook of mathematics for engineers and scienteists part 32 potx
... to both spaces L1 and L2.Suchelements form a subspace of V.The sum of subspaces L1 and L2 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 λj and μk, respec-tively. Then the spectrum of the Kronecker product A ⊗ B is the set of all products λjμk.The spectrum of the direct sum of ... ,xm.5.3.2-2. Sum and intersection of subspaces.The intersection of subspaces L1 and L2 of one and the same linear space V is, by definition,the set of all elements x of V that belong simultaneously...
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