Handbook of mathematics for engineers and scienteists part 4 potx
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, ... 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, ... the Department of Applied Mathematics at Bauman Moscow State Technical University, professor of mathematics at MoscowState University of Engineering and Computer Science, vice-chairman of Mathematics and...
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Handbook of mathematics for engineers and scienteists part 5 potx
... 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 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 ... Alexei Zhurov for fruitful discussions and valuable remarks. We also appreciate the help of Vladimir Nazaikinskii and GrigoriiYosifian for translating several chapters of this book and are thankful...
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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 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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Handbook of mathematics for engineers and scienteists part 38 potx
... 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 orthogonalgroup of dimension n and ... product of two cosets aH and bH(as subsets of G)isthecosetabH. Consider the set Q whose elements are cosets of thesubgroup H in G, and define the product of the elements of Q as the product of cosets.Endowed...
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Handbook of mathematics for engineers and scienteists part 52 potx
... S(t) is the boundary of the domain U(t), n is the unit normal to S(t), and v is thevelocity of motion of the points of S(t).7.3.5-4. Some geometric and physical applications of the triple integral.1. ... neither thepartition Lnnor the selection of the points (xi, yi, zi), then it is called the line integral of the first kind of the function f(x, y, z) over the curveAB and is denotedABf(x, ... integral of the first kind does not depend of the direction the pathAB is traced; its properties aresimilar to those of the definite integral.7.4.1-2. Computation of the line integral of the...
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Handbook of mathematics for engineers and scienteists part 60 potx
... vectors of the tangent, of the binormal, and of the principal normal.9.2. THEORY OF SURFACES 387Varying the parameters u and v arbitrarily, we obtain the position vector and the coordi-nates of ... the natural parameter s of the position vector of a point of a curve is equal to the first derivative of the unit vector rs, i.e., of a vector of constant length, and hence it is perpendicular ... equation of the surface near thispoint can be solved for one of the coordinates. Condition (9.2.1.2) also guarantees a one-to-one correspondence between points on the surface and pairs of values of...
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Handbook of mathematics for engineers and scienteists part 100 potx
... variables, x and t,wheret plays therole of time and x is a spatial coordinate.15.3.2. Traveling-Wave Solutions. Invariance of Equations UnderTranslations15.3.2-1. General form of traveling-wave ... solution of equation (15.2.3.8). Then the formulas (15.2.3.5) definethe corresponding solution of equation (15.2.3.7) in parametric form.Remark. The Legendre transformation may result in the loss of ... corresponding partial derivatives.The Euler transformation (15.2.3.9) is employed in finding solutions and linearization of certain nonlinear partial differential equations.The Euler transformation...
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Handbook of mathematics for engineers and scienteists part 109 potx
... successive application(composition) of two transformations of the form (15.8.4.1) with parameters ε1 and ε2isequivalent to a single transformation of the same form with parameter ε1+ ε2.Further ... systemFigure 15.4. An algorithm for constructing invariant solutions for evolution second-order equations. Notation:ODE stands for ordinary differential equation and PDE stands for partial differential ... solutions.Suppose a particular solution,w = g(x, y), (15.8.3.1) of a given equation is known. Let us show that any symmetry of the equation definedby a transformation of the form (15.8.1.1) generates...
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Handbook of mathematics for engineers and scienteists part 112 potx
... generality of itsstatements and the necessity of selecting differential constraints suitable for specific classes of equations. Thisis why for the construction of exact solutions of nonlinear ... w), η = η(x, y, w), and ζ = ζ(x, y, w) are unknown functions, and thecoordinates of the first and the second prolongations ζi and ζijare defined by formu-las (15.8.1.9) and (15.8.1.14). The ... twoindependent variables, x and t, and one dependent variable, w, explicitly independent of x and t.A solution sought in a small neighborhood of a manifold x – x0(t)=0 is the form of thefollowing series...
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Handbook of mathematics for engineers and scienteists part 125 potx
... (16.4.6.2) has the formY(u)=F(u)1 + K(u).(16.4.6.3)Formula (16.4.6.3) gives the transform of the solution of the original integral equation interms of the transforms of the known functions, ... λ.(16.4.5.24)Here the collections ϕk(x )and kform the system of eigenfunctions and characteristicvalues of (16.4.5.22). It follows from formula (16.4.5.24) that the resolvent of a symmetrickernel has only ... transform we reduce the solution of the original integralequation (16.4.6.1) to the solution of the algebraic equation (16.4.6.2) for the transform of the unknown function. The solution of equation...
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Handbook of mathematics for engineers and scienteists part 139 potx
... bilinear functional equation of the form (17.5.5.8). Usingformulas (17.5.5.9) for the construction of its solution, one can first obtain a system of ODEs and then find solutions of the original equation ... A. D. and Zaitsev, V. F., Handbook of Nonlinear Partial Differential Equations (Sections S.4 and S.5), Chapman & Hall/CRC Press, Boca Raton, 2004.Polyanin, A. D., Zaitsev, V. F., and Zhurov, ... Institute of Com-putational Mathematics and Mathematical Geophysics, Novosibirsk, 1997.934 DIFFERENCE EQUATIONS AND OTHER FUNCTIONAL EQUATIONSwhere B1, , B4are arbitrary constants and k1and...
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Handbook of mathematics for engineers and scienteists part 157 potx
... spots on top of the first die, and let X2be the number of spots on top of the second die. We consider the random variables Y1= X1+ X2 and Y2= X1– X2(the sum and difference of points ... The correlation (or correlationcoefficient) of random variables X1 and X2is the ratio of the covariance of X1 and X2tothe product of their standard deviations,ρ(X1, X2)=Cov(X1, ... some random variables and continuous in the otherrandom variables.20.2.5-9. Numerical characteristics of multivariate random variables.The expectation of a function g(X) of a multivariate random...
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Handbook of mathematics for engineers and scienteists part 159 potx
... conditions of the problem satisfy the assumptions of thetheory of birth–death processes. In this problem, λk= λ for any k, υk= kυ for k ≤ n ,and k= nυ for k ≥ n.By formulas (20.4.2.19) and (20.4.2.20), ... Outline of Probability, Random Variables, and Random Processes, McGraw-Hill, NewYork, 1996.Kokoska, S. and Zwillinger, D. (Editors), CRC Standard Probability and Statistics Tables and Formulae,Student ... thatλk=(n – k)λ for 0 ≤ k < n,0 for k = n,υk=kυ for 1 ≤ k < l,lυ for l ≤ k ≤ n.By formulas (20.4.2.19) and (20.4.2.20), we havepk=⎧⎪⎨⎪⎩n!k!(n – k)!ρkp0 for 1 ≤ k ≤ l,n!ln–kl!(n...
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Handbook of mathematics for engineers and scienteists part 166 potx
... Press,New York, 2000.Hansen,E.R.,A Table of Series and Products, Printice Hall, Englewood Cliffs, London, 1975.Mangulis, V., Handbook of Series for Scientists and Engineers, Academic Press, New York, ... π.References for Chapter T1Dwight, H. B., Tables of Integrals and Other Mathematical Data, Macmillan, New York, 1961.Gradshteyn,I.S.andRyzhik,I.M.,Tables of Integrals, Series, and Products, ... Brychkov, Yu. A., and Marichev, O. I., Integrals and Series, Vol. 1, Elementary Functions,Gordon & Breach, New York, 1986.Zwillinger, D., CRC Standard Mathematical Tables and Formulae, 31st...
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