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LEADER: 08078cam 2200577Ii 4500
001 ocn496951742
003 OCoLC
005 20201005020735.0
008 100106s2011 si a b 001 0 eng d
010 $a 2011288115
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019 $a595136719
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020 $a981429926X
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035 $a(OCoLC)496951742$z(OCoLC)595136719
050 4 $aQA331.5$b.M67 2011
082 04 $a515.84$222
100 1 $aMoskowitz, Martin A.,$eauthor.
245 10 $aFunctions of Several Real Variables /$cMartin Moskowitz, Fotios Paliogiannis.
264 1 $aSingapore ;$aHackensack, N.J. :$bWorld Scientific,$c[2011]
264 4 $c©2011
300 $axv, 716 pages :$billustrations ;$c24 cm
336 $atext$btxt$2rdacontent
337 $aunmediated$bn$2rdamedia
338 $avolume$bnc$2rdacarrier
546 $aText in English.
504 $aIncludes bibliographical references (pages 705-707) and index.
520 $aThis Book begins with the basics of geometry and topology of Euclidean space and continues with the main topics in the theory of functions of several real variables including limits, continuity, differentiation and integration. All topics and in particular, differentiation and integration, are treated in depth and with mathematical rigor. The classical theorems of differentiation and integration such as the Inverse and Implicit Function theorems, Lagrange's multiplier rule, Fubini's theorem, the change of variables formula, Green's, Stokes' and Gauss' theorems are proved in detail and many of them with novel proofs. The authors develop the theory in a logical sequence building one result upon the other, enriching the development with numerous explanatory remarks and historical footnotes. A number of well chosen illustrative examples and counter-examples clarify matters and teach the reader how to apply these results, and solve problems in mathematics, the other sciences and economics.
520 $aEach of the chapters concludes with groups of exercises and problems, many of them with detailed solutions while others with hints or final answers. More advanced topics, such as Morse's lemma, Sardis theorem, the Weierstrass approximation theorem, the Fourier transform, Vector fields on spheres, Brouwer's fixed point theorem, Whitney's embedding theorem, Picard's theorem, and Hermite polynomials are discussed in starred sections. --Book Jacket.
505 0 $a1.1. Real numbers -- 1.1.1. Convergence of sequences of real numbers -- 1.2. Rn as a vector space -- 1.3. Rn as an inner product space -- 1.3.1. The inner product and norm in IV -- 1.3.2. Orthogonality -- 1.3.3. The cross product in R3 -- 1.4. IV as a metric space -- 1.5. Convergence of sequences in Rn -- 1.6.Compactness -- 1.7. Equivalent norms (*) -- 1.8. Solved problems for Chapter 1 -- 2.1. Functions from Rn to Rm -- 2.2. Limits of functions -- 2.3. Continuous functions -- 2.4. Linear transformations -- 2.5. Continuous functions on compact sets -- 2.6. Connectedness and convexity -- 2.6.1. Connectedness -- 2.6.2. Path-connectedness -- 2.6.3. Convex sets -- 2.7. Solved problems for Chapter 2 -- 3.1. Differentiable functions -- 3.2. Partial and directional derivatives, tangent space -- 3.3. Homogeneous functions and Enler's equation -- 3.4. The mean value theorem -- 3.5. Higher order derivatives -- 3.5.1. The second derivative -- 3.6. Taylor's theorem -- 3.6.1. Taylor's theorem in one variable -- 3.6.2. Taylor's theorem in several variables -- 3.7. Maxima and minima in several variables -- 3.7.1. Local extrema for functions in several variables -- 3.7.2. Degenerate critical points -- 3.8. The inverse and implicit function theorems -- 3.8.1. The Inverse Function theorem -- 3.8.2. The Implicit Function theorem -- 3.9. Constrained extrema, Lagrange multipliers -- 3.9.1. Applications to economics -- 3.10. Functional dependence -- 3.11. Morse's lemma (*) -- 3.12. Solved problems for Chapter 3 -- 4.1. The integral in Rn -- 4.1.1. Darboux sums. Integrability condition -- 4.1.2. The integral over a hounded set -- 4.2. Properties of multiple integrals -- 4.3. Fubini's theorem -- 4.3.1. Center of mass, centroid, moment of inertia -- 4.4. Smooth Urysohn's lemma and partition of unity (*) -- 4.5. Sard's theorem (*) -- 4.6. Solved problems for Chapter 4 -- 5.1. Change of variables formula -- 5.1.1. Change of variables; linear case -- 5.1.2. Change of variables; the general case -- 5.1.3. Applications, polar and spherical coordinates -- 5.2. Improper multiple integrals -- 5.3. Functions defined by integrals -- 5.3.1. Functions defined by improper integrals -- 5.3.2. Convolution of functions -- 5.4. The Weierstrass approximation theorem (*) -- 5.5. The Fourier transform (*) -- 5.5.1. The Schwartz space -- 5.5.2. The Fourier transform on Rn -- 5.6. Solved problems for Chapter 5 -- 6.1. Arc-length and Line integrals -- 6.1.1. Paths and curves -- 6.1.2. Line integrals -- 6.2. Conservative vector fields and Poincare's lemma -- 6.3. Surface area and surface integrals -- 6.3.1. Surface area -- 6.3.2. Surface integrals -- 6.4. Green's theorem and the divergence theorem in R2 -- 6.4.1. The divergence theorem in R2 -- 6.5. The divergence and curl -- 6.6. Stokes' theorem -- 6.7. The divergence theorem in R3 -- 6.8. Differential forms (*) -- 6.9. Vector fields on spheres and Brouwer fixed point theorem (*) -- 6.9.1. Tangential vector fields on spheres -- 6.9.2. The Brouwer fixed point theorem -- 6.10. Solved problems for Chapter 6 -- 7.1. Introduction -- 7.2. First order differential equations -- 7.2.1. Linear first order ODE -- 7.2.2. Equations with variables separated -- 7.2.3. Homogeneous equations -- 7.2.4. Exact equations -- 7.3. Picard's theorem (*) -- 7.4. Second order differential equations -- 7.4.1. Linear second order ODE with constant coefficients -- 7.4.2. Special types of second order ODE; reduction of order -- 7.5. Higher order ODE and systems of ODE -- 7.6. Some more advanced topics in ODE (*) -- 7.6.1. The method of Frobenius; second order equations with variable coefficients -- 7.6.2. The Hermite equation -- 7.7. Partial differential equations -- 7.8. Second order PDE in two variables -- 7.8.1. Classification and general solutions -- 7.8.2. Boundary value problems for the wave equation -- 7.8.3. Boundary value problems for Laplace's equation -- 7.8.4. Boundary value problems for the heat equation -- 7.8.5.A note on Fourier series -- 7.9. The Fourier transform method (*) -- 7.10. Solved problems for Chapter 7 -- 8.1. Simple variational problems -- 8.1.1. Some classical problems -- 8.1.2. Sufficient conditions -- 8.2. Generalizations -- 8.2.1. Geodesics on a Riemannian surface -- 8.2.2. The principle of least action -- 8.3. Variational problems with constraints -- 8.4. Multiple integral variational problems -- 8.4.1. Variations of double integrals -- 8.4.2. The case of n variables -- 8.5. Solved problems for Chapter 8 -- B.1. Differential calculus -- B.2. Integral calculus -- B.2.1.Complex-valued functions -- B.3. Series -- C.1. The Stone-Weierstrass theorem.
650 0 $aFunctions of real variables.
650 0 $aMathematical analysis.
650 7 $aFunctions of real variables.$2fast$0(OCoLC)fst00936120
650 7 $aMathematical analysis.$2fast$0(OCoLC)fst01012068
650 7 $aFonctions d'une variable réelle.$2ram
650 7 $aAnalyse mathématique.$2ram
700 1 $aPaliogiannis, Fotios,$eauthor.
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