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Applied Calculus of Variations for Engineers, Third edition 3rd edition [Kõva köide]

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  • Formaat: Hardback, 292 pages, kõrgus x laius: 234x156 mm, kaal: 880 g, 4 Tables, black and white; 33 Illustrations, black and white
  • Ilmumisaeg: 02-Dec-2019
  • Kirjastus: CRC Press
  • ISBN-10: 0367376091
  • ISBN-13: 9780367376093
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Applied Calculus of Variations for Engineers, Third edition 3rd editionSuurem pilt
  • Formaat: Hardback, 292 pages, kõrgus x laius: 234x156 mm, kaal: 880 g, 4 Tables, black and white; 33 Illustrations, black and white
  • Ilmumisaeg: 02-Dec-2019
  • Kirjastus: CRC Press
  • ISBN-10: 0367376091
  • ISBN-13: 9780367376093
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Märksõnad:

Calculus of variations has a long history. Its fundamentals were laid down by icons of mathematics like Euler and Lagrange. It was once heralded as the panacea for all engineering optimization problems by suggesting that all
one needed to do was to state a variational problem, apply the appropriate Euler-Lagrange equation and solve the resulting differential equation.

This, as most all encompassing solutions, turned out to be not always true and the resulting differential equations are not necessarily easy to solve. On the other hand, many of the differential equations commonly used in various fields of engineering are derived from a variational problem. Hence it is an extremely important topic justifying the new edition of this book.

This third edition extends the focus of the book to academia and supports both variational calculus and mathematical modeling classes. The newly added sections, extended explanations, numerous examples and exercises aid the students in learning, the professors in teaching, and the engineers in applying variational concepts.

Preface xi
Acknowledgments xiii
Author xv
Introduction xvii
I Mathematical foundation
1(138)
1 The foundations of calculus of variations
3(28)
1.1 The fundamental problem and lemma of calculus of variations
3(5)
1.2 The Legendre test
8(2)
1.3 The Euler-Lagrange differential equation
10(3)
1.4 Minimal path problems
13(14)
1.4.1 Shortest curve between two points
14(2)
1.4.2 The brachistochrone problem
16(4)
1.4.3 Fermat's principle
20(2)
1.4.4 Particle moving in a gravitational field
22(5)
1.5 Open boundary variational problems
27(2)
1.6 Exercises
29(2)
2 Constrained variational problems
31(20)
2.1 Algebraic boundary conditions
31(5)
2.1.1 Transversality condition computation
33(3)
2.2 Lagrange's solution
36(1)
2.3 Isoperimetric problems
37(9)
2.3.1 Maximal area under curve with given length
38(4)
2.3.2 Optimal shape of curve of given length under gravity
42(4)
2.4 Closed-loop integrals
46(2)
2.5 Exercises
48(3)
3 Multivariate functionals
51(18)
3.1 Functionals with several functions
51(3)
3.1.1 Euler--Lagrange system of equations
52(2)
3.2 Variational problems in parametric form
54(5)
3.2.1 Maximal area by closed parametric curve
57(2)
3.3 Functionals with two independent variables
59(1)
3.4 Minimal surfaces
60(3)
3.4.1 Minimal surfaces of revolution
62(1)
3.5 Functional with three independent variables
63(4)
3.6 Exercises
67(2)
4 Higher order derivatives
69(12)
4.1 The Euler--Poisson equation
69(3)
4.2 The Euler--Poisson system of equations
72(3)
4.3 Algebraic constraints on the derivative
75(2)
4.4 Linearization of second order problems
77(3)
4.5 Exercises
80(1)
5 The inverse problem
81(14)
5.1 Linear differential operators
81(1)
5.2 The variational form of Poisson's equation
82(1)
5.3 The variational form of eigenvalue problems
83(3)
5.3.1 Orthogonal eigensolutions
85(1)
5.4 Sturm--Liouville problems
86(8)
5.4.1 Legendre's equation and polynomials
89(5)
5.5 Exercises
94(1)
6 Analytic solutions
95(20)
6.1 Laplace transform solution
95(2)
6.2 d'Alembert's solution
97(5)
6.3 Complete integral solutions
102(4)
6.4 Poisson's integral formula
106(5)
6.5 Method of gradients
111(3)
6.6 Exercises
114(1)
7 Approximate methods
115(24)
7.1 Euler's method
115(2)
7.2 Ritz's method
117(4)
7.3 Galerkin's method
121(2)
7.4 Approximate solutions of Poisson's equation
123(3)
7.5 Kantorovich's method
126(5)
7.6 Boundary integral method
131(4)
7.7 Finite element method
135(3)
7.8 Exercises
138(1)
II Modeling applications
139(124)
8 Differential geometry
141(14)
8.1 The geodesic problem
141(5)
8.1.1 Geodesies of a sphere
143(1)
8.1.2 Geodesic polyhedra
144(2)
8.2 A system of differential equations for geodesic curves
146(4)
8.2.1 Geodesies of surfaces of revolution
147(3)
8.3 Geodesic curvature
150(4)
8.3.1 Geodesic curvature of helix
152(2)
8.4 Generalization of the geodesic concept
154(1)
9 Computational geometry
155(20)
9.1 Natural splines
155(3)
9.2 B-spline approximation
158(6)
9.3 B-splines with point constraints
164(2)
9.4 B-splines with tangent constraints
166(4)
9.5 Generalization to higher dimensions
170(1)
9.6 Weighting and non-uniform parametrization
171(1)
9.7 Industrial applications
172(3)
10 Variational equations of motion
175(26)
10.1 Legendre's dual transformation
175(1)
10.2 Hamilton's principle
176(1)
10.3 Hamilton's canonical equations
177(5)
10.3.1 Conservation of energy
179(1)
10.3.2 Newton's law of motion
180(2)
10.4 Lagrange's equations of motion
182(5)
10.4.1 Mechanical system modeling
183(2)
10.4.2 Electro-mechanical analogy
185(2)
10.5 Orbital motion
187(9)
10.5.1 Conservation of angular momentum
191(2)
10.5.2 The 3-body problem
193(3)
10.6 Variational foundation of fluid motion
196(5)
11 Analytic mechanics
201(36)
11.1 Elastic string vibrations
201(5)
11.2 The elastic membrane
206(7)
11.2.1 Circular membrane vibrations
209(2)
11.2.2 Non-zero boundary conditions
211(2)
11.3 Bending of a beam under its own weight
213(9)
11.3.1 Transverse vibration of beam
219(3)
11.4 Buckling of a beam under axial load
222(6)
11.4.1 Axial vibration of a beam
225(3)
11.5 Simultaneous axial and transversal loading of beam
228(2)
11.6 Heat diffusion in a beam
230(7)
11.6.1 Dimensionless heat equation
233(4)
12 Computational mechanics
237(26)
12.1 The finite element technique
237(14)
12.1.1 Finite element meshing
238(1)
12.1.2 Shape functions
239(4)
12.1.3 Element matrix generation
243(5)
12.1.4 Element matrix assembly and solution
248(3)
12.2 Three-dimensional elasticity
251(3)
12.3 Mechanical system analysis
254(4)
12.4 Heat conduction
258(2)
12.5 Fluid mechanics
260(3)
Solutions to selected exercises 263(4)
Notations 267(2)
References 269(2)
Index 271
Dr. Louis Komzsik worked in the industry as an engineering mathematician for 42 years and during those years also lectured as a Visiting Professor at various southern California colleges and universities. Since his retirement he is lecturing in the Mathematics Department of the University of California at Irvine.