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E-raamat: Generalized Calculus with Applications to Matter and Forces

(Instituto Superior Technico, Lisboa, Portugal)
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Campos completes his trilogy on the theory of functions by considering generalized functions such as the Heaviside unit jump, Dirac unit impulse, and its derivatives of all orders. The two main approaches to generalized functions he presents are a non-uniform limit of a family of ordinary functions, and a functional over a set of test functions from which properties are inherited. He develops the latter more extensively to include multi-dimensional generalized functions whose arguments are ordinary functions of several variables, which allows integration along curves, surfaces, and hypersurfaces and relates to the invariant differential operator's gradient, curl, and divergence. The first two volumes cover complex analysis with applications to flows and fields, and transcendental representations with applications to solids and fluids. Annotation ©2014 Ringgold, Inc., Portland, OR (protoview.com)

Combining mathematical theory, physical principles, and engineering problems, Generalized Calculus with Applications to Matter and Forces examines generalized functions, including the Heaviside unit jump and the Dirac unit impulse and its derivatives of all orders, in one and several dimensions. The text introduces the two main approaches to generalized functions: (1) as a nonuniform limit of a family of ordinary functions, and (2) as a functional over a set of test functions from which properties are inherited. The second approach is developed more extensively to encompass multidimensional generalized functions whose arguments are ordinary functions of several variables.

As part of a series of books for engineers and scientists exploring advanced mathematics, Generalized Calculus with Applications to Matter and Forces presents generalized functions from an applied point of view, tackling problem classes such as:

  • Gauss and Stokes’ theorems in the differential geometry, tensor calculus, and theory of potential fields
  • Self-adjoint and non-self-adjoint problems for linear differential equations and nonlinear problems with large deformations
  • Multipolar expansions and Green’s functions for elastic strings and bars, potential and rotational flow, electro- and magnetostatics, and more

This third volume in the series Mathematics and Physics for Science and Technology is designed to complete the theory of functions and its application to potential fields, relating generalized functions to broader follow-on topics like differential equations. Featuring step-by-step examples with interpretations of results and discussions of assumptions and their consequences, Generalized Calculus with Applications to Matter and Forces enables readers to construct mathematical–physical models suited to new observations or novel engineering devices.

Arvustused

"... strongly application oriented. ... The language (text, figures) of the book is precise, attractive, inspiring, and very readable. The book is strongly recommended for students, teachers, and scientists. It is very good for self-study." -Zentralblatt MATH 1297 "Engineers looking for detailed approaches to the use of distributions in solving problems will certainly want to dip into this text. It certainly satisfies its stated aim..." -MAA Reviews

List of Tables, Notes, Diagrams, Classifications, and Lists
xix
Series Preface xxvii
Preface xxix
Acknowledgments xxxi
Author xxxiii
Mathematical Symbols xxxv
Physical Symbols xliii
Introduction xlix
1 Limit of a Sequence of Functions
1(48)
1.1 Evaluation of Integrals of Gaussian Functions
1(8)
1.1.1 Evaluation of the Basic Gaussian Integral
1(2)
1.1.2 Alternative Evaluation of the Gaussian Integral
3(1)
1.1.3 Gaussian Integrals with Trigonometric Factors
4(1)
1.1.4 Gaussian Integrals with Products of Power and Trigonometric Factors
5(1)
1.1.5 Gaussian Integrals with Power Factors
6(1)
1.1.6 Evaluation of Gaussian Integrals in terms of Hermite Polynomials
7(1)
1.1.7 Summary and Examples of Gaussian Integrals
8(1)
1.2 Unit Jump as a Generalized Function
9(2)
1.2.1 Unit Jump as the Limit of Error Functions
9(1)
1.2.2 Unit Jump as the Limit of Hyperbolic Tangents
10(1)
1.3 Unit Impulse as a Generalized Function
11(2)
1.3.1 Definition of Derivative of a Generalized Function
11(1)
1.3.2 Unit Impulse as the Derivative of the Unit Step
12(1)
1.4 First Derivative of the Unit Impulse
13(3)
1.4.1 Unit Impulse (Derivative of) as a Monopole (Dipole)
13(2)
1.4.2 Derivative of Unit Impulse as a Limit of Families of Derivatives
15(1)
1.5 Derivative of Order N of the Unit Impulse
16(2)
1.5.1 Multipole of Order N as the Nth Derivative of the Unit Impulse
16(2)
1.5.2 Derivatives of All Orders of the Unit Impulse
18(1)
1.6 Integration, Substitution, and Product Properties
18(3)
1.6.1 Integration Property for the Derivatives of the Unit Impulse
19(1)
1.6.2 Substitution and Product Properties of the Derivatives of the Unit Impulse
20(1)
1.7 Sign and Modulus and Related Generalized Functions
21(3)
1.7.1 Symmetric and Skew-Symmetric Generalized Functions
21(1)
1.7.2 Sign and Its Derivatives as Generalized Functions
22(1)
1.7.3 Modulus and Its Derivates as Generalized Functions
23(1)
1.8 Nonremovable, Isolated, and Finite Discontinuities
24(3)
1.8.1 Classification of Discontinuities of an Ordinary Function
24(2)
1.8.2 Function with a Finite Number of Finite Discontinuities
26(1)
1.9 Jump of a Function and of Its Successive Derivates
27(20)
1.9.1 Derivative of a Discontinuous Function
27(1)
1.9.2 Derivatives and Jumps of Higher Order
28(19)
1.10 Conclusion
47(2)
2 Shape of a Loaded String
49(88)
2.1 Tangential Tension and Transverse Force
49(5)
2.1.1 Tangential and Normal Components of the Tension
50(1)
2.1.2 Transverse Force and Shear Stress
51(1)
2.1.3 Conditions for Linear and Nonlinear Deflection
52(1)
2.1.4 One (Two)-Dimensional Elastic String (Membrane)
53(1)
2.2 Shear Stress and Elastic Energy
54(2)
2.2.1 Quadratic and Higher-Order Terms in the Elastic Energy
54(1)
2.2.2 Balance Equation and Boundary Conditions
55(1)
2.3 Influence Function, Superposition, and Reciprocity
56(10)
2.3.1 Linear Influence Function of a String
56(3)
2.3.2 String with Supports at Different Heights
59(1)
2.3.3 Reaction Forces at the Two Supports
60(1)
2.3.4 Concentrated Force at Equal Distance from the Supports
61(1)
2.3.5 Principles of Reciprocity and Superposition
62(1)
2.3.6 Original and Reciprocal Influence Function
63(1)
2.3.7 Extended Form of the Reciprocity Principle
64(1)
2.3.8 Alternate Extended Reciprocity Principle
64(1)
2.3.9 Linear Deflection under an Arbitrary Load
65(1)
2.4 Linear Deflection under Own Weight
66(5)
2.4.1 Deflection under a Uniform Load
66(1)
2.4.2 Deflection, Slope, and Their Extrema
67(1)
2.4.3 Parabolic Shape for a Homogeneous String
68(1)
2.4.4 Small Total Weight Compared with the Tension
68(1)
2.4.5 Linear Deflection under Uniform or Concentrated Loads
69(2)
2.5 Multiple Concentrated and Distributed Loads
71(10)
2.5.1 Unit Staircase and Impulse Haircomb
71(1)
2.5.2 Two Concentrated Loads with the Same Direction
72(1)
2.5.3 Differential Equation and Matching Conditions
73(1)
2.5.4 Heaviside Parapet and Dirac Alternating Haircomb
74(1)
2.5.5 Two Concentrated Forces with Opposite Directions
74(2)
2.5.6 Boundary and Continuity Conditions
76(1)
2.5.7 Regular/Irregular Sawtooth Generalized Functions
77(1)
2.5.8 Mixed, Distributed, and Concentrated Loads
78(2)
2.5.9 Comparison of Distributed, Concentrated, and Mixed Loads
80(1)
2.6 Nonlinear Deflection by a Concentrated Force
81(14)
2.6.1 Shape of a String under Arbitrary Loading
82(1)
2.6.2 Nonlinear Influence or Green Function for a String
83(1)
2.6.3 Maximum Nonlinear Deflection by a Concentrated Force
84(1)
2.6.4 Linear Limit of a Small Slope
85(1)
2.6.5 Quartic Equation for the Nonlinear Unsymmetric Elastic Deflection
86(1)
2.6.6 Existence of Real Roots of the Quartic Polynomial
87(1)
2.6.7 Inequalities for the Reaction Forces at the Supports
87(1)
2.6.8 Balance of the Tension and Reaction Forces
88(1)
2.6.9 Linear Deflection or Symmetric Case
89(1)
2.6.10 Balance of Forces and Moments
90(1)
2.6.11 Vertical/Horizontal Reaction Forces at the Supports
90(1)
2.6.12 Nonlinear Extension and Deflection
91(2)
2.6.13 Elastic Energy and Slope
93(1)
2.6.14 Effect of the Nonlinearity Parameter
93(2)
2.7 Large Deflection by a Uniform Load
95(9)
2.7.1 Nonlinear Deflection by Own Weight
95(1)
2.7.2 Nonlinear Deflection with Constant Shear Stress
96(1)
2.7.3 Exact and First-Order Nonlinear Corrections
97(1)
2.7.4 Comparison of Linear and Nonlinear Deflections
98(1)
2.7.5 Deflection, Slope, and Extrema
99(1)
2.7.6 Elastic Energy and Extension
100(1)
2.7.7 Effect of the Nonlinearity Parameter
101(1)
2.7.8 Shape of the String as a Circular Arc
101(3)
2.8 Nonlinear Deflection of a Heavy Elastic String
104(8)
2.8.1 Support and Dimensional/Dimensionless Apex Coordinates
104(1)
2.8.2 Solution of a Nonlinear Ordinary Differential Equation
105(1)
2.8.3 Nonlinear Deflection under Own Weight
106(1)
2.8.4 Maximum Deflection and Apex/Support Coordinates
107(1)
2.8.5 Nonlinear Correction to the Linear Shape
108(1)
2.8.6 Nonlinear Shape and Maximum Deflection
108(1)
2.8.7 Exact/Maximum Slope in Apex/Support Coordinates
109(1)
2.8.8 Effect of the Nonlinearity Parameter
110(2)
2.9 Comparison with an Inextensible String: The Catenary
112(23)
2.9.1 Shape of a Heavy, Inextensible String
112(1)
2.9.2 Length of the String and Distance from the Supports
113(1)
2.9.3 Exact and Approximate Nonlinearity Parameter
114(1)
2.9.4 Deflection, Slope, and Their Extrema
114(2)
2.9.5 Potential Energy in the Gravity Field
116(1)
2.9.6 Comparison of the Elastic and Gravity Energies
116(1)
2.9.7 Effect of the Nonlinearity Parameter
117(1)
2.9.8 Second Dimensionless Apex Coordinates
118(1)
2.9.9 Linear/Nonlinear Deflection under Own Weight
119(1)
2.9.10 Catenary in the First Dimensionless Apex Coordinates
120(1)
2.9.11 Nonlinear Deflection by a Uniform or Concentrated Load
121(1)
2.9.12 Comparison of Five Loading Cases
122(2)
2.9.13 Sequence of Increasing Deflections/Slopes
124(1)
2.9.14 Effects of the Spatial Load Distribution
125(1)
2.9.15 Linear/Nonlinear Deflection of Elastic/Inextensible Strings
125(10)
2.10 Conclusion
135(2)
3 Functionals over Test Functions
137(48)
3.1 Unit Jump and Unit Impulse
138(3)
3.1.1 Unit Jump and Integrable Functions
138(1)
3.1.2 Fundamental Property of the Unit Jump
139(1)
3.1.3 Unit Impulse and Continuous Functions
140(1)
3.1.4 Translations and Change of Scale
141(1)
3.2 Derivates of All Orders of the Unit Impulse
141(2)
3.2.1 Derivate of the Unit Impulse as a Functional
141(1)
3.2.2 nth Derivative of the Unit Impulse
142(1)
3.3 Growth/Decay and Support of Test Functions
143(5)
3.3.1 Fairly Good, Good, and Very Good Functions
143(1)
3.3.2 Compact Support for Ordinary or Generalized Functions
144(1)
3.3.3 Functions with Slow/Fast Decay/Growth
144(1)
3.3.4 Smooth Functions with Compact Support
145(1)
3.3.5 Inclusion of Classes of Test Functions
146(1)
3.3.6 Excellent and Superlative Test Functions
147(1)
3.4 Generalized Function as a Continuous Functional
148(5)
3.4.1 Temperate Generalized Functions
148(1)
3.4.2 Infinitely Differentiable Generalized Functions
148(1)
3.4.3 Even/Odd Test and Generalized Functions
149(1)
3.4.4 Relation with Inner Product and Norm of Square-Integrable Functions
149(1)
3.4.5 Derivatives of the Logarithm on the Branch-Cut
150(1)
3.4.6 Test and Other Alternative Reference Functions
151(1)
3.4.7 Advantages and Limitations of Generalized Functions
152(1)
3.5 Series of Impulses, Jumps, and Ramps
153(2)
3.5.1 Taylor Series for the Unit Jump and Impulse
153(1)
3.5.2 Expansion in a Series of Ramp Functions
153(2)
3.6 Product of a Power and a Derivate of an Impulse
155(4)
3.6.1 Product of a Nonintegral Power by a Unit Impulse
155(1)
3.6.2 Product of a Nonintegral Power by a Derivative Unit Impulse
156(1)
3.6.3 Product of an Integral Power by a Unit Impulse or Derivative Impulse
156(2)
3.6.4 Generalized Substitution Rule for the Derivative Impulse
158(1)
3.6.5 Product of a Derivative Impulse by a Polynomial
158(1)
3.7 Algebraic Equations Involving Generalized Functions
159(2)
3.7.1 Product of a Generalized Function by Its Variable
159(1)
3.7.2 Product of a Generalized Function by a Power of Its Variable
160(1)
3.8 Products of Moduli, Powers, and Logarithms
161(3)
3.8.1 Differentiation with regard to the Variable and a Parameter
161(1)
3.8.2 Extension of Powers to Negative Exponents
162(1)
3.8.3 Extension of Inverse Powers to the Origin
163(1)
3.8.4 Powers with Negative Nonintegral and Integral Exponents
163(1)
3.9 Finite Part of an Integral (Hadamard)
164(21)
3.9.1 Integral with Power-Type Singularity
164(1)
3.9.2 Extension to All Negative Nonintegral Powers
165(20)
4 Bending of Bars and Beams
185(110)
4.1 Bending Moment and Curvature of the Elastica (Bernoulli, 1744; Euler, 1744)
186(5)
4.1.1 Radius of Curvature and Material Properties
186(1)
4.1.2 Linear and Nonlinear Bending Moment
186(2)
4.1.3 Transverse Force and Shear Stress
188(1)
4.1.4 Bending of Nonuniform and Uniform Beams
189(1)
4.1.5 Linear and Nonlinear Bending of a Beam
189(1)
4.1.6 Longitudinal Tension Associated with Fixed Ends
190(1)
4.1.7 Bar and String as Opposite Limits of a Beam
191(1)
4.2 Deformation and Displacement of the Cross Section (Saint-Venant, 1856) and Elastic Energy (Green, 1837)
191(9)
4.2.1 Strain and Stress Tensors for Simple Bending
192(1)
4.2.2 Longitudinal and Cross-Sectional Displacement Vectors
193(1)
4.2.3 Elastic Energy and Moment of Inertia of the Cross Section
194(1)
4.2.4 Principle of Virtual Work Applied to the Bending of a Bar
195(1)
4.2.5 Work of the Transverse Force and Bending Moment
196(1)
4.2.6 Principle of Virtual Work for Weak and Strong Bending
197(1)
4.2.7 Three Distinct Boundary Conditions for a Beam
198(1)
4.2.8 Isostatic, Hypostatic, and Hyperstatic Beams
199(1)
4.2.9 Four Static Combinations of Three Boundary Conditions at Two Supports
199(1)
4.3 Weak Bending of a Heavy Bar
200(8)
4.3.1 Heavy Bar Clamped at Both Ends
200(3)
4.3.2 Clamped-Pinned Heavy Bar
203(1)
4.3.3 Deflection by Own Weight of a Pinned-Pinned Bar
204(1)
4.3.4 Heavy Cantilever with One Clamped and One Free End
205(1)
4.3.5 Heavy Bar with Four Combinations of Supports
206(1)
4.3.6 Reaction Forces and Moments at the Supports
207(1)
4.4 Influence Functions for Pinned/Clamped/Free Ends
208(18)
4.4.1 Clamped-Free Bar with a Concentrated Force
209(3)
4.4.2 Pinned-Pinned Bar with a Concentrated Force
212(3)
4.4.3 Clamped-Pinned Bar with a Concentrated Force
215(3)
4.4.4 Clamped-Clamped Bar with a Concentrated Force
218(3)
4.4.5 Determinate, Indeterminate, or Incompatible Problems
221(2)
4.4.6 Concentrated Force at Mid- or Extreme Positions
223(2)
4.4.7 Influence Function for Four Sets of Boundary Conditions
225(1)
4.5 Weak Bending by a Concentrated Torque
226(14)
4.5.1 Derivatives of the Impulse and Influence Functions
227(1)
4.5.2 Clamped-Free Bar with a Concentrated Torque
227(4)
4.5.3 Pinned-Pinned Bar with a Concentrated Torque
231(2)
4.5.4 Clamped-Pinned Bar with a Concentrated Torque
233(2)
4.5.5 Clamped-Clamped Bar with a Concentrated Torque
235(2)
4.5.6 Comparison of a Concentrated Torque with Different Supports
237(3)
4.6 Tangential Tension along a Beam with Pinned Ends
240(5)
4.6.1 Shape of a Beam under Longitudinal Tension
240(1)
4.6.2 Relation between the Transverse Load and the Longitudinal Tension
241(1)
4.6.3 Longitudinal Tension due to Bending with Fixed Supports
242(2)
4.6.4 Second (Third)-Order Approximation to the Elastica (Tension)
244(1)
4.7 Nonlinear Bending by Concentrated Moments
245(11)
4.7.1 General Solution for Strong Bending of a Nonuniform Bar
245(1)
4.7.2 Bar with One Clamped and One Free End
246(1)
4.7.3 Maximum Torque That the Bar Can Withstand
247(1)
4.7.4 Two Methods of Exact Solution of the Nonlinear Equation of the Elastica
248(1)
4.7.5 Concentrated Moment at the Free End or at an Intermediate Position
249(1)
4.7.6 Nonlinearity Parameter and Exact Shape
250(2)
4.7.7 Nonlinear Matching of Multiple Deflections
252(1)
4.7.8 Strong Bending of a Bar by Two Concentrated Moments
253(1)
4.7.9 Effect of the Relative Magnitude of the Two Moments
254(2)
4.8 Nonlinear Influence or Green Function
256(17)
4.8.1 Nonlinear Influence Function due to a Concentrated Force
256(2)
4.8.2 Nonlinear Bending of Pinned-Pinned and Clamped-Free Bars
258(2)
4.8.3 Relation between Fixed Cartesian and Local Tangent Coordinates
260(2)
4.8.4 Strong Bending by Moments, Forces, and Weight
262(1)
4.8.5 Nonlinear Bending or a Heavy Cantilever
263(2)
4.8.6 Linear Approximation in Cartesian and Local Coordinates
265(1)
4.8.7 First-Order Nonlinear Correction
265(1)
4.8.8 Second-Order Nonlinear Correction
266(1)
4.8.9 Comparison of Linear and Nonlinear Terms
267(1)
4.8.10 Clamped-Free Bar with a Concentrated Force at the Tip
268(1)
4.8.11 Evaluation of Elliptic Integrals by Double Series
269(1)
4.8.12 Nonlinear Corrections to All Orders
270(1)
4.8.13 Linear Approximation to Nonlinear Bending
271(1)
4.8.14 Lowest-Order Nonlinear Correction
272(1)
4.8.15 Comparison of Weak and Strong Bending of a Bar
273(1)
4.9 Elastic Instability: Buckling and Collapse
273(20)
4.9.1 Nonlinear Bending by an Axial Force
274(1)
4.9.2 Linear and Nonlinear Buckling Load
275(2)
4.9.3 Linear Shape of a Buckled Bar
277(1)
4.9.4 Strong Bending with Sliding Supports
278(2)
4.9.5 Shape of the Elastica for Linear Bending
280(1)
4.9.6 Nonlinear Effect on the Arc Length
281(1)
4.9.7 Collapse of a Bar between Sliding Supports
282(1)
4.9.8 Dimensional Scalings of the Displacement, Slope, and Curvature
283(1)
4.9.9 Buckling/Collapse/Displacement/Twist Instabilities
284(9)
4.10 Conclusion
293(2)
5 Differential Operators and Geometry
295(110)
5.1 Generalized Function with an Ordinary Function as the Argument
296(5)
5.1.1 Unit Jump of an Ordinary Function
296(1)
5.1.2 Ordinary Function as the Argument of a Unit Impulse
297(2)
5.1.3 Derivative Impulse of an Ordinary Function
299(1)
5.1.4 Unit Jump of Intervals of the Real Line
300(1)
5.2 Argument a Function of Two Variables
301(6)
5.2.1 Unit Jump of a Two-Dimensional Domain
301(2)
5.2.2 Unit Impulse of a Function of Two Variables
303(1)
5.2.3 Decomposition of the Impulse of a Function into Simple Impulses
304(1)
5.2.4 Two Equivalent Decompositions into Simple Impulses
305(1)
5.2.5 Gradient of the Unit Jump of an Ordinary Function
306(1)
5.3 Products of Generalized Functions of Different Variables
307(4)
5.3.1 Two-Dimensional Unit Jump
308(1)
5.3.2 Two-Dimensional Cartesian and Polar Impulse
309(1)
5.3.3 Product of a Unit Jump by a Unit Impulse
310(1)
5.4 Two-Dimensional Generalized Functions with Two Ordinary Functions as Arguments
311(4)
5.4.1 Two Linearly Independent Functions of Two Variables
311(1)
5.4.2 Product of Unit Jumps of Functions of Two Variables
312(1)
5.4.3 Two-Dimensional Impulse of Two Functions of Two Variables
313(1)
5.4.4 Amplitudes of the Decomposition into Two-Dimensional Unit Impulses
314(1)
5.5 Multidimensional Generalized Functions with Several Ordinary Functions as Arguments
315(12)
5.5.1 Multidimensional Linearly Independent Functions
317(1)
5.5.2 One-Dimensional Generalized Function of an Ordinary Function of Several Variables
317(3)
5.5.3 Multidimensional Generalized Function with the Coordinates as the Arguments
320(1)
5.5.4 Multidimensional Generalized Function with Ordinary Functions as Arguments
321(4)
5.5.5 Generalized Function over a Subspace of Dimension N - M of an N-Dimensional Space
325(2)
5.6 Generalized Functions for Hypersurface and Line Integration
327(6)
5.6.1 Generalized Function for Integration along a Hypercurve
327(2)
5.6.2 Generalized Function for Integration over a Hypersurface
329(3)
5.6.3 Generalized Function for Multipoles in Subspaces
332(1)
5.7 Divergence, Gradient, and Curl Theorems
333(17)
5.7.1 Divergence Integral Theorem (Gauss, 1809; Ostrogradski, 1828; Hankel, 1861; Thomson, 1869)
333(3)
5.7.2 Gradient Integral Theorem
336(2)
5.7.3 Curl Volume Theorem for a Domain
338(1)
5.7.4 Permutation Symbol and Polar/Axial Vectors
339(2)
5.7.5 Invariant Integral Theorems in Three Dimensions
341(1)
5.7.6 Regular Surface Supported on a Loop
342(2)
5.7.7 Curl Theorem for a Loop
344(2)
5.7.8 Divergence and Curl Integral Theorems in the Plane
346(2)
5.7.9 First and Second Green Identities
348(2)
5.8 Scalar/Vector Potentials for Irrotational/Solenoidal Fields
350(11)
5.8.1 Scalar Potential for an Irrotational Field
352(2)
5.8.2 Vector Potential of a Solenoidal Field
354(1)
5.8.3 Two-Dimensional Curl and Field Function
355(1)
5.8.4 Scalar and Vector Potentials of a General Field
356(1)
5.8.5 Classification of Continuously Differentiable Vector Fields
357(1)
5.8.6 Specification of a Vector Field by Its Curl and Divergence
358(2)
5.8.7 Vector Field with Constant Curl and Divergence
360(1)
5.8.8 Decomposition of a Vector into Irrotational and Solenoidal Parts
361(1)
5.9 Cylindrical, Spherical, and Hyperspherical Symmetry
361(41)
5.9.1 Area and Volume of the W-Dimensional Hypersphere
362(2)
5.9.2 Impulse with Cylindrical, Spherical, and Hyperspherical Symmetry
364(38)
5.10 Conclusion
402(3)
6 Axisymmetric Flows and Four Sphere Theorems
405(90)
6.1 Invariant Differential Operators in Space
406(9)
6.1.1 Arc Length and Scale Factor
406(2)
6.1.2 Cartesian, Cylindrical, and Spherical Coordinates
408(2)
6.1.3 Gradient of a Scalar in Orthogonal Curvilinear Coordinates
410(1)
6.1.4 Divergence of a Vector in Two and Three Dimensions
411(1)
6.1.5 Two- and Three-Dimensional Curl of a Vector
412(2)
6.1.6 Cartesian, Cylindrical, and Spherical Laplacian
414(1)
6.1.7 Relation between the Vector and Scalar Laplacians
414(1)
6.2 Stream Function for an Axisymmetric Flow
415(13)
6.2.1 Coordinates and Velocity for an Axisymmetric Flow
416(1)
6.2.2 Stream Function in Cylindrical/Spherical Coordinates
417(1)
6.2.3 Vector Potential for an Incompressible Flow
418(1)
6.2.4 Scalar Potential for an Irrotational Flow
419(1)
6.2.5 Potential Flow: Irrotational and Incompressible
420(1)
6.2.6 Laplace and Modified Laplace Operators
420(1)
6.2.7 Scalar and Vector Poisson Equations
421(1)
6.2.8 Scalar and Vector Laplace Operators
422(1)
6.2.9 Streamlines, Vortex Lines, and Stream Surfaces
423(1)
6.2.10 Relation of the Stagnation Enthalpy or Pressure to the Stream Function
424(2)
6.2.11 Bernoulli Equation and Extension to Viortical Flows
426(1)
6.2.12 Scalar, Modified, and Vector Poisson Equations
427(1)
6.3 Point Source/Sink in a Uniform Stream
428(6)
6.3.1 Relation between the Potential and the Stream Function
429(1)
6.3.2 Uniform Flow Parallel to the Axis
430(1)
6.3.3 Radial Flow from a Point Source or Sink
430(1)
6.3.4 Rankine Semi-Infinite Axisymmetric Fairing
431(2)
6.3.5 Pressure Distribution and Drag Force
433(1)
6.4 Fairings, Bodies, and Multipoles
434(10)
6.4.1 Source and Sink Pair in a Uniform Stream
434(2)
6.4.2 Rankine Oval Finite Body
436(1)
6.4.3 Dipole as a Limit of Opposite Monopoles
437(1)
6.4.4 Quadrupole as a Limit of Opposite Dipoles
438(1)
6.4.5 Potential for an Axial Dipole and Quadrupole
439(1)
6.4.6 Axial, Transverse, and Mixed Multipoles
440(1)
6.4.7 Potential and Velocity for Triaxial Multipoles
441(1)
6.4.8 Dipole/Quadrupole in Cartesian/Cylindrical/Spherical Coordinates
442(2)
6.5 Sphere in a Stream and in a Large Cavity
444(7)
6.5.1 Dipole Moment and Blockage Effect
444(1)
6.5.2 Aiming Distance and Pressure Distribution
445(1)
6.5.3 Wall Effect for a Spherical Cavity
446(3)
6.5.4 Kinetic Energy and Added Mass of Entrained Fluid
449(1)
6.5.5 Balance of Inertia and External and Drag Forces
450(1)
6.6 Spherical Vortex in a Uniform Flow
451(7)
6.6.1 Modified Laplace and Laplace Axisymmetric Operators
451(1)
6.6.2 Stream Function for a Spherical Vortex (Hill 1894)
452(2)
6.6.3 Stagnation Circle and Toroidal Stream Surfaces
454(2)
6.6.4 Matching a Spherical Vortex to a Uniform Stream
456(1)
6.6.5 Pressure and Stagnation Pressure inside a Vortex
457(1)
6.7 Four Axisymmetric and Nonaxisymmetric Sphere Theorems
458(7)
6.7.1 First Sphere Theorem for the Reciprocal Stream Function (Butler, 1953)
458(3)
6.7.2 Second Sphere Theorem for the Reciprocal Potential (Kelvin)
461(2)
6.7.3 Third Sphere Theorem for the Integral Reciprocal Potential (Weiss, 1945)
463(1)
6.7.4 Fourth Sphere Theorem for the Integral Nonreciprocal Potential
464(1)
6.8 Electric Charges and Currents on a Sphere
465(7)
6.8.1 Insulating Sphere in an Electrostatic Field
465(1)
6.8.2 Electric Charges on a Conducting Sphere
466(2)
6.8.3 Ohm's Law and Electrical Conductivity
468(1)
6.8.4 Hydrodynamic and Electric Charge/Current Monopoles
468(1)
6.8.5 Electric Currents Flowing between the Poles of a Sphere
469(1)
6.8.6 Boundary Condition for Surface Electric Currents
470(1)
6.8.7 Spherical Surface Potential for the Electric Current
470(2)
6.8.8 Singularities of the Electric Field at the Poles
472(1)
6.9 Two Spheres Moving Orthogonal to the Line of Centers (Stokes)
472(21)
6.9.1 Infinite Set of Pairs of Images in Two Spheres
473(2)
6.9.2 Isolated and Perturbation Potentials for Each Sphere
475(1)
6.9.3 Values of the Total Potentials on the Two Spheres
476(1)
6.9.4 Kinetic Energy and Added Mass
477(1)
6.9.5 Sphere Moving Parallel to a Wall
478(1)
6.9.6 Added Mass and Fluid Entrainment by Bodies
479(1)
6.9.7 Dipole Moment for a Cylinder and a Sphere
480(1)
6.9.8 Flow Blockage in Two and Three' Dimensions
480(1)
6.9.9 Added Mass of a Cylinder Moving in a Cylindrical Cavity
481(12)
6.10 Conclusion
493(2)
7 Convolution, Reciprocity, and Adjointness
495(66)
7.1 Norm of a Function and Metric Spaces
495(2)
7.1.1 Functions with Integrable Power of the Modulus and Normed Spaces
495(1)
7.1.2 Triangular and Projective Inequalities in a Metric Space
496(1)
7.2 Schwartz and Holder Projective Inequalities and Equalities
497(4)
7.2.1 Extension from Schwartz to Holder Projective Inequalities
497(2)
7.2.2 Proof of the Holder Inequality in a Normed Space
499(2)
7.3 Discrete/Integral Triangular and Minkowski (1896) Inequalities
501(2)
7.3.1 Triangular Inequality for Absolutely or Square Integrable Functions
501(1)
7.3.2 Minkowski Triangular and Stronger Inequality in Normed Spaces
502(1)
7.4 Convolution Integral for Ordinary and Generalized Functions
503(4)
7.4.1 Convolution of Absolutely or Square Integrable Functions
503(1)
7.4.2 Convolution of Ordinary Functions in Normed Spaces
504(1)
7.4.3 Convolution of Generalized Functions with Compact Support
505(2)
7.5 Commutative, Associative, and Impulsive Properties
507(2)
7.5.1 Commutative and Associative Properties of the Convolution of Ordinary Functions
507(1)
7.5.2 Convolution of the Unit Impulse and Their Derivatives
508(1)
7.5.3 Identity and Addition Properties for the Convolution of Impulses
508(1)
7.6 Principles of Superposition and Reciprocity
509(4)
7.6.1 Fundamental Solution of an Ordinary or Partial Differential Equation
509(2)
7.6.2 Discrete and Integral Principle of Linear Superposition
511(1)
7.6.3 Influence or Green (1837) Function and Self-Adjoint Operator
511(2)
7.7 Operator, Adjoint, and Bilinear Concomitant
513(10)
7.7.1 General Second-Order Linear Ordinary Differential Operator
513(1)
7.7.2 Second-Order Linear Self-Adjoint Operator
514(1)
7.7.3 Elastic String under Nonuniform Tension Supported on Springs
514(2)
7.7.4 Multiplying Factor and Transformation to Self-Adjoint Form
516(1)
7.7.5 Undamped and Damped Harmonic Oscillators
516(2)
7.7.6 High-Order Linear Operators with Variable Coefficients (Lagrange)
518(1)
7.7.7 Linear Differential Operator with Constant Coefficients
519(1)
7.7.8 High-Order Self-Adjoint Operator with Variable Coefficients
520(2)
7.7.9 Bending of a Nonuniform Beam under Tension
522(1)
7.8 Anisotropic, Intermediate, and Isotropic Operators
523(10)
7.8.1 General Second-Order Linear Partial Differential Operator
523(1)
7.8.2 Self-Adjoint Partial Differential Operator of the Second Order
524(1)
7.8.3 Existence of Transformation from General to Self-Adjoint Operator
525(1)
7.8.4 Electrostatic Field in an Anisotropic Inhomogeneous Dielectric
526(1)
7.8.5 Homogeneous/Inhomogeneous Laplace and Helmholtz Operators
527(1)
7.8.6 Elastic Membrane with Nonuniform Tension
528(1)
7.8.7 Self-Adjoint and Non-Self-Adjoint Differential Operators
529(1)
7.8.8 High-Order Partial Differential Operator
530(1)
7.8.9 Self-Adjoint Operator with Variable Coefficients
530(1)
7.8.10 High-Order Operator with Constant Coefficients
531(1)
7.8.11 Isotropic and Intermediate High-Order Operators
531(2)
7.8.12 Bending of a Plate under In-Plane Tension Supported on Springs
533(1)
7.9 Equations of Mathematical Physics in Space-Time
533(26)
7.9.1 Linear Second-Order Differential Operator in Space-Time
533(1)
7.9.2 Anisotropic Self-Adjoint Operator in Space-Time
534(1)
7.9.3 Laplace, Wave, and Klein-Gordon Equations
535(1)
7.9.4 Diffusion, Telegraphy, and Schrodinger Equations
536(1)
7.9.5 Intermediate Self-Adjoint Differential Operator
536(1)
7.9.6 Inhomogeneous Diffusion, Telegraphy, and Schrodinger Equations
537(1)
7.9.7 Transverse Vibrations of an Inhomogeneous String/Membrane
538(1)
7.9.8 Space-Time Method for High-Order Operators
538(2)
7.9.9 Transverse Vibrations of Bars and Plates
540(19)
7.10 Conclusion
559(2)
8 Electric/Magnetic Multipoles and Images
561(86)
8.1 Electric Dipole as the Limit of Opposite Monopoles
562(6)
8.1.1 Electric Charge Density of a Monopole and a Dipole
562(1)
8.1.2 Electrostatic Potential of a Dipole
563(1)
8.1.3 Influence Function for the Electrostatic Field
564(2)
8.1.4 Electric Field due to a Monopole and a Dipole
566(2)
8.2 Longitudinal, Transverse, and Cross Quadrupoles
568(3)
8.2.1 Electric Quadrupole as Limit of Opposite Dipoles
568(1)
8.2.2 Electrostatic Potential and Field of a Quadrupole
569(2)
8.3 Multipolar Expansion for an Irrotational Field
571(9)
8.3.1 Rules of Spatial Derivation of the Inverse Distance
571(1)
8.3.2 Taylor Series for the Influence Function
572(1)
8.3.3 Multipolar Expansion for an Arbitrary Electric Charge Distribution
573(1)
8.3.4 Multipole of Any Order and Asymptotic Decay
574(2)
8.3.5 Rule of Successive Generation of Multipoles of All Orders
576(1)
8.3.6 Spherical Harmonics and Legendre Polynomials
577(1)
8.3.7 Potentials and Field Functions for Multipoles of All Orders
578(1)
8.3.8 Monopole and Axial Dipole, Quadrupole, and Octupole
579(1)
8.4 Vector/Scalar Potential and Solenoidal/Irrotational Field
580(3)
8.4.1 Vector Potential for a Solenoidal Field
580(1)
8.4.2 Magnetic Field due to a Distribution of Electric Currents
581(2)
8.4.3 Multipole Expansion for the Magnetostatic Field
583(1)
8.5 Point Current and Magnetic Dipole
583(2)
8.5.1 Dipolar Vector Potential and Magnetic Field
583(1)
8.5.2 Electric Current due to a Moving Electric Charge
584(1)
8.6 Quadrupolar Magnetic Potential and Field
585(3)
8.6.1 Time Average of the Vector Potential
585(1)
8.6.2 Magnetic Field and Moment of a Quadrupole
586(2)
8.7 Image on a Conducting or Insulating Plane
588(6)
8.7.1 Point Electric Charge near a Conducting or Insulating Plane
588(2)
8.7.2 Flow Source/Sink near a Rigid Impermeable Plane
590(2)
8.7.3 Identical or Opposite Images of a Point Electric Current in a Plane
592(1)
8.7.4 Point Vortex near a Rigid Impermeable Wall
593(1)
8.8 Source/Sink Images on Perpendicular or Parallel Planes
594(11)
8.8.1 Point Charge near Orthogonal Conducting/Insulating Planes
594(4)
8.8.2 Identical or Alternating Images on Parallel Planes
598(3)
8.8.3 Flow due to a Source/Sink between Parallel Planes
601(1)
8.8.4 Source/Sink near Orthogonal Walls
602(1)
8.8.5 Monopole at Equal Distance from Parallel Planes
603(2)
8.9 Discrete or Continuous Images in Spheres
605(40)
8.9.1 Image of a Point Charge on a Conducting Sphere
605(2)
8.9.2 Induced Charge Distribution and Asymptotic Field
607(2)
8.9.3 Point Flow Source/Sink near a Rigid Sphere
609(3)
8.9.4 Point Source and Line Sink as Images
612(1)
8.9.5 Continuous Sink along a Finite Line
612(1)
8.9.6 Velocity Field due to a Source near a Sphere
613(2)
8.9.7 Exact and Approximate Velocity on the Sphere
615(2)
8.9.8 Line Sink in a Uniform Stream
617(2)
8.9.9 Source Distribution Representing an Axisymmetric Fairing
619(1)
8.9.10 Source Distribution Representing a Finite Body
619(1)
8.9.11 Source Distribution for an Arbitrary Body of Revolution
620(1)
8.9.12 Flow in a Rotating Cylinder with Arbitrary Cross Section
621(1)
8.9.13 Kinetic Energy of a Flow in a Rotating Elliptic Cylinder
622(2)
8.9.14 Pressure Distribution on a Sphere due to a Source Sink
624(2)
8.9.15 Force on an Impermeable Sphere due to a Monopole
626(1)
8.9.16 Analogy between the Potential Flow and Electrostatics
627(1)
8.9.17 Force Exerted by a Point Charge on an Insulating/Conducting Sphere
628(17)
8.10 Conclusion
645(2)
9 Multidimensional Harmonic Potentials
647(98)
9.1 Boundary and Asymptotic Conditions for Unicity
648(6)
9.1.1 Divergence Theorem, the Two Green Identities, and the Kinetic Energy
648(1)
9.1.2 Cauchy-Dirichlet, Neumann, and Robin Problems
649(1)
9.1.3 Inner Unicity Problem for a Compact Region
650(1)
9.1.4 Asymptotic Condition for an Unbounded Region
651(1)
9.1.5 Outer Unicity Problem for a Noncompact Region
652(2)
9.2 Influence Function and Source Distributions
654(4)
9.2.1 Laplace Operator with Hyperspherical Symmetry
654(1)
9.2.2 Green Function for the Laplace Operator in Any Dimension
655(1)
9.2.3 Arbitrary Source Distribution in Free Space
656(1)
9.2.4 Poisson Equation in a Compact versus Noncompact Region
656(2)
9.2.5 Volume and Surface Source Distributions
658(1)
9.3 Mean Value, Extrema, and Constancy Theorems
658(4)
9.3.1 Volume Sources and Surface Flux
658(2)
9.3.2 Linear, Circular, Spherical, and Hyperspherical Mean Value
660(1)
9.3.3 Lemma of the Maximum and the Minimum on the Boundary
661(1)
9.3.4 Identity, Constancy, and Nullity Theorems
661(1)
9.4 Irrotational Fields and the Newton/Coulomb Laws
662(8)
9.4.1 Electrostatics in One/Two/Three Dimensions
662(2)
9.4.2 Ramp Potential due to a Charged Plane
664(1)
9.4.3 Logarithmic Potential due to a Line Charge
665(2)
9.4.4 Inverse-Distance Potential due to a Point Charge
667(1)
9.4.5 Gravity and Electric (Coulomb) Forces
668(2)
9.5 Solenoidal Fields and Biot-Savart Force
670(4)
9.5.1 Vector/Scalar Potential and Irrotational/Solenoidal Field
670(1)
9.5.2 Magnetic Field and Biot-Savart Force
671(1)
9.5.3 Continuous and Discrete Electric Current Distributions
672(1)
9.5.4 Gauge Condition for the Vector Potential
672(1)
9.5.5 Field Function for Plane Magnetostatics
673(1)
9.6 Hyperspherical or Generalized Legendre Polynomials (Campos and Cunha, 2012)
674(11)
9.6.1 Multidimensional Potential, Field, and Force
674(1)
9.6.2 Definition of Generalized or Hyperspherical Legendre Polynomials
675(2)
9.6.3 Hyperspherical Legendre Generating Function
677(1)
9.6.4 Particular Values of the Hyperspherical Legendre Polynomials
677(1)
9.6.5 Explicit Coefficients for the Generalized Legendre Polynomials
678(1)
9.6.6 Explicit Expressions in terms of Cosines of Angles and Multiple Angles
679(2)
9.6.7 Recurrence Formula for the Hyperspherical Harmonics
681(1)
9.6.8 Four Differentiation Formulas for the Hyperspherical Legendre Polynomials
682(1)
9.6.9 Hyperspherical Legendre Differential Equation
683(1)
9.6.10 Rodrigues and Schlaffi Integrals
684(1)
9.7 Multipoles in Hyperspherical and Hypercylindrical Coordinates
685(8)
9.7.1 Multidimensional Multipolar Expansion
685(1)
9.7.2 Radial, Longitudinal, and Multicolatitude Coordinates
686(2)
9.7.3 Hyperspherical and Hypercylindrical Coordinates
688(1)
9.7.4 Potential and Field of a Multidimensional Monopole
688(1)
9.7.5 Potential and Field of a Multidimensional Dipole
689(1)
9.7.6 Potential and Field of a Multidimensional Quadrupole
690(1)
9.7.7 Potential of an Octupole and Other Multipoles
691(1)
9.7.8 Multidimensional Nonaxisymmetric Multipoles
692(1)
9.8 Hypersphere Theorem and Insertion in a Uniform Field
693(5)
9.8.1 Insertion of a Hypersphere in a Uniform Field
693(1)
9.8.2 Hypersphere Theorem for the Potential
694(1)
9.8.3 Reciprocal Hyperpotential as a Harmonic Function
695(2)
9.8.4 Equipotential Hypersphere in a Uniform Field
697(1)
9.9 Images on Hyperplanes and Hyperspheres
698(45)
9.9.1 Equal or Opposite Images on a Hyperplane
698(1)
9.9.2 Far Field of a Monopole near a Hyperplane
699(1)
9.9.3 Reciprocal Point and Image on Hypersphere
699(2)
9.9.4 Force Exerted by a Source on a Hypersphere
701(42)
9.10 Conclusion
743(2)
10 Twenty Examples
745(70)
10.1 Examples 10.1 through 10.20
745(67)
10.2 Conclusion
812(3)
Bibliography 815(8)
Index 823
Considered one of Europes leading applied mathematicians, Luis Manuel Braga de Costa Campos has authored 9 books, 129 papers in 56 journals, and 203 communications to symposia. He holds 2 patents in the US and UK and has been involved in over 40 research projects with international external funding. Dr. Campos also received the Von Karman Medal from the NATO Advisory Group for Aerospace Research and Development (AGARD) and Research and Technology Organization (RTO). Participation in AGARD/RTO included a vice-chairmanship of the System Concepts and Integration Panel, chairmanship of the Flight Mechanics and Flight Vehicle Integration Panels, and membership in the Flight Test Techniques Working Group.
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