Analysis in Mechanical Engineering
© Leon van Dommelen
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List of Figures
Contents
Dedication
List of Figures
List of Tables
Preface
To the Student
Acknowledgments
Comments and Feedback
I. Calculus
1. Graphs
1.1 Introduction
1.2 Example
1.2.1 Using reasoning
1.2.2 Using brute force
1.3 Example
1.3.1 Solution
2. Optimization
2.1 Introduction
2.2 Example
2.2.1 Definition
2.2.2 Reduction
2.2.3 Further reduction
2.2.4 Finding the length
2.2.5 Finding the optimum angle
2.2.6 Finding the optimum length
2.3 General Approach
2.3.1 Formulation
2.3.2 Interior minima
2.3.3 Boundary minima
3. Approximations
3.1 Introduction
3.2 Example
3.2.1 Identification
3.2.2 Results
3.2.3 Other way
3.3 Example
3.3.1 Identification
3.3.2 Finish
4. Limits
4.1 Introduction
4.2 Example
4.2.1 Observations
4.2.2 L'Hopital
4.2.3 Better
4.3 Example
4.3.1 Grinding it out
4.3.2 Using insight
5. Combined Changes in Variables
5.1 Introduction
5.2 Example
5.2.1 Identification
5.2.2 Results
5.3 Example
5.3.1 Solution
6. Curvilinear Motion
6.1 Introduction
6.2 Example
6.2.1 Position
6.2.2 Velocity
6.2.3 Acceleration
7. Line Integrals
7.1 Introduction
7.2 Example
7.2.1 Identification
7.2.2 Solution
8. Surface and Volume Integrals
8.1 Introduction
8.2 Example
8.2.1 Region
8.2.2 Approach
8.2.3 Results
8.3 Example
8.3.1 Approach
8.3.2 Results
9. Numerical Integration
9.1 Introduction
9.2 Example
9.2.1 Solution
10. Geometry using vectors
10.1 Introduction
10.2 Example
10.2.1 Identification
10.2.2 Solution
10.3 Example
10.3.1 Identification
10.3.2 Solution
11. Vector Analysis
11.1 Coordinate Changes
11.1.1 General
11.1.2 Orthogonal coordinates
II. Linear Algebra
12. Gaussian Elimination
12.1 Elimination Procedure
12.2 Partial Pivoting
12.3 Back substitution
12.4 LU Theorem
12.5 Row Canonical Form
12.6 Null Spaces and Solution Spaces
13. Inverse Matrices
13.1 Finding inverses using GE
13.2 Finding inverses using minors
13.3 Finding inverses using transposing
14. Eigenvalues and Eigenvectors
14.1 Finding Eigenvalues
14.2 Eigenvectors of nonsymmetric matrices
14.3 Eigenvectors of symmetric matrices
15. Change of Basis
15.1 General Procedure
15.2 Diagonalization of nonsymmetric matrices
15.3 Diagonalization of symmetric matrices
III. Ordinary Differential Equations
16. Laplace Transformation
16.1 Partial Fractions
16.2 Completing the square
17. More on Systems
17.1 Solution of systems using diagonalization
17.2 Solving Partial Differential Equations
17.3 More details on the extension
IV. Partial Differential Equations
18. Introduction
18.1 Basic Concepts
18.1.1 The prevalence of partial differential equations
18.1.2 Definitions
18.1.3 Typical boundary conditions
18.2 The Standard Examples
18.2.1 The Laplace equation
18.2.2 The heat equation
18.2.3 The wave equation
18.3 Properly Posedness
18.3.1 The conditions for properly posedness
18.3.2 An improperly posed parabolic problem
18.3.3 An improperly posed elliptic problem
18.3.4 Improperly posed hyperbolic problems
18.4 Energy methods
18.4.1 The Poisson equation
18.4.2 The heat equation
18.4.3 The wave equation
18.5 Variational methods [None]
18.6 Classification
18.6.1 Introduction
18.6.2 Scalar second order equations
18.7 Changes of Coordinates
18.7.1 Introduction
18.7.2 The formulae for coordinate transformations
18.7.3 Rotation of coordinates
18.7.4 Explanation of the classification
18.8 Two-Dimensional Coordinate Transforms
18.8.1 Characteristic Coordinates
18.8.2 Parabolic equations in two dimensions
18.8.3 Elliptic equations in two dimensions
19. Green’s Functions
19.1 Introduction
19.1.1 The one-dimensional Poisson equation
19.1.2 More on delta and Green’s functions
19.2 The Poisson equation in infinite space
19.2.1 Overview
19.2.2 Loose derivation
19.2.3 Rigorous derivation
19.3 The Poisson or Laplace equation in a finite region
19.3.1 Overview
19.3.2 Intro to the solution procedure
19.3.3 Derivation of the integral solution
19.3.4 Boundary integral (panel) methods
19.3.5 Poisson’s integral formulae
19.3.6 Derivation
19.3.7 The integral formula for the Neumann problem
19.3.8 Smoothness of the solution
20. First Order Equations
20.1 Classification and characteristics
20.2 Numerical solution
20.3 Analytical solution
20.4 Using the boundary or initial condition
20.5 The inviscid Burgers’ equation
20.5.1 Wave steepening
20.5.2 Shocks
20.5.3 Conservation laws
20.5.4 Shock relation
20.5.5 The entropy condition
20.6 First order equations in more dimensions
20.7 Systems of First Order Equations (None)
21. D'Alembert Solution of the Wave equation
21.1 Introduction
21.2 Extension to finite regions
21.2.1 The physical problem
21.2.2 The mathematical problem
21.2.3 Dealing with the boundary conditions
21.2.4 The final solution
22. Separation of Variables
22.1 A simple example
22.1.1 The physical problem
22.1.2 The mathematical problem
22.1.3 Outline of the procedure
22.1.4 Step 1: Find the eigenfunctions
22.1.5 Should we solve the other equation?
22.1.6 Step 2: Solve the problem
22.2 Comparison with D'Alembert
22.3 Understanding the Procedure
22.3.1 An ordinary differential equation as a model
22.3.2 Vectors versus functions
22.3.3 The inner product
22.3.4 Matrices versus operators
22.3.5 Some limitations
22.4 Handling Periodic Boundary Conditions
22.4.1 The physical problem
22.4.2 The mathematical problem
22.4.3 Outline of the procedure
22.4.4 Step 1: Find the eigenfunctions
22.4.5 Step 2: Solve the problem
22.4.6 Summary of the solution
22.5 Finding the Green's function
22.6 Inhomogeneous boundary conditions
22.6.1 The physical problem
22.6.2 The mathematical problem
22.6.3 Outline of the procedure
22.6.4 Step 0: Fix the boundary conditions
22.6.5 Step 1: Find the eigenfunctions
22.6.6 Step 2: Solve the problem
22.6.7 Summary of the solution
22.7 Finding the Green's functions
22.8 An alternate procedure
22.8.1 The physical problem
22.8.2 The mathematical problem
22.8.3 Step 0: Fix the boundary conditions
22.8.4 Step 1: Find the eigenfunctions
22.8.5 Step 2: Solve the problem
22.8.6 Summary of the solution
22.9 A Summary of Separation of Variables
22.9.1 The form of the solution
22.9.2 Limitations of the method
22.9.3 The procedure
22.9.4 More general eigenvalue problems
22.10 More general eigenfunctions
22.10.1 The physical problem
22.10.2 The mathematical problem
22.10.3 Step 0: Fix the boundary conditions
22.10.4 Step 1: Find the eigenfunctions
22.10.5 Step 2: Solve the problem
22.10.6 Summary of the solution
22.10.7 An alternative procedure
22.11 A Problem in Three Independent Variables
22.11.1 The physical problem
22.11.2 The mathematical problem
22.11.3 Step 1: Find the eigenfunctions
22.11.4 Step 2: Solve the problem
22.11.5 Summary of the solution
23. Fourier Transforms [None]
24. Laplace Transforms
24.1 Overview of the Procedure
24.1.1 Typical procedure
24.1.2 About the coordinate to be transformed
24.2 A parabolic example
24.2.1 The physical problem
24.2.2 The mathematical problem
24.2.3 Transform the problem
24.2.4 Solve the transformed problem
24.2.5 Transform back
24.3 A hyperbolic example
24.3.1 The physical problem
24.3.2 The mathematical problem
24.3.3 Transform the problem
24.3.4 Solve the transformed problem
24.3.5 Transform back
24.3.6 An alternate procedure
V. Supplementary Information
A. Addenda
A.1 Distributions
D. Derivations
D.1 Orthogonal coordinate derivatives
D.2 Harmonic functions are analytic
D.3 Some properties of harmonic functions
D.4 Coordinate transformation derivation
D.5 2D coordinate transformation derivation
D.6 2D elliptical transformation
N. Notes
N.1 Why this book?
N.2 History and wish list
Web Pages
References
Notations
Index
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