Calculus 관련 글 목차

2000, Jan 01    

• 아래 내용은 대학 수학에 기본적으로 필요한 수학인 Kreyszig 공업수학 10판 내용과 Vector Calculus, Differential Geometry 입니다.

Advanced Engineering Mathematics

• 본 내용은 Kreyszig 공업수학으로 알려진 Advanced Engineering Mathematics의 내용을 책 순서대로 공부하여 정리한 글 목록 입니다.

Part A Ordinary Differential Equations (ODEs)

Chapter 1 First-Order ODEs

• 1.1 Basic Concepts. Modeling
• 1.2 Geometric Meaning of y’= ƒ(x, y). Direction Fields, Euler’s Method
• 1.3 Separable ODEs. Modeling
• 1.4 Exact ODEs. Integrating Factors
• 1.5 Linear ODEs. Bernoulli Equation. Population Dynamics
• 1.6 Orthogonal Trajectories. Optional
• 1.7 Existence and Uniqueness of Solutions for Initial Value Problems

Chapter 2 Second-Order Linear ODEs 46

• 2.1 Homogeneous Linear ODEs of Second Order
• 2.2 Homogeneous Linear ODEs with Constant Coefficients
• 2.3 Differential Operators. Optional
• 2.4 Modeling of Free Oscillations of a Mass–Spring System
• 2.5 Euler–Cauchy Equations
• 2.6 Existence and Uniqueness of Solutions. Wronskian
• 2.7 Nonhomogeneous ODEs
• 2.8 Modeling: Forced Oscillations. Resonance
• 2.9 Modeling: Electric Circuits
• 2.10 Solution by Variation of Parameters

Chapter 3 Higher Order Linear ODEs

• 3.1 Homogeneous Linear ODEs
• 3.2 Homogeneous Linear ODEs with Constant Coefficients
• 3.3 Nonhomogeneous Linear ODEs

Chapter 4 Systems of ODEs. Phase Plane. Qualitative Methods

• 4.0 For Reference: Basics of Matrices and Vectors
• 4.1 Systems of ODEs as Models in Engineering Applications
• 4.2 Basic Theory of Systems of ODEs. Wronskian
• 4.3 Constant-Coefficient Systems. Phase Plane Method
• 4.4 Criteria for Critical Points. Stability
• 4.5 Qualitative Methods for Nonlinear Systems
• 4.6 Nonhomogeneous Linear Systems of ODEs

Chapter 5 Series Solutions of ODEs. Special Functions

• 5.1 Power Series Method
• 5.2 Legendre’s Equation. Legendre Polynomials Pn(x)
• 5.3 Extended Power Series Method: Frobenius Method
• 5.4 Bessel’s Equation. Bessel Functions Jv(x)
• 5.5 Bessel Functions of the Yv(x). General Solution

Chapter 6 Laplace Transforms

• 6.1 Laplace Transform. Linearity. First Shifting Theorem (s-Shifting)
• 6.2 Transforms of Derivatives and Integrals. ODEs
• 6.3 Unit Step Function (Heaviside Function). Second Shifting Theorem (t-Shifting)
• 6.4 Short Impulses. Dirac’s Delta Function. Partial Fractions
• 6.5 Convolution. Integral Equations
• 6.6 Differentiation and Integration of Transforms. ODEs with Variable Coefficients
• 6.7 Systems of ODEs
• 6.8 Laplace Transform: General Formulas
• 6.9 Table of Laplace Transforms

Part B Linear Algebra. Vector Calculus

Chapter 7 Linear Algebra: Matrices, Vectors, Determinants. Linear Systems

• 7.1 Matrices, Vectors: Addition and Scalar Multiplication
• 7.2 Matrix Multiplication
• 7.3 Linear Systems of Equations. Gauss Elimination
• 7.4 Linear Independence. Rank of a Matrix. Vector Space
• 7.5 Solutions of Linear Systems: Existence, Uniqueness
• 7.6 For Reference: Second- and Third-Order Determinants
• 7.7 Determinants. Cramer’s Rule
• 7.8 Inverse of a Matrix. Gauss–Jordan Elimination
• 7.9 Vector Spaces, Inner Product Spaces. Linear Transformations. Optional

Chapter 8 Linear Algebra: Matrix Eigenvalue Problems

• 8.1 The Matrix Eigenvalue Problem. Determining Eigenvalues and Eigenvectors
• 8.2 Some Applications of Eigenvalue Problems
• 8.3 Symmetric, Skew-Symmetric, and Orthogonal Matrices
• 8.4 Eigenbases. Diagonalization. Quadratic Forms
• 8.5 Complex Matrices and Forms. Optional

Chapter 9 Vector Differential Calculus. Grad, Div, Curl

• 9.1 Vectors in 2-Space and 3-Space
• 9.2 Inner Product (Dot Product)
• 9.3 Vector Product (Cross Product)
• 9.4 Vector and Scalar Functions and Their Fields. Vector Calculus: Derivatives
• 9.5 Curves. Arc Length. Curvature. Torsion
• 9.6 Calculus Review: Functions of Several Variables. Optional
• 9.7 Gradient of a Scalar Field. Directional Derivative
• 9.8 Divergence of a Vector Field
• 9.9 Curl of a Vector Field

Chapter 10 Vector Integral Calculus. Integral Theorems

• 10.1 Line Integrals
• 10.2 Path Independence of Line Integrals
• 10.3 Calculus Review: Double Integrals. Optional
• 10.4 Green’s Theorem in the Plane
• 10.5 Surfaces for Surface Integrals
• 10.6 Surface Integrals
• 10.7 Triple Integrals. Divergence Theorem of Gauss
• 10.8 Further Applications of the Divergence Theorem
• 10.9 Stokes’s Theorem

Part C Fourier Analysis. Partial Differential Equations (PDEs)

Chapter 11 Fourier Analysis

• 11.1 Fourier Series
• 11.2 Arbitrary Period. Even and Odd Functions. Half-Range Expansions
• 11.3 Forced Oscillations
• 11.4 Approximation by Trigonometric Polynomials
• 11.5 Sturm–Liouville Problems. Orthogonal Functions
• 11.6 Orthogonal Series. Generalized Fourier Series
• 11.7 Fourier Integral
• 11.8 Fourier Cosine and Sine Transforms
• 11.9 Fourier Transform. Discrete and Fast Fourier Transforms
• 11.10 Tables of Transforms

Chapter 12 Partial Differential Equations (PDEs)

• 12.1 Basic Concepts of PDEs
• 12.2 Modeling: Vibrating String, Wave Equation
• 12.3 Solution by Separating Variables. Use of Fourier Series
• 12.4 D’Alembert’s Solution of the Wave Equation. Characteristics
• 12.5 Modeling: Heat Flow from a Body in Space. Heat Equation
• 12.6 Heat Equation: Solution by Fourier Series. Steady Two-Dimensional Heat Problems. Dirichlet Problem
• 12.7 Heat Equation: Modeling Very Long Bars. Solution by Fourier Integrals and Transforms
• 12.8 Modeling: Membrane, Two-Dimensional Wave Equation
• 12.9 Rectangular Membrane. Double Fourier Series
• 12.10 Laplacian in Polar Coordinates. Circular Membrane. Fourier–Bessel Series
• 12.11 Laplace’s Equation in Cylindrical and Spherical Coordinates. Potential
• 12.12 Solution of PDEs by Laplace Transforms

Part D Complex Analysis

Chapter 13 Complex Numbers and Functions. Complex Differentiation

• 13.1 Complex Numbers and Their Geometric Representation
• 13.2 Polar Form of Complex Numbers. Powers and Roots
• 13.3 Derivative. Analytic Function
• 13.4 Cauchy–Riemann Equations. Laplace’s Equation
• 13.5 Exponential Function
• 13.6 Trigonometric and Hyperbolic Functions. Euler’s Formula
• 13.7 Logarithm. General Power. Principal Value

Chapter 14 Complex Integration

• 14.1 Line Integral in the Complex Plane
• 14.2 Cauchy’s Integral Theorem
• 14.3 Cauchy’s Integral Formula
• 14.4 Derivatives of Analytic Functions

Chapter 15 Power Series, Taylor Series

• 15.1 Sequences, Series, Convergence Tests
• 15.2 Power Series
• 15.3 Functions Given by Power Series
• 15.4 Taylor and Maclaurin Series
• 15.5 Uniform Convergence. Optional

Chapter 16 Laurent Series. Residue Integration

• 16.1 Laurent Series
• 16.2 Singularities and Zeros. Infinity
• 16.3 Residue Integration Method
• 16.4 Residue Integration of Real Integrals

Chapter 17 Conformal Mapping

• 17.1 Geometry of Analytic Functions: Conformal Mapping
• 17.2 Linear Fractional Transformations (Möbius Transformations)
• 17.3 Special Linear Fractional Transformations
• 17.4 Conformal Mapping by Other Functions
• 17.5 Riemann Surfaces. Optional

Chapter 18 Complex Analysis and Potential Theory

• 18.1 Electrostatic Fields
• 18.2 Use of Conformal Mapping. Modeling
• 18.3 Heat Problems
• 18.4 Fluid Flow
• 18.5 Poisson’s Integral Formula for Potentials
• 18.6 General Properties of Harmonic Functions. Uniqueness Theorem for the Dirichlet Problem

Part E Numeric Analysis

Chapter 19 Numerics in General

• 19.1 Introduction
• 19.2 Solution of Equations by Iteration
• 19.3 Interpolation
• 19.4 Spline Interpolation
• 19.5 Numeric Integration and Differentiation

Chapter 20 Numeric Linear Algebra

• 20.1 Linear Systems: Gauss Elimination
• 20.2 Linear Systems: LU-Factorization, Matrix Inversion
• 20.3 Linear Systems: Solution by Iteration
• 20.4 Linear Systems: Ill-Conditioning, Norms
• 20.5 Least Squares Method
• 20.6 Matrix Eigenvalue Problems: Introduction
• 20.7 Inclusion of Matrix Eigenvalues
• 20.8 Power Method for Eigenvalues
• 20.9 Tridiagonalization and QR-Factorization

Chapter 21 Numerics for ODEs and PDEs

• 21.1 Methods for First-Order ODEs
• 21.2 Multistep Methods
• 21.3 Methods for Systems and Higher Order ODEs
• 21.4 Methods for Elliptic PDEs
• 21.5 Neumann and Mixed Problems. Irregular Boundary
• 21.6 Methods for Parabolic PDEs
• 21.7 Method for Hyperbolic PDEs

Part F Optimization, Graphs

Chapter 22 Unconstrained Optimization. Linear Programming

• 22.1 Basic Concepts. Unconstrained Optimization: Method of Steepest Descent
• 22.2 Linear Programming
• 22.3 Simplex Method
• 22.4 Simplex Method: Difficulties

Chapter 23 Graphs. Combinatorial Optimization

• 23.1 Graphs and Digraphs
• 23.2 Shortest Path Problems. Complexity
• 23.3 Bellman’s Principle. Dijkstra’s Algorithm
• 23.4 Shortest Spanning Trees: Greedy Algorithm
• 23.5 Shortest Spanning Trees: Prim’s Algorithm
• 23.6 Flows in Networks
• 23.7 Maximum Flow: Ford–Fulkerson Algorithm
• 23.8 Bipartite Graphs. Assignment Problems

Vector Calculus

Differential Geometry

Calculus 관련 기타 내용

• 적분 기초 계산 정리 (치환 적분, 부분 적분, 삼각함수 적분, 분수함수 적분)
• Gradient (그래디언트), Jacobian (자코비안) 및 Hessian (헤시안)
• 극좌표계 (Polar Coordinate)
• 원통 좌표계와 구면 좌표계
• 오일러 공식 (Euler formula)
• 복소 평면 (complex plane)
• 페이저 (Phasor)
• 삼각함수 공식 모음
• atan과 atan2 비교
• Asymmetric Exponential