Differential Equations and Linear Algebra

This text is intended for engineering, science, and math majors who have completed a three semester calculus sequence.

Mathematics

Tác giả: Steven Zook
Nhà xuất bản: Reedley College
Nguồn: LibreTexts
Giấy phép: CC BY-NC-SA
Ngôn ngữ: EN
Danh mục: Khoa học cơ bản

Đọc trực tuyến Tải PDF (nguồn mở)

Mục lục· 133 mục

01: Systems of Equations
1.01: Systems of Equations Geometry
1.02: Systems of Equations Algebraic Procedures
1.03: Gaussian Elimination
1.04: Uniqueness of the Reduced Row-Echelon Form
1.05: Rank and Homogeneous Systems
1.06: Exercises
02: Matrices
2.01: Matrix Arithmetic
2.02: Multiplication of Matrices
2.03: The ijth Entry of a Product
2.04: Properties of Matrix Multiplication
2.05: The Transpose
2.06: The Identity and Inverses
2.07: Finding the Inverse of a Matrix
2.08: Elementary Matrices
2.09: More on Matrix Inverses
2.10: LU Factorization
2.11: Exercises
03: Determinants
3.01: Basic Techniques
3.02: Properties of Determinants
3.03: Finding Determinants using Row Operations
3.04: Applications of the Determinant
3.05: Exercises
04: R
4.01: Vectors in R
4.02: Vector Algebra
4.03: Geometric Meaning of Vector Addition
4.04: Length of a Vector
4.05: Geometric Meaning of Scalar Multiplication
4.06: Parametric Lines
4.07: Spanning Linear Independence and Basis in R
4.08: Orthogonality
4.09: Applications
4.10: Exercises
05: Linear Transformations
5.01: Linear Transformations
5.02: The Matrix of a Linear Transformation I
5.03: Properties of Linear Transformations
5.04: Special Linear Transformations in R
5.05: One-to-One and Onto Transformations
5.06: The Kernel and Image of A Linear Map
5.07: The Matrix of a Linear Transformation II
5.08: The General Solution of a Linear System
5.09: Exercises
06: Spectral Theory
6.01: Eigenvalues and Eigenvectors of a Matrix
6.02: Diagonalization
6.03: Applications of Spectral Theory
6.04: Orthogonality
6.05: Exercises
07: What are Differential Equations
7.01: Applications Leading to Differential Equations
7.02: Basic Concepts
7.2.01: Basic Concepts (Exercises)
7.03: Direction Fields for First Order Equations
7.3.01: Direction Fields for First Order Equations (Exercises)
08: First Order Equations
8.01: Linear First Order Equations
8.1.01: Linear First Order Equations (Exercises)
8.02: Separable Equations
8.2.01: Separable Equations (Exercises)
8.03: Existence and Uniqueness of Solutions of Nonlinear Equations
8.3.01: Existence and Uniqueness of Solutions of Nonlinear Equations (Exercises)
8.04: Transformation of Nonlinear Equations into Separable Equations
8.4.01: Transformation of Nonlinear Equations into Separable Equations (Exercises)
8.05: Exact Equations
8.5.01: Exact Equations (Exercises)
8.06: Integrating Factors
8.6.01: Integrating Factors (Exercises)
09: Applications of First Order Equations
9.01: Growth and Decay
9.1.01: Growth and Decay (Exercises)
9.02: Cooling and Mixing
9.2.01: Cooling and Mixing (Exercises)
9.03: Elementary Mechanics
9.3.01: Elementary Mechanics (Exercises)
10: Linear Second Order Equations
10.01: Homogeneous Linear Equations
10.1.01: Homogeneous Linear Equations (Exercises)
10.02: Constant Coefficient Homogeneous Equations
10.2.01: Constant Coefficient Homogeneous Equations (Exercises)
10.03: Nonhomogeneous Linear Equations
10.3.01: Nonhomgeneous Linear Equations (Exercises)
10.04: The Method of Undetermined Coefficients I
10.4.01: The Method of Undetermined Coefficients I (Exercises)
10.05: The Method of Undetermined Coefficients II
10.5.01: The Method of Undetermined Coefficients II (Exercises)
10.06: Reduction of Order
10.6.01: Reduction of Order (Exercises)
11: Applications of Linear Second Order Equations
11.01: Spring Problems I
11.1.01: Spring Problems I (Exercises)
11.02: Spring Problems II
11.2.01: Spring Problems II (Exercises)
12: Series Solutions of Linear Second Order Equations
12.01: Prelude to Series Solutions of Linear Second Order Equations
12.02: Review of Power Series
12.2.01: Review of Power Series (Exercises)
12.03: Series Solutions Near an Ordinary Point I
12.3.01: Series Solutions Near an Ordinary Point I (Exercises)
13: Laplace Transforms
13.01: Introduction to the Laplace Transform
13.1.01: Introduction to the Laplace Transform (Exercises)
13.02: The Inverse Laplace Transform
13.2.01: The Inverse Laplace Transform (Exercises)
13.03: Solution of Initial Value Problems
13.3.01: Solution of Initial Value Problems (Exercises)
13.04: The Unit Step Function
13.4.01: The Unit Step Function (Exercises)
13.05: Constant Coefficient Equations with Piecewise Continuous Forcing Functions
13.5.01: Constant Coefficient Equations with Piecewise Continuous Forcing Functions (Exercises)
13.06: A Brief Table of Laplace Transforms
14: Linear Higher Order Differential Equations
14.01: Introduction to Linear Higher Order Equations
14.1.01: Introduction to Linear Higher Order Equations (Exercises)
14.02: Higher Order Constant Coefficient Homogeneous Equations
14.2.01: Higher Order Constant Coefficient Homogeneous Equations (Exercises)
15: Linear Systems of Differential Equations
15.01: Introduction to Systems of Differential Equations
15.1.01: Introduction to Systems of Differential Equations (Exercises)
15.02: Linear Systems of Differential Equations
15.2.01: Linear Systems of Differential Equations (Exercises)
15.03: Basic Theory of Homogeneous Linear Systems
15.3.01: Basic Theory of Homogeneous Linear Systems (Exercises)
15.04: Constant Coefficient Homogeneous Systems I
15.4.01: Constant Coefficient Homogeneous Systems I (Exercises)
16: Appendices
16.01: Trigonometric Identities
16.02: Review of Derivative Rules
16.03: Table of Derivatives
16.04: Table of Integrals