Xiong-Fei Du

| Home | MATHLAB | Math Notes | Elections |

This site stores some of my personal math notes on introductory mathematics courses, such as single variable calculus, multivariable calculus, differential equations, and linear algebra. If you find errors here or there as I do every so often, feel free to send me an email.

Single Variable Calculus

- Limits
- Concepts of Calculus
- Differentiation
- Rules of Differentiation
- Differentiation Techniques
- Analysis of Functions with Differentiation
- Rolle's Theorem and Mean Value Theorem
- Applications of Differentiation
- Differentiation and Approximation
- Integration
- Integration and Approximation
- Integration Techniques
- Volumes of Revolution
- Geometric Applications of Integration
- Parametric Equations and Vectors
- Polar Equations and Transformations
- Series and Sequences
- Series: Convergence and Divergence Tests
- Taylor Polynomials and Power Series
- First-Order Differential Equations

Practice Examination


Multivariable Calculus

I’ve somewhat tried to make these notes more about multivariable calculus than just three-dimensional calculus, so some of these notes encompass more than what is typical in a multivariable calculus course, which at most institutions, only goes up to three dimensions. Basic knowledge of matrices and linear algebra is assumed. Also, just a heads up, my understanding of the concept of manifolds in space and integration on a manifold is a bit shaky, so if you notice any errors, feel free to email me. If you are really interested in more multivariable calculus and analysis with a more rigourous approach, I highly recommend the following notes from MIT's 18.101 Analysis II.

- Geometry of Space
- Vectors
- Linear Geometry in Space
- Curves in Space
- Multivariable Functions
- Partial Differentiation
- Approximation
- First-Order Partial Differentiation
- Second-Order Partial Differentiation
- Multivariable Optimization
- Iterated Integrals
- Change of Variables
- Scalar Integration in Space
- Vector Fields in Space
- Vector Integration in Space
- Theorems of Vector Calculus

MATLAB code for three-dimensional graphing:
- Cartesian Coordinates
- Cylindrical Coordinates
- Spherical Coordinates


Differential Equations

These notes are a bit longer in comparison to my previous ones, but overall, I tried to cover a broad range of topics. These notes assume a working knowledge of complex numbers and basic linear algebra.

- Differential Equations and Modeling
- First-Order Differential Equations
- Separable First-Order Differential Equations
- First-Order Linear Differential Equations
- Approximation and Numerical Methods
- Superposition Principle and Wronskian Determinants (Proof)
- Homogeneous Linear Differential Equations
- Second-Order Homogeneous Linear ODE with Constant Coefficients
- Non-Homogeneous Linear Differential Equations
- Particular Solutions of Nonhomogeneous ODEs
- Resonance and Beats
- Fourier Series
- Common Discontinuous Functions (Periodic)
- Laplace Transform (Table)
- Convolution
- Eigenvalues and Eigenvectors
- System of Ordinary Differential Equations
- First-Order Linear Homogeneous Systems of ODEs
- First-Order Linear Non-Homogeneous Systems of ODEs
- Matrix Exponential
- Eigenvalues and Eigenfunctions (Worked Example)
- Boundary Value Problems and Sturm-Liouville Theory
- Partial Differential Equations
- Separation of Variables

Background Notes:
- Complex Analysis
- Vectors
- Matrices
- Matrix Operations
- Eigenvalues and Eigenvectors

MATLAB code for ordinary differential equations:
- First-Order
- Second-Order

For more MATLAB codes, Prof. Gilbert Strang of MIT has a pretty good collection on his Computational Science and Engineering site, which can be found here.


Linear Algebra

I'm still working on these.

- Vectors
- Matrices
- Square Matrices
- Matrix Operations
- Determinants
- Inverse Matrices
- Systems of Linear Equations
- Gaussian Elimination and Row-Echelon Form
- LU Decomposition and Cramer's Rule
- Block Matrices
- Linear Geometry
- Linear Independence
- Vector Spaces and Subspaces
- Basis and Dimension
- Inner Product and Orthogonality
- Vector Projections and the Gram-Schmidt Process
- Linear Transformations
- Eigenvalues and Eigenvectors
- QR-Decomposition and Diagonalization
- Jordan Normal Form and Similar Matrices
- First-Order Linear Homogeneous Systems of ODEs
- Matrix Exponential
- Matrix Calculus
- Definiteness of a Matrix
- Perron-Frobenius Theorem and Stochastic Matrices
- Singular Value Decomposition