Linear Algebra
Definition of a Vector Space24Subspaces30Linear Independence32Spanning Sets49Basis and Dimension51Definition of a Linear Transformation54Matrix Representation of Linear Transformations46Invertible Linear Transformations45Eigenvalues and Eigenvectors47Diagonalization53Definition of an Inner Product Space55Orthogonality51Gram-Schmidt Process58Projection49
Differential Equations
Introduction to First-Order Differential Equations50Separable Equations52Linear Equations55Exact Equations55Applications of First-Order Differential Equations51Introduction to Higher-Order Differential Equations59Homogeneous Linear Equations43Nonhomogeneous Linear Equations47Applications of Higher-Order Differential Equations32Introduction to Systems of Differential Equations46Solving Systems of Linear Equations59Eigenvalues and Eigenvectors53Applications of Systems of Differential Equations57Introduction to Partial Differential Equations52First-Order Partial Differential Equations38Second-Order Partial Differential Equations53Applications of Partial Differential Equations51
Complex Analysis
Definition and Representation of Complex Numbers51Algebraic Operations with Complex Numbers19Complex Functions44Cauchy-Riemann Equations56Complex Integration44Power Series and Taylor Series55Conformal Transformations56Applications of Conformal Mappings51Complex Analysis in Physics56Complex Analysis in Engineering55
Numerical Analysis
Definition and Scope of Numerical Analysis51Errors in Numerical Computations52Floating-Point Arithmetic51Direct Methods for Solving Linear Systems51Iterative Methods for Solving Linear Systems50Applications of Linear Systems42Root-Finding Methods53Systems of Nonlinear Equations57Applications of Nonlinear Equations55Polynomial Interpolation39Spline Interpolation51Least Squares Approximation31Numerical Differentiation51Numerical Integration50Applications of Numerical Differentiation and Integration55Initial Value Problems51Boundary Value Problems54Applications of Ordinary Differential Equations52Finite Difference Methods52Finite Element Methods59Applications of Partial Differential Equations53
Mathematical Modeling
What is Mathematical Modeling?16Steps in the Modeling Process26Types of Mathematical Models10Model Validation and Evaluation19Linear Equations and Systems28Linear Regression29Applications of Linear Models34Limitations of Linear Models40Introduction to Nonlinear Functions35Nonlinear Regression22Applications of Nonlinear Models57Challenges in Nonlinear Modeling10Introduction to Optimization0Linear Programming28Nonlinear Programming20Applications of Optimization Models17Introduction to Probability and Statistics19Stochastic Processes28Monte Carlo Simulation9Applications of Stochastic Models9Introduction to Differential Equations10Modeling with Ordinary Differential Equations16Modeling with Partial Differential Equations28Numerical Methods for Solving Differential Equations10Modeling in Finance0Modeling in Biology and Medicine44Modeling in Engineering48Modeling in Social Sciences56