Lesson Directory
Modeling and Analysis develops methods for continuous change, multidimensional systems, approximation, optimization, simulation, and applied mathematics. Lessons are grouped below by section.
Sections
5.1 Mathematical Modeling
- Quantities, Relationships, and Mechanisms
- Defining Variables and Parameters
- Assumptions, Constraints, and Domains
- Selecting Equations, Functions, Systems, and Algorithms
- Deterministic and Stochastic Models
- Static and Dynamic Models
- Calibration and Validation
- Sensitivity and Uncertainty
- Interpretation and Communication
- Model Limitations and Revision
5.2 Quantities, Parameters, Scale, and Dimension
5.3 Limits and Continuity
5.4 Differential Analysis
- Difference Quotients
- Derivatives from Limits
- Derivative Rules
- Product, Quotient, and Chain Rules
- Implicit and Parametric Differentiation
- Higher Derivatives
- Tangent and Normal Relationships
- Related Rates
- Linearization and Differentials
- Critical Points, Monotonicity, and Concavity
- Extrema and Curve Sketching
- Sensitivity as Derivative Information
5.5 Integral Analysis and Accumulation
5.6 Infinite Processes, Sequences, and Series
- Sequence Convergence
- Infinite Series
- Geometric and Telescoping Series
- Positive-Term Convergence Tests
- Alternating Series
- Absolute and Conditional Convergence
- Power Series
- Taylor and Maclaurin Series
- Error and Remainder Estimates
- Function Approximation by Series
- Products and Continued Processes
- Discrete-to-Continuous Connections
5.7 Multivariable Relationships
- Functions of Several Variables
- Domains, Ranges, and Level Sets
- Partial Derivatives
- Directional Derivatives
- Gradients
- Tangent Planes and Linear Approximation
- Multiple Integrals
- Coordinate Changes and Jacobians
- Constrained Extrema
- Implicit Relationships
- Vector-Valued Functions
- Multivariable Sensitivity
5.8 Vector Calculus and Fields
5.9 Linear Algebra and Multidimensional Systems
- Vector Spaces and Subspaces
- Linear Combinations, Span, and Independence
- Bases and Dimension
- Linear Transformations
- Matrix Representations
- Systems and Elimination
- Determinants
- Eigenvalues and Eigenvectors
- Diagonalization
- Inner Products and Orthogonality
- Least Squares
- Singular Values and Dimensional Reduction
5.10 Differential Equations
- First-Order Differential Equations
- Separable and Linear Equations
- Exact Equations
- Growth and Decay
- Logistic Models
- Second-Order Equations
- Oscillations and Forcing
- Systems of Differential Equations
- Initial and Boundary Conditions
- Phase-Plane Representations
- Numerical Solution Methods
- Introductory Partial Differential Equations
5.11 Dynamical Systems, Stability, and Chaos
5.12 Optimization
5.13 Approximation and Numerical Analysis
- Floating-Point Representation
- Sources and Propagation of Error
- Conditioning and Stability
- Root-Finding
- Numerical Linear Algebra
- Interpolation
- Approximation of Functions
- Numerical Differentiation
- Numerical Integration
- Numerical Differential Equations
- Convergence Rates
- Adaptive and Error-Controlled Methods
5.14 Transform and Spectral Methods
5.15 Complex Analysis
- Complex Numbers and the Complex Plane
- Polar and Exponential Forms
- Complex Functions
- Complex Limits and Continuity
- Complex Differentiation
- Analytic Functions
- Cauchy–Riemann Relationships
- Complex Integration
- Cauchy's Theorem and Integral Formula
- Series and Singularities
- Residues
- Conformal Mapping and Applications
5.16 Real, Functional, and Generalized Analysis
- Real-Number Completeness
- Sequences and Function Convergence
- Metric Spaces
- Normed and Inner-Product Spaces
- Continuity and Compactness
- Rigorous Differentiation and Integration
- Measure and Measurable Functions
- Lebesgue Integration
- Function Spaces
- Operators and Linear Functionals
- Distributions and Generalized Functions
- Introductory Functional Analysis
5.17 Uncertainty Within Models
- Measurement and Parameter Uncertainty
- Probability Distributions in Models
- Error Propagation
- Confidence and Prediction Intervals
- Monte Carlo Methods
- Stochastic Differential Ideas
- Reliability and Risk
- Scenario Analysis
- Sensitivity and Uncertainty Quantification
- Bayesian Parameter Treatment
- Robust Modeling
- Communication of Model Uncertainty
5.18 Computational Modeling and Simulation
5.19 Inverse Problems and Parameter Estimation
5.20 Applied Modeling Domains
- Mechanics and Engineering
- Waves, Heat, and Fluid Flow
- Electricity, Magnetism, and Fields
- Population, Ecology, and Epidemiology
- Chemistry and Reaction Systems
- Economics, Finance, and Operations
- Networks, Transportation, and Logistics
- Climate, Environment, and Geoscience
- Data Science and Machine Learning
- Control Systems and Robotics
- Imaging and Signal Processing
- Social and Policy Systems