Unit 3073 Comprehensive Outline

1. Multivariable Calculus

1.1 Functions of Several Variables

• Definition and examples
• Domain and range in multivariable contexts

1.2 Partial Derivatives

• Concept and notation
• Higher-order partial derivatives
• Chain rule for multiple variables
• Gradient vector and directional derivatives

1.3 Multiple Integrals

• Double and triple integrals
• Change of variables and Jacobians
• Applications: volume, mass, and center of mass calculations

1.4 Vector Fields

• Definition and examples
• Gradient, divergence, and curl

1.5 Integral Theorems

• Green's Theorem
• Stokes' Theorem
• Divergence (Gauss) Theorem
• Applications of theorems in physics and engineering

2. Linear Algebra and Matrix Theory

2.1 Vector Spaces and Subspaces

• Definitions and examples
• Basis and dimension

2.2 Linear Transformations

• Definitions and matrix representation
• Kernel and image

2.3 Eigenvalues and Eigenvectors

• Characteristic polynomial
• Diagonalization and its criteria
• Applications in systems of differential equations

2.4 Matrix Applications

• Matrix decompositions (LU, QR)
• Applications in real-world problems: computer graphics, systems modeling

3. Differential Equations

3.1 Ordinary Differential Equations (ODEs)

• First-order ODEs: separable, linear, exact
• Higher-order linear ODEs
• Existence and uniqueness theorems

3.2 Partial Differential Equations (PDEs)

• Classification: elliptic, parabolic, hyperbolic
• Solution methods: separation of variables, Fourier series

3.3 Applications

• Physics: mechanics, heat conduction
• Engineering: control systems, signal processing

4. Complex Analysis

4.1 Complex Functions and Differentiation

• Complex plane and complex functions
• Cauchy-Riemann equations

4.2 Contour Integration

• Line integrals in the complex plane
• Cauchy's integral theorem and formula

4.3 Series Expansions

• Taylor series
• Laurent series

4.4 Residue Theorem

• Calculation of residues
• Application to evaluate integrals

5. Numerical Methods

5.1 Interpolation

• Polynomial interpolation
• Spline interpolation

5.2 Numerical Integration

• Trapezoidal and Simpson’s rules
• Gaussian quadrature

5.3 Numerical Solutions to Differential Equations

• Euler’s method
• Runge-Kutta methods

5.4 Error Analysis

• Types of errors
• Stability and convergence

6. Discrete Mathematics

6.1 Combinatorics

• Permutations and combinations
• Principle of inclusion-exclusion

6.2 Graph Theory

• Graph types and terminology
• Eulerian and Hamiltonian paths
• Graph coloring

6.3 Set Theory and Logic

• Basic set operations
• Propositional and predicate logic

6.4 Algorithms

• Complexity and Big O notation
• Fundamental algorithms and their applications

7. Topology

7.1 Topological Spaces

• Definitions and examples
• Open and closed sets

7.2 Continuity

• Continuous functions between topological spaces

7.3 Connectedness and Compactness

• Definitions and properties
• Applications in analysis

7.4 Convergence

• Convergent sequences and nets
• Limit points

8. Probability Theory and Statistics

8.1 Probability Theory

• Probability axioms
• Conditional probability and independence

8.2 Random Variables and Distributions

• Discrete and continuous random variables
• Common distributions: Binomial, Normal, Poisson

8.3 Statistical Inference

• Hypothesis testing
• Confidence intervals

8.4 Regression Analysis

• Linear regression
• Correlation and causation