Unit 2953: Advanced Calculus Topics

1. Sequences and Series

1.1 Introduction to Sequences

• Definition of sequences
• Arithmetic sequences: definition, general term, sum
• Geometric sequences: definition, general term, sum

1.2 Infinite Series

• Definition and notation
• Convergence and divergence
• Tests for convergence:
  • Ratio test
  • Integral test
  • Comparison test (brief mention)

1.3 Power Series

• Definition and radius of convergence
• Interval of convergence
• Operations on power series

1.4 Taylor and Maclaurin Series

• Concept and derivation
• Examples of expansions (e.g., exponential, trigonometric, logarithmic functions)
• Applications of Taylor and Maclaurin series

2. Techniques of Integration

2.1 Integration by Parts

• Formula and derivation
• Choosing u and dv
• Examples

2.2 Trigonometric Integrals

• Integrals involving powers of sine and cosine
• Integrals involving powers of tangent and secant

2.3 Trigonometric Substitution

• Substitutions for �6022+x^2,22-x^2,22-x^2�60 forms
• Examples

2.4 Partial Fractions Decomposition

• Method for rational functions
• Types of partial fractions
• Examples

2.5 Improper Integrals

• Definition and types (infinite limits, discontinuities)
• Convergence tests
• Examples

2.6 Applications of Integration

• Area between curves
• Volumes of solids of revolution
• Work and fluid pressure problems

3. Differential Equations

3.1 First-Order Differential Equations

• Separable equations
• Linear first-order equations
• Applications (growth and decay)

3.2 Second-Order Linear Differential Equations with Constant Coefficients

• Homogeneous equations
• Characteristic equation and solutions

3.3 Methods of Solving Nonhomogeneous Equations

• Method of undetermined coefficients
• Variation of parameters

3.4 Applications

• Modeling growth and decay
• Mechanical and electrical systems (basic introduction)

4. Parametric Equations and Polar Coordinates

4.1 Parametric Equations

• Definition and examples
• Eliminating the parameter
• Calculus with parametric curves (derivatives, arc length)

4.2 Polar Coordinates

• Definition and relationship to Cartesian coordinates
• Plotting polar curves

4.3 Calculus in Polar Coordinates

• Area inside a polar curve
• Arc length of polar curves

4.4 Conic Sections in Polar Form

• Equations of conics
• Eccentricity and directrix

5. Vectors and Vector Calculus

5.1 Vectors in 2D and 3D

• Definition and notation
• Vector operations: addition, scalar multiplication

5.2 Dot Product

• Definition and properties
• Applications (work, angle between vectors)

5.3 Cross Product

• Definition and properties
• Applications (area of parallelogram, torque)

5.4 Vector-Valued Functions

• Definition and examples
• Derivatives and integrals

5.5 Line Integrals

• Definition and computation
• Physical interpretation

5.6 Green's Theorem

• Statement and explanation
• Applications

5.7 Surface Integrals and Vector Field Operators

• Surface integrals overview
• Divergence and curl definitions
• Physical interpretations

6. Multivariable Calculus

6.1 Multivariable Functions

• Domain and range
• Graphing and level curves

6.2 Partial Derivatives

• Definition and computation
• Gradient vector

6.3 Optimization Using Partial Derivatives

• Critical points
• Lagrange multipliers (brief introduction)

6.4 Multiple Integrals

• Double integrals over rectangular and general regions
• Triple integrals

6.5 Coordinate Systems

• Cylindrical coordinates
• Spherical coordinates
• Conversion formulas

6.6 Applications

• Volume, mass, center of mass calculations
• Applications in physics and engineering

6.7 Vector Calculus Theorems

• Divergence theorem
• Stokes' theorem
• Physical and geometric interpretations