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Documents

Lecture notes: Notes

Exercise sheets:

• TD1 - Introduction and Basic Concepts (Worksheet)
• TD2 - Parametric Curves (Worksheet & Solutions)
• TD3 - Geometry and Topology in $\mathbb{R}^n$ (Worksheet & Solutions)
• TD4 - Differential Calculus (Worksheet & Solutions)
• TD5 - Multiple Integrals (Worksheet & Solutions)

Exam preparations:

• Midterm 1 - Parametric Curves, Topology, Taylor Expansion (Exam & Solutions)
• Pop Quiz - Topology of Normed Spaces, Multivariable Functions (Quiz)
• Midterm 2.1 - Differential Calculus (Exam & Solutions)
• Midterm 2.2 - Multivariable functions (Exam & Solutions)

Additional ressources:

• Python script to plot surfaces and level curves for two-variable functions (Code)

Contents

1) Introduction and Basic Concepts

• Introduction
• Multivariable Functions: Definitions, Examples, Graphs, Level Sets, Partial Functions
• Vector Functions and Parametric Curves: Basics and Vector Fields

2) Parametric Curves

• Single-Variable Vector Functions: Limits, Continuity, Differentiability
• Parametric Curves: Definitions, Examples, Kinematic Interpretation, Length, and Curvature
• Analyzing Parametric Arcs: Local Study, Infinite Branches, Study Plan

3) Geometry and Topology in $\mathbb{R}^n$

• Norms and Distances in $\mathbb{R}^n$: Definitions, Examples, Open/Closed Balls, Equivalent Norms
• Limits of Sequences and Functions
• Continuity: Definitions, Examples, Sequential Characterization, Operations
• Elementary Topology: Open/Closed Sets, Compactness, Arc Connectivity
• Scalar Product and Euclidean Norm: Bilinear and Quadratic Forms, Scalar Product, Euclidean Norm, Quadratic Form Signature

4) Differential Calculus in $\mathbb{R}^n$

• Differentiability: Partial Functions, Partial Derivatives, Differentiability, Gradient Vector, Jacobian Matrix
• $\mathcal{C}^1$ Class Functions: Definitions, Properties, Diffeomorphisms, Implicit Functions
• Higher-Order Derivatives: Definitions, Properties, Taylor’s Formula, Hessian Matrix, Local Extrema Study

5) Multiple Integrals

• Review of Single Integrals: Riemann Integral Construction, Parameter Integrals
• Double Integrals: Over Rectangles, Elementary and Simple Regions, Properties
• Triple Integrals: Over Cuboids, Summation Methods
• Change of Variables Formula: For Double and Triple Integrals
• Vector Field Circulation: Definitions, Properties, Gradient Fields, Green-Riemann Formula