Vector Calculus Overview

Author

Sandro Vitenti

Overview

This module introduces the mathematical framework of vector calculus, essential for understanding physics in three or more dimensions. It covers fundamental concepts, operations, and applications frequently used across physics courses, especially electromagnetism and mechanics.

Contents

  • Vector Spaces: Definitions, linear independence, and the structure of spaces where vectors live.
  • Inner Product: Properties of the dot product, including the Kronecker delta notation.
  • Vector Product: Cross product, exterior product, and their properties.
  • Triple Products: Relations involving scalar and vector triple products.
  • Euclidean Space: Cartesian coordinates and tangent vectors.
  • Differential Calculus: Gradient, divergence, curl, and higher-order derivatives.
  • Integral Calculus: Line, surface, and volume integrals.
  • Curvilinear Coordinates: Basis vectors, differential operators, and orthogonal coordinate systems.
  • Dirac Delta: Introduction and properties of the delta distribution.
  • Vector Field Decomposition: Techniques to break down vector fields into simpler components.

This course aims to build a solid foundation for applying these mathematical tools throughout your physics studies.

Feel free to create issues, ask questions, or suggest improvements in the GitHub repository.