Bibliographic summary
A concise academic handbook intended for university students studying physics, as well as readers interested in the mathematical apparatus of general relativity.
Tensorial Differential Operations in Curvilinear Coordinates
A focused introduction to the geometric foundations of tensor analysis, developed from familiar three-dimensional Euclidean geometry toward Riemannian geometry and general relativity.
This page is designed as an academic entry point for students, instructors, and readers interested in covariant and contravariant vectors, Christoffel symbols, covariant differentiation, and vector calculus in curvilinear coordinates.
Supplementary information
Abstract
The handbook clarifies the origins of fundamental notions of Riemannian geometry by introducing arbitrary curvilinear coordinates in conventional three-dimensional Euclidean space. It explains the appearance of contravariant and covariant vectors, the genesis of Christoffel symbols, and the role of covariant differentiation, while also deriving the differential operations of vector analysis in curvilinear coordinates, especially in orthogonal, spherical, and cylindrical coordinates.
Who this page is for
- University students in physics and applied mathematics
- Readers studying tensor analysis and differential geometry
- Those seeking a bridge from Euclidean vector calculus to general relativity
- Readers looking for formulas and derivations in curvilinear and orthogonal coordinates
Core topics
- Curvilinear coordinates in Euclidean space
- Covariant and contravariant vectors
- Metric tensor
- Christoffel symbols
- Covariant differentiation
- Vector differential operations in orthogonal coordinates
- Spherical and cylindrical coordinates
- Conceptual transition to Riemannian geometry
Why this work is distinctive
Conceptual approach
The exposition begins from familiar geometry and shows how tensor structures emerge naturally from the use of curvilinear coordinates, rather than introducing them as purely formal abstractions.
Reference value
In addition to conceptual clarification, the handbook serves as a compact reference for vector differential operations and related formulas in important coordinate systems.