Practical guide to coordinate transformations

This practical and comprehensive guide is designed for students, engineers, and scientists who wish to deepen their understanding of coordinate transformations in Euclidean space. With a clear focus on curvilinear coordinate systems and tensor calculus, the book builds from foundational concepts to advanced mathematical tools essential for theoretical and applied physics. Key topics include the Einstein Summation Convention, vector calculus in index notation, the role of the metric tensor and Christoffel symbols, and the transition from Cartesian to curvilinear coordinates. Readers are introduced to differential operators (gradient, divergence, curl, Laplacian) and their generalizations, as well as the principles of tensor algebra and calculus. The guide features: Over 60 fully solved exercises and numerous worked examples. A structured introduction to covariant/contravariant vectors and dual bases. Detailed treatment of the Jacobi matrix, tensor operations, and coordinate transformations. Applications in integration using polar, cylindrical, and spherical coordinates. A reference section listing more than 15 coordinate systems with their metric properties and transformation tools. Supplementary SageMath notebooks are available for download, providing interactive support for key calculations and visualizations. Written in a clear and pedagogical style, this volume lays the groundwork for further study in differential geometry and General Relativity, to be explored in Part II.