Vector Analysis

Abstract

This chapter is dedicated to the extension of the ordinary differential calculus to general curvilinear coordinates in a Euclidean point space. After defining the metric in general coordinates, the Christoffel symbols are introduced to define the covariant derivative of any tensor field in general coordinates. All these concepts are applied to derive the expressions of the differential operators (gradient, divergence, curl, and Laplacian) in any system of coordinates. Then, the above results are analyzed in coordinates that are often used in the applications. Gauss’ and Stokes’ theorems are extended to vector fields that exhibit a jump in their derivatives across a surface (singular surface). The fundamental Hadamard’s theorem is proved for this singular surfaces. The last sections contain exercises and the presentation of the program Operator, written with Mathematica^® [69], which allows the user to obtain by computer all the results exposed in this chapter.