Two Textbooks on Vector Analysis: Gibbs and Runge

The goal of this work is to do a critical analysis and comparative study of two textbooks on vector analysis: One written in 1901 in the United States by Edwin Wilson based on lectures by J. Willard Gibbs, and the other written by Carl Runge in Germany in 1919, and translated into English in 1923. Both were well-known scientists: Gibbs is considered as a father of the modern school of vector analysis, while Runge kept his German Grassmannian roots. First, we present biographical sketches of Gibbs and Runge and describe their scientific styles. Then we comment upon their ideas chapter by chapter, with particular emphasis on vector operators and the integral calculus of vectors. We discuss in detail the Gibbs and Wilson book because it was the first book on vector analysis. We finish with a brief introduction to tensors from Runge’s first volume.