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This text introduces an approach for mastering the concepts of vectors and vector analysis which not only brings together many loose ends of the traditional treatment, but also leads directly into the practical use of vectors in general curvilinear co-ordinates. It separates those relationships which are topologically invariant from those which are not. Based on the essentially geometric nature of the subject, this approach builds consistently on students' prior knowledge and geometrical intuition. Written in an informal and personal style, this text provides a guide for any student of vector analysis. Clear line drawings illustrate key points in the text, and problem sets as well as physical examples are provided.