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This text covers all the major concepts and techniques in an elementary and clear style and explores some fields usually dealt with only in the advanced literature, also in a simplified manner. This edition includes new discussions on the three basic numerical methods, an introduction to computational fluid dynamics, and an introduction to Lie group methods for partial differential equations. All topics are treated as brief descriptions of needed results, not as condensed short courses. The text's practical approach will motivate students to pursue many areas of study, such as Fourier series, complex analysis, Lebesgue integration and functional analysis.