
Math is Exciting. We are living in the greatest age of mathematics ever
seen. In the 1930s, there were some people who feared that the rising
abstractions of the early twentieth century would either lead to mathematicians
working on sterile, silly intellectual exercises or to mathematics
splitting into sharply distinct subdisciplines, similar to the way natural
philosophy split into physics, chemistry, biology and geology. But the very
opposite has happened. Since World War II, it has become increasingly
clear that mathematics is one unified discipline. What were separate areas
now feed off of each other. Learning and creating mathematics is indeed a
worthwhile way to spend one's life..
Few beginning graduate students in mathematics and other quantitative subjects possess the daunting breadth of mathematical knowledge expected of them when they begin their studies. This book will offer students a broad outline of essential mathematics and will help to fill in the gaps in their knowledge. The author explains the basic points and a few key results of all the most important undergraduate topics in mathematics, emphasizing the intuitions behind the subject. The topics include linear algebra, vector calculus, differential and analytical geometry, real analysis, pointset topology, probability, complex analysis, set theory, algorithms, and more. An annotated bibliography offers a guide to further reading and to more rigorous foundations. 


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