An Introduction to Higher Mathematics 2 Volume Set

The self-taught mathematician Hua Loo-Keng (1910–1985) has been credited with inspiring generations of mathematicians, while his papers on number theory are regarded as some of the most significant contributions made to the subject during the first half of the twentieth century. An Introduction to Higher Mathematics is based on the lectures given by Hua at the University of Science and Technology of China from 1958. The course reflects Huas instinctive technique, using the simplest tools to tackle even the most difficult problems, and contains both pure and applied mathematics, emphasising the interdependent relationships between different branches of the discipline. With hundreds of diagrams, examples and exercises, this is a wide-ranging reference text for university mathematics and a testament to the teaching of one of the most eminent mathematicians of his generation.

Spis treści:
Volume I
1. Real and complex numbers
2. Vector algebra
3. Functions and graphs
4. Limits
5. The differential calculus
6. Applications of the derivative
7. Taylor expansions
8. Approximate solutions to equations
9. Indefinite integrals
10. Definite integrals
11. Applications of integral calculus
12. Functions of several variables
13. Sequences, series and integrals with variables
14. Differential properties of curves
15. Multiple integral
16. Curvilinear integral and surface integral
17. Scalar field and vector field
18. Differential properties of curved surfaces
19. Fourier series
20. System of ordinary differential equations. Volume II
1. Geometry of the complex plane
2. Non-Euclidean geometry
3. Definitions and examples of analytic and harmonic functions
4. Harmonic functions
5. Some basic concepts in point set theory and topology
6. Analytic functions
7. Residues and their application to definite integral
8. Maximum modulus principle and the family of functions
9. Entire function and meromorphic function
10. Conformal transformation
11. Summation
12. Harmonic functions under various boundary conditions
13. Weierstrass elliptic function theory
14. Jacobis elliptic functions
15. Systems of linear equations and determinants (review outline)
16. Equivalence of matrices
17. Functions, sequences and series of square matrices
18. Difference equations with constant coefficients and ordinary differential equations
19. Asymptotic property of solutions
20. Quadratic form
21. Orthogonal groups and pair of quadratic forms
22. Volumes
23. Non-negative square matrices. Volume III
1. The geometry of the complex plane
2. Non-Euclidean geometry
3. Definitions and examples of analytic functions and harmonic functions
4. Harmonic functions
5. Point set theory and preparations for topology
6. Analytic functions
7. The residue and its application to evaluation of definite integrals
8. Maximum modulus theorem and families of functions
9. Integral functions and metamorphic functions
10. Conformal transformations
11. Summability methods
12. Harmonic functions satisfying various types of boundary conditions
13. Weierstrass elliptic function theory
14. Jacobian elliptic function theory. Volume IV
1. Linear systems and determinants (review)
2. Equivalence of matrices
3. Functions, sequences and series of square matrices
4. Difference and differential equations with constant coefficients
5. Asymptotic properties of solutions
6. Quadratic forms
7. Orthogonal groups corresponding to quadratic forms
8. Volumes
9. Non-negative square matrices.