Tentative Course Outline

Textbook: Patrick M. Fitzpatrick, Advanced Calculus (sec. ed.), AMS, 2006.

• 1. Thie field, the positivity and the completenes axioms. Principle of mathematical induction. Ineuqlities and identites (sec. 11 -1.3)
  
• 2. Convergence of sequences. Sequences and sets.(sec. 2.1, 2.2)
  
• 3. The monotone convergence and sequential compactness theorems. Continuity. (sec.2.3, 2.4, 3.1)
  
• 4. The extreme value and intermediate value theorems. Uniform continuity. (sec.3.2, 3.3, 3.4, 3.5)
  
• 5. Monotone functions. Limits. Tangent line and derivatives. (sec.3.6, 3.7, 4.1)
  
• 6. Derivative of the inverse function and composition of two functions. The mean value theorem for derivative and consequences. Extreme values. The Cauchy mean value theorem (sec. 4.2, 4.3, 4.4)
• 7. Taaylor’s formula. Darboux Teorem. Sequences and series of numbers (sec.9,1 )
• 8. Sequences and series of functions: Pointwise convergence (sec 9.1, 9.2)
• 9. Uniform convegence. Weierstrass M-test. The linear structure of Rn. Cauchy – Schwarz inequality (sec. 9.3, 9.4, 10.1)
• 10. Triangle inequality. Convergence of sequences in Rn. Open and closed sets. (sec. 10.1, 10.2, 10.3)
• 11. Continuous functiona and mappings. Sequential compactness. Exteme vlaues. Connected sets (sec. 11.1, 11.2, 11.3)
• 12. Limits. Partial derivatives and extreme value theorem for functions of two variables (13.1,13.2 and Lectures 19, 20)
• 13. The mean value theorem and directional derivative. Local approxmation of functions. (13.3, 14.1)
• 14. Quadratic forms, Hessan. Extreme values,second-derivative test ( 14.2, 14.3)