Mat-1.443 Mathematics P3, Passing the Course, Fall '97

Passing the course can be done either by taking three mid-term examinations and doing homework or by taking the final examination. For passing the couse with the mid-term exams, doing all three exams is required. The maximum number of points given from the mid-term exams is 3 x 24 = 72. The number of points required for different grades will be decided after all exams have been marked.

The highest grade (5) can be obtained by doing only the mid-term examinations, but a considerable number of extra points are given for home work.

Mid-term Exam Dates

1st mid-term exam Mon 13 Oct  16-19  Halls ABCDEFGHJKLN
2nd mid-term exam Mon 17 Nov  16-19  Halls ABCDEFGHJKLN
3rd mid-term exam Tue 16 Dec  16-19  Halls ABCDEFGHJKLN

Exam Requirements

• 1st mid-term exam:
  • Kreyszig 7 ed: Ch 7 all except: Cramer's rule. In addition to Kreyszig, Gram-Schmidt-ortogonalzation is included (Greenberg p. 56) (there will be a www-link to an explanation also) The actual course material starts at 7.4, but 7.1-7.3 will be assumed to be known.
  • Alternatively: Greenberg Chapters 1,2,3,4,5 without: 3.6 (Cramer), 3.7 (Change of basis) 4.5 (Jordan form), 4.6 (Appl. do DE)
  • Lectures up to 2.10.
  • Exercises 1-3.
  The mathematical contents of "end-of-the week" exercises are of course also included, Maple or Matlab syntax is not required. (It is possible that output of some Maple commands are given but that should be self-explanatory.)

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• 2nd mid-term exam:
  • Numerical methods: KRE, the following parts of chapter 18, and sections 19.1, 19.4
    • 18.2: Newton's method; also multidimensional Newton's method and fixed point iteration which are not covered by KRE
    • 18.3: (pp. -> 940) Lagrange's method, for Newton's method only the principle using which it is formed, not divided differences
    • 18.4: understanding the definition of the cubic spline and ability to do calculations like those in Example 1 on pp. 952-953 and the excercises that have been dealt with. Derivations of the general form of of splines will not be required.
  • Differential equations and systems
    • Chapter 2 will be assumed known as prerequisites, especially oscillations and electrical circuits may be useful
    • All of chapter 4. In addition, the exponential function for matrices which is presented in Greenberg's book and (in Finnish) in Timo Eirola's paper starting from page 8. If I have time, I will write something about it on the lecture pages (in Finnish language / Maple syntax)
    • Chapter 6 (Laplace transforms), more spefically:
      • Sections 6.1 - 6.4 are required
      • Sections 6.5 - 6.6 superficially
      • Section 6.7 (partial fractions) is not required; if special partial fraction tricks are needed, the forms in question will be given.
      • Section 6.8 (periodic functions) is not required.
  • Excercises 4 - 8.
  Laplace transform tables: This table will be given with the exam paper.

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• 3rd mid-term exam:
  • According to KRE
    • Fourier series: 10.1 - 10.8
    • Partial differential equations: 11.1, 11.5, 11.11
    • Numerical methods for ordinary differential equations: 20.1
      (Instead of section 20.3 will be required the ability to write the algorithms of section 20.1 in vector function form, i.e. for a general system of n equations.)
    • Numerical methods for partial differential equations: 20.4, 20.6
    • Complex analysis: chapter 12
  • Alternatively, according to GREE
    • Fourier series: 10.1 - 10.3, 10.5
    • Partial differential equations: 13, 14.1, 14.2
    • Numerical methods for ordinary differential equations: Not found in GREE, see KRE
    • Numerical methods for partial differential equations: 14.5, 15.5
    • Complex analysis: chapter 18
  • Excercises 9 - 12.
  (Although the convolution theorem was not required for the second mid-term exam, Laplace transforms will not be required in this exam.)

Which formulas have to be remembered?

• Numerical methods for ODE's: Heun, Runge-Kutta
• Fourier coefficients (same formulas as on "Harj. 11")
• The difference formulas for Laplace/Poisson and heat equ (both difference and Crank-Nicholson).

• Numerical methods for ODE's: Euler, Backward Euler, Taylor
• The form of the heat-equation and Laplace/Poisson-equ., the procedure of separation of variables. (It is recommended to recall the procedure, not the final formula.)
• Cauchy-Riemann-equations, definitions of elementary functions (like exp, sin, cos, log, sinh, cosh, ...)