Calculus 2 for Informaticians

Chapter 1: Ordinary differential equations (2.5 weeks)

Week 1: Basic concepts. Separable differential equations.

Week 2: First order linear differential equations. Changing variables. Slope field, isocline.

Week 3: Higher order linear differential equations. External and internal resonance.

Chapter 2: Linear recursion (0.5 week)

Week 3: Fibonacci sequence. Fibonacci-type sequences.

Chapter 3: Numerical end function series (5 weeks)

3.1 Numerical series (2 weeks)

Week 4: Sum of a series. Examples: geometric series, telescopic series, harmonic series. Arithmetic rules. Alternating series.

Week 5: Absolute and conditional convergence. Convergence tests: comparison, ratio, root and integral tests.

3.2 Basic properties of function series (1 week)

Week 6: Domain of convergence, sum of a function series, examples. Uniform and absolute convergence. Weierstrass' criterion. Sufficient condition for the continuity of the sum, for the termwise integrability and differentiability of the series.

3.3 Power series (2 weeks)

Week 7: Radius of convergence. Cauchy-Hadamard formula. Taylor polynomial.

Week 8: Taylor series. Taylor expansion of common functions. Binomial series.

Chapter 4: Multivariable functions (3.5 weeks)

4.1 Limit, continuity (0.5 week)

Week 9: Visualizing multivariable functions. Limit and continuity of multivariable functions.

4.2 Differentiation (1.5 weeks)

Week 9: Partial derivatives, total derivative (gradient), tangent plane, directional derivative.

Week 10: Young's theorem. Local extrema.

4.3 Integration (1.5 weeks)

Week 11: Double and iterated integrals over rectangular, type I and type II regions.

Week 12: Integral transformation. Planar polar, cylindrical and spherical polar coordinates.

Chapter 5: Fourier analysis (1.5 weeks)

Week 12: The trigonometric system. Fourier series. Examples.

Week 13: Fourier transformation. (Definition, properties, examples.)