Course Outline / Syllabus

40 lectures plus 6 problems classes.

• Elementary Functions (mainly revision)
  Manipulation of algebraic equations, powers, exponentials and logarithms, inverse functions, trigonometric functions, sine, cosine and tangent for special angles, hyperbolic functions.
• Vectors
  Definition, addition, subtraction, scalar and vector multiplication. Vector and scalar triple products, vector equations (Third order determinants only very briefly).
• Differentiation (mainly revision) Definition, product rule, function of a function rule, implicit functions, logarithmic derivative, parametric differentiation, maxima and minima.
• Integration (mainly revision)
  Integration as converse of differentiation, changing variables, integration by parts, partial fractions, trigonometric and other substitutions, definite integral, integral as the area under a curve, trapezium rule, integral of odd and even functions. The Gaussian, Feynman, Breit Wigner (Lorentzian) integrals. Further definite integrals obtained by differentiation w. r. t. a parameter.
• Partial Differentiation
  definition, surface representation of functions of two variables, total differentials, chain rule, change of variables, second order derivatives. Maxima, minima and saddle points for functions of two variables. Stationary values of functions subject to constraints.
• More Vectors
  Vector geometry - straight lines and planes. Vector differentiation, vectors in plane polar, cylindrical, and spherical polar coordinates.
• Series
  Summation of arithmetic, geometric and other simple series. Sequences and series, convergence of infinite series. Power series, radius of convergence, simple examples including the binomial series. Taylor and Maclaurin series, L'Hopital's rule.
• Complex Numbers
  Representation, addition, subtraction, multiplication, division, Cartesian, polar exponential forms, De Moivre's theorem, powers and roots, complex equations.

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