Week 2.
Lecture 3 on complex numbers.
Motivation/need for them. Definition. Arithmetic operations. Geometric interpretation. Polar/exponential form.
Week 2.
Lecture 4 on complex numbers.
de Moivre's theorem. Euler's formula and identity. Roots of complex numbers. Further identities.
Week 3.
Lecture 5 on differentiation.
Definition and notations. Use of limits. Higher derivatives. Linearity. Product rule. Product of more than two functions.
Week 3.
Lecture 6 on differentiation.
Chain rule and proof. Functions of functions of functions etc.. Advanced cases. Quotient rule.
Week 4.
Lecture 7 on differentiation.
Critical points. Primary and secondary criteria. Checklist.
Week 4.
Lecture 8 on differentiation.
Definition and examples. Clairaut's theorem. Critical points of surfaces. Don't forget the definition of H and the criteria for
classification!
Week 5.
Lecture 9 on integration.
From sums to integrals. Definite and indefinite. Integration by substitution - various cases.
f ′/f form.
Week 5.
Lecture 10 on integration.
Ratios of polynomials and partial fractions. Top heavy ratios. Repeated factors. Irreducible quadratics.
Week 6.
Lecture 11 on integration.
Integration by parts. Derivation of the Rees method - rules of implementation. Miscellaneous cases.
Week 6.
Week 7.
Lecture 12 on integration.
Applications 1. Means and RMS. Recurrence relations. Volumes under surfaces in Cartesians and in polar coordinates.
Week 7.
Lecture 13 on integration.
Applications 2. Length of a line. Volumes and surfaces of revolution. Triple integral for mass given the density profile.
Week 8.
Lecture 14 on series.
Difference between series and sequences. Binomial theorem and Pascal's triangle. Why does 0!=1? Binomial series and examples.
Week 8.
Lecture 15 on series.
Taylor's series and Maclaurin. Derivation/justification. Two forms of Taylor's series. Examples.
Week 9.
Lecture 16 on series.
Convergence and d'Alembert's test. Examples of numerical and of power series. Radius of convergence. l'Hôpital's rule. Derivations. Examples.
Week 9.
Week 10.
Lecture 17 on vectors.
Revision: definition and elementary concepts. Unit vectors. Scalar product. Angle between two vectors.
Week 10.
Lecture 18 on vectors.
Vector product. Lexicographical ordering! Area of a triangle using vector products. Coplanarity of three vectors or four points.
Week 11.
Lecture 19 on vectors.
Equation of a line. Closest approach of a line to a given point. Closest approach of two lines to one another.
Week 11.
Lecture 20 on vectors.
Distance of a point from a plane. Equation of a line - two forms. The direction of the intersection
of two planes.