Mathematics MA1132, Advanced Calculus
Notes
Some (not all) parts of the course notes will be in the form of a handout or will be available here. All will be in PDF format and require a programme such as Adobe Acrobat Reader to read them.
- Lecture 1, January 16
- Limits of sequences.
- Lecture 2, January 20
- More on limits of sequences: link to continuity; monotone sequences.
- Lecture 3, January 23
- Series (of positive terms), examples, comparison test (version 1).
- Lecture 4, January 27
- Examples of use of the comparison test. Integral test. Absolute convergence (implies convergence).
- Lecture 5, January 30
- Comparison test (more general version). Alternating series test, ration test, examples. Power series.
- Lecture 6, February 3
- Differentiation and integration of power series. Example series of the expoential function. use of integrating factors to solve linear first order differential equations (constant coefficent case). Geometric series as a further example. Taylor polynomials.
- Lecture 7, February 6
- Starting towards functions of several variables. (First some recap on coordinates and vectors in space; planes and lines; dot products; cross products. Assumed known from linear algebra!) Vector valued functions of one variable (parametric curves).
- Lecture 8, February 10
- Parametric curves. Graphs of functions of two variables.
- Lecture 9, February 13
- Partial derivatives. Directional derivatives.
- Lecture 10, February 17
- Tangent plane to a graph. Linear approximation for functions of two variables, (total) derivatives.
- Lecture 11, March 5
- Gradient vector, its relation to directional derivatives; level curves (for functions of two variables).
- Lecture 12, March 9
- Gradient vector perpendicular to level curve; idea of the implicit function theorem.
- Lecture 13, March 12
- Functions of 3 variables.
- Lecture 14, March 16, 23 & 26
- Double integrals.
- Lecture 15, March 26, 30 & April 6
- Triple integrals. Change of variables in multiple integrals.