Mathematics MA1131, 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, September 30
- Introductory remarks, some notation, functions, beginning of limits.
- Lecture 2, October 1
- Limits of polynomials and rational functions. Definition of derivative.
- Lecture 3, October 7
- Interpretations of the derivative; rules for differentiation.
- Lecture 4, October 8
- Implicit differentiation; rational powers; trigonometric functions; sign of the derivative and monotone functions.
- Lecture 5, October 14
- Exponential function.
- Lecture 6, October 15
- Inverse functions, the natural logarithm; arbitrary powers of positive numbers; Inverse trigonometric functions; hyperbolic functions.
- Lecture 7, October 21
- Inverse hyperbolic functions.
- Lecture 8, October 22
- 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 9, October 28
- Parametric curves. Graphs of functions of two variables.
- Lecture 10, October 29
- Partial derivatives. Directional derivatives. Tangent plane to a graph. Linear approximation for functions of two variables, (total) derivatives.
- Lecture 11, November 18
- Gradient vector, its relation to directional derivatives, a chain rule for partial derivatives, level curves and the fact that the gradient is perpendicular to them.
- Lecture 12, November 19
- Statement of the implicit function theorem and relation to implicit differentiation. Functions of 3 variables, level surfaces, partials, directional derivatives, gradient vector, gradient perpendicular to level surface, gradient and direction of fastest inrease.
- Lecture 13, November 25
- Integrals in one variable, both as antiderivatives and as definite integrals (defined by a limit of sums).
- Lecture 15, December 2 & 3
- More on techniques of integration (integration by parts, trigonometric integrals, more examples of substitutions, and partial fractions).
- Lecture 16, December 9, 10 & 16
- Double and triple integrals. How to compute them (Fubini theorem) via iterated single integrals. Double integrals in polar coordinates.