Mathematics 2E1 (for SF Engineering & MSISS students)
Notes
Some (not all) parts of the course notes will be in the form of a handout or will be available here.
- Chapter 1: Vector valued functions
- These notes are in PDF format and require a programme such as Adobe Acrobat Reader to read them. They deal with vector valued functions of a single real variable (parametric curves in n dimensional space), tangent vectors, chain rule and product rules, unit tangent vectors, velocity, accerlation, arclength or unut speed paramatrisation, curvature, tangential and normal components of acceleration. We also mention the paraemtric form of the equations of a line, dot product, cross product.
- Chapter 2: Functions of several variables
- This chapter deals with calculus of (scalar valued) functions of two and three variable. We start with some information on planes and lines in space, as these don't seem to have been covered elsewhere in any great detail. We deal with graphs, partial derivatives, directional derivatives, linear approximation, the (total) derivative, tangent planes to graphs of functions of two variables, the chain rule, gradient vectors and properties of the gradient. Then we consider three variable versions of most of these. Finally we look at maxima and minima for functions of two or three variables, critical points, Lagrange multipliers and Taylor's theorem for the remainder in linear approximation in two variables.
- Chapter 3: Double and triple integrals
- (This chapter is not proof read yet.) Definitions of double and triple integrals, how to work them out using Fubini's theorem. Polar coordinates, cylindrical coordinates and spherical coordinates. Change of variables in multiple integrals (Jacobian factor). Applications to areas, volumes, mass, centre of mass, average of a functio, inertia matrix.