In this course we shall be concerned only with three dimensional but the basic definitions of points, difference vectors and distances are the same for all with of course, in dimensions higher than 3, the extra directions will arise from other features than ordinary space--such as time, temperature, pressure etc. The important fact to hang onto is that consists of points represented by coordinates p=(p1,p2,p3) while the directed difference between a pair of such points p,q is a vector with components (q1-p1,q2-p2,q3-p3). In modern mathematics, it is customary to omit the overbar when writing vectors and this will be our usual practice; we identify vectors with their sets of components and points with their sets of coordinates.
The space has one particularly important feature: the availability of the vector cross product on which simplifies many geometrical proofs.
Our main interest in this course is to develop the geometry of curves and surfaces in The basic ideas are very simple: a curve is a continuous image of an interval and a surface is a continuous image of a product of intervals; in each case the intervals may be open or closed or neither.Difference vectors and distances
The difference map gives the vector arrow from one point to
another and is defined by
The standard unit sphere
in a Euclidean n-space is
the set of points unit distance from the origin; we shall often
A parametric equation for the unit 2-sphere
is given by