Multivariable Calculus

Multivariable Calculus Euclid 1.1 Rectangular Coordinates in R3 Throughout the course, we will use an ordered triple (x, y, z) to represent a point in the three dimensional space. The real numbers x, y and z in an ordered triple (x, y, z) are respectively the x-, y- and z-coordinates which, by convention, are defined according to the following diagram: z y x Figure 1.1: Rectangular coordinates system in 3-space Any point on the x-axis has the form (x, 0, 0), i.e. y = 0 and z = 0. Similarly, points on the y-axis are of the form (0, y, 0), and points on the z-axis are of the form (0, 0, z). The three coordinate axes meet at a point with coordinates (0, 0, 0) which is called the origin. A vector in R3 is an arrow which is based at one point and is pointing at another point. If a vector v is based at (x0, y0, z0) and points toward (x1, y1, z1), then the vector is written as: v = (x1 − x0)i + (y1 − y0)j + (z1 − z0)k. For example, the vector based at (3, 2, −1) pointing at (5, 2, 0) is expressed as (5 − 3)i + (2 − 2)j + (0 − (−1))k = 2i + k.