Page 17 - Standard Handbook Of Petroleum & Natural Gas Engineering

6 Mathematics

rectangular parallelpiped is a right rectangular prism. A cube is a rectangular
parallelpiped all of whose edges are congruent. A triangular pyramid has a
triangular base, etc. A circular cylinder is a prism whose base is a circle and a
circular cone is a pyramid whose base is a circle.
Coordinate Systems

Each point on a plane may be defined by a pair of numbers. The coordinate
system is represented by a line X in the plane (the x-axis) and by a line Y (the
y-axis) perpendicular to line X in the plane, constructed so that their intersection,
the origin, is denoted by zero. Any point P on the plane can now be described
by its two coordinates which form an ordered pair, so that P(x,,y,) is a point
whose location corresponds to the real numbers x and y on the x-axis and
the y-axis.
If the coordinate system is extended into space, a third axis, the z-axis,
perpendicular to the plane of the xI and y, axes, is needed to represent the
third dimension coordinate defining a point P(x,,y,,z,). The z-axis intersects the
x and y axes at their origin, zero. More than three dimensions are frequently
dealt with mathematically, but are difficult to visualize.
The slope m of a line segment in a plane with end points P,(x,,y,) and P,(x,,y,)
is determined by the ratio of the change in the vertical (y) coordinates to the
change in the horizontal (x) coordinates or

m = (Y' - YI)/(X2 - XI)
except that a vertical line segment (the change in x coordinates equal to zero)
has no slope, i.e., m is undefined. A horizontal segment has a slope of zero.
Two lines with the same slope are parallel and two lines whose slopes are
negative reciprocals are perpendicular to each other.
Since the distance between two points P,(x,,y,) and P,(x,,y,) is the hypotenuse
of a right triangle, the length of the line segment PIP, is equal to

Graphs

A graph is a figure, i.e., a set of points, lying in a coordinate system and a
graph of a condition (such as x = y + 2) is the set of all points that satisfy the
condition. The graph of the slope-intercept equation, y = mx + b, is the line which
passes through the point (O,b), where b is the y-intercept (x = 0) and m is the
slope. The graph of the equation
(x - a)' + (y - b)' = r2
is a circle with center (a,b) and radius r.

Vectors
A vector is described on a coordinate plane by a directed segment from its initial
point to its terminal point. The directed segment represents the fact that every
vector determines not only a magnitude, but also a direction. A vector v is not