Advanced Mathematics (Revised Edition)

Author: Bai Hong, Liu Rui / Country: Mainland China
Publisher:
Publish Date: 2000-09-01
Features: To determine the position of any point on a plane, a Cartesian coordinate system was once used. Now, to determine the position of a point in space, a spatial Cartesian coordinate system must be introduced. From a fixed point O in space, three mutually perpendicular number axes—Ox, Oy, and Oz—are drawn, called coordinate axes. The point of intersection of these three lines is called the origin. By convention, the x-axis and y-axis are arranged in the horizontal plane, while the z-axis is a vertical line. It is also specified that the positive direction of the x-axis is forward, the positive direction of the y-axis is from left to right, and the positive direction of the z-axis is upward (Figure 9.1.1).
Let M be a known point in space. The projections of M onto the coordinate axes are made by drawing three planes perpendicular to the x-axis, y-axis, and z-axis, respectively. Let the points of intersection with the x-axis, y-axis, and z-axis be P, Q, and R, respectively. Then, P, Q, and R are the projections of point M onto the three coordinate axes, respectively. Conversely, if three points P, Q, and R are given on the axes, a unique point M in space can be determined, with P, Q, and R as its projections onto the axes. Thus, the determination of the position of point M is reduced to the determination of the positions of its projections P, Q, and R on the axes. However, the positions of the three points P, Q, and R on the axes are determined by the values of the directed line segments OP, OQ, and OR, respectively. If these three values are denoted by x, y, and z, respectively, then:
\[ x = \text{OP}, \quad y = \text{OQ}, \quad z = \text{PR} \]
Thus, there is a one-to-one correspondence between the point M in space and the ordered triple of numbers (x, y, z). Such an ordered triple (x, y, z) is called the coordinates of point M, where x is the abscissa, y is the ordinate, and z is the vertical coordinate. The notation M(x, y, z) represents the point M with abscissa x, ordinate y, and vertical coordinate z.
When studying certain practical problems, it is first necessary to establish a differential equation and then find a function that satisfies the differential equation.
Definition 2: A function that satisfies a differential equation is one that, when substituted along with its derivatives into the differential equation, makes the equation an identity. Such a function is called a solution of the differential equation. For an nth-order differential equation, a solution that contains n independent arbitrary constants is called the general solution of the equation. A solution in which the arbitrary constants in the general solution are assigned specific values is called a particular solution. The condition that specifies the values of the unknown function and its derivatives when the independent variable takes a certain value is called an initial condition (or an auxiliary condition). (P164)