18) The unit vectors 00, 00, 00 of cylindrical
coordinate syste
O a) Are constant vectors
O b) are not constant vectors
O c) are constant in time only
O d) none of the above
Answers
Answer:
The Cartesian coordinate system provides a straightforward way to describe the location of points in space. Some surfaces, however, can be difficult to model with equations based on the Cartesian system. This is a familiar problem; recall that in two dimensions, polar coordinates often provide a useful alternative system for describing the location of a point in the plane, particularly in cases involving circles. In this section, we look at two different ways of describing the location of points in space, both of them based on extensions of polar coordinates. As the name suggests, cylindrical coordinates are useful for dealing with problems involving cylinders, such as calculating the volume of a round water tank or the amount of oil flowing through a pipe. Similarly, spherical coordinates are useful for dealing with problems involving spheres, such as finding the volume of domed structures.
Cylindrical Coordinates
When we expanded the traditional Cartesian coordinate system from two dimensions to three, we simply added a new axis to model the third dimension. Starting with polar coordinates, we can follow this same process to create a new three-dimensional coordinate system, called the cylindrical coordinate system. In this way, cylindrical coordinates provide a natural extension of polar coordinates to three dimensions.
Definition
In the cylindrical coordinate system, a point in space ((Figure)) is represented by the ordered triple \left(r,\theta ,z\right), where
\left(r,\theta \right) are the polar coordinates of the point’s projection in the xy-plane
z is the usual z\text{-coordinate} in the Cartesian coordinate system
The right triangle lies in the xy-plane. The length of the hypotenuse is r and \theta is the measure of the angle formed by the positive x-axis and the hypotenuse. The
z-coordinate describes the location of the point above or below the xy-plane.
This figure is the first octant of the 3-dimensional coordinate system. There is a point labeled “(x, y, z) = (r, theta, z).” In the x y-plane, there is a line segment extending to underneath the point. This line segment is labeled “r.” The angle between the line segment and the x-axis is theta. There is a line segment perpendicular to the x-axis. Along with the line segment labeled r, this line segment and the x-axis form a right triangle.
In the xy-plane, the right triangle shown in (Figure) provides the key to transformation between cylindrical and Cartesian, or rectangular, coordinates.
Conversion between Cylindrical and Cartesian Coordinates
The rectangular coordinates \left(x,y,z\right) and the cylindrical coordinates \left(r,\theta ,z\right) of a point are related as follows:
\begin{array}{cccccc}\hfill x& \hfill =\hfill & r\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta \hfill & & & \text{These equations are used to convert from}\hfill \\ \hfill y& \hfill =\hfill & r\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta \hfill & & & \text{cylindrical coordinates to rectangular}\hfill \\ \hfill z& \hfill =\hfill & z\hfill & & & \text{coordinates.}\hfill \\ & \hfill \text{and}\hfill & & & & \\ \hfill {r}^{2}& \hfill =\hfill & {x}^{2}+{y}^{2}\hfill & & & \text{These equations are used to convert from}\hfill \\ \hfill \text{tan}\phantom{\rule{0.2em}{0ex}}\theta & \hfill =\hfill & \frac{y}{x}\hfill & & & \text{rectangular coordinates to cylindrical}\hfill \\ \hfill z& \hfill =\hfill & z\hfill & & & \text{coordinates.}\hfill \end{array}