Some Curvilinear Coordinate Systems 若干曲线坐标系

本页译自 LibreTexts · A First Course in Linear Algebra (Kuttler) 第 8 章「Some Curvilinear Coordinate Systems」（极坐标、柱面坐标与球面坐标）。公式经 MathJax 渲染，自定义宏已注入；正文与译文段段对照。

8.1 Polar Coordinates and Polar Graphs 极坐标与极图

Outcomes.

1. Understand polar coordinates.
2. Convert points between Cartesian and polar coordinates.

1. 理解极坐标。
2. 在直角坐标与极坐标之间转换点。

You have likely encountered the Cartesian coordinate system in many aspects of mathematics. There is an alternative way to represent points in space, called polar coordinates. The idea is suggested in the following picture.

Consider the point above, which would be specified as $(x,y)$ in Cartesian coordinates. We can also specify this point using polar coordinates, which we write as $\left( r, \theta \right)$. The number $r$ is the distance from the origin $\left( 0,0\right)$ to the point, while $\theta$ is the angle shown between the positive $x$ axis and the line from the origin to the point. In this way, the point can be specified in polar coordinates as $\left( r, \theta \right)$.

Now suppose we are given an ordered pair $\left( r,\theta \right)$ where $r$ and $\theta$ are real numbers. We want to determine the point specified by this ordered pair. We can use $\theta$ to identify a ray from the origin as follows. Let the ray pass from $\left( 0,0\right)$ through the point $\left( \cos \theta ,\sin \theta \right)$ as shown.

The ray is identified on the graph as the line from the origin, through the point $\left(\mbox{cos}(\theta),\mbox{sin}(\theta)\right)$. Now if $r>0,$ go a distance equal to $r$ in the direction of the displayed arrow starting at $\left(0,0\right)$. If $r<0,$ move in the opposite direction a distance of $\left\vert r\right\vert$. This is the point determined by $\left( r,\theta \right)$.

It is common to assume that $\theta$ is in the interval $[0,2\pi )$ and $r>0.$ In this case, there is a very simple relationship between the Cartesian and polar coordinates, given by $x=r\cos \left( \theta \right) ,$ $y=r\sin \left( \theta \right)$.

These equations demonstrate how to find the Cartesian coordinates when we are given the polar coordinates of a point. They can also be used to find the polar coordinates when we know $\left( x, y \right)$. A simpler way to do this is the following equations:

In the next example, we look at how to find the Cartesian coordinates of a point specified by polar coordinates.

Example 1: Finding Cartesian Coordinates 例 1：求直角坐标

The polar coordinates of a point in the plane are $\left( 5,\pi /6\right)$. Find the Cartesian coordinates of this point.

Solution. The point is specified by the polar coordinates $\left( 5,\pi /6\right)$. Therefore $r=5$ and $\theta = \pi /6$. From the above equations $x= r \cos \left( \theta \right)= 5\cos \left( \frac{\pi }{6}\right) = \frac{5}{2}\sqrt{3} $ $y= r \sin \left( \theta \right) = 5\sin \left( \frac{\pi }{6}\right) = \frac{5}{2} $ Thus the Cartesian coordinates are $\left( \frac{5}{2}\sqrt{3}, \frac{5}{2}\right)$. The point is shown in the below graph.

Consider the following example of the case where $r < 0$.

Example 2: Finding Cartesian Coordinates 例 2：求直角坐标

The polar coordinates of a point in the plane are $\left( -5,\pi /6\right) .$ Find the Cartesian coordinates.

Solution. For the point specified by the polar coordinates $\left( -5, \pi /6 \right)$, $r=-5$, and $\theta = \pi /6$. From the above equations $x= r \cos \left( \theta \right)= -5\cos \left( \frac{\pi }{6}\right) = -\frac{5}{2}\sqrt{3} $ $y= r \sin \left( \theta \right) = -5\sin \left( \frac{\pi }{6}\right) = -\frac{5}{2} $ Thus the Cartesian coordinates are $\left( -\frac{5}{2}\sqrt{3}, -\frac{5}{2}\right)$. The point is shown in the following graph.

Recall from the previous example that for the point specified by $\left( 5, \pi /6 \right)$, the Cartesian coordinates are $\left( \frac{5}{2}\sqrt{3}, \frac{5}{2}\right)$. Notice that in this example, by multiplying $r$ by $-1$, the resulting Cartesian coordinates are also multiplied by $-1$.

The following picture exhibits both points in the above two examples to emphasize how they are just on opposite sides of $\left( 0,0\right)$ but at the same distance from $\left( 0,0\right)$.

In the next two examples, we look at how to convert Cartesian coordinates to polar coordinates.

Example 3: Finding Polar Coordinates 例 3：求极坐标

Suppose the Cartesian coordinates of a point are $\left( 3,4\right)$. Find a pair of polar coordinates which correspond to this point.

Solution. Using the above equations, we can find $r$ and $\theta$. Hence $r=\sqrt{3^{2}+4^{2}}=5$. It remains to identify the angle $\theta$ between the positive $x$ axis and the line from the origin to the point. Since both the $x$ and $y$ values are positive, the point is in the first quadrant. Therefore, $\theta$ is between $0$ and $\pi /2$. Using this and the above equations, we have to solve: $\mbox{tan}\left(\theta \right)=\frac{4}{3}$. Conversely, we can use the above equations as follows: $3=5\cos \left( \theta \right) $ $4 = 5\sin \left( \theta \right) $. Solving these equations, we find that, approximately, $\theta = 0.927$ radians.

Example 4: Finding Polar Coordinates 例 4：求极坐标

Suppose the Cartesian coordinates of a point are $\left( -\sqrt{3},1\right)$. Find the polar coordinates which correspond to this point.

Solution. Given the point $\left( -\sqrt{3}, 1\right)$, $$ \begin{aligned} r &= \sqrt{ 1^2 + (-\sqrt{3})^2}\\ &= \sqrt{1 + 3}\\ &=2\end{aligned} $$ In this case, the point is in the second quadrant since the $x$ value is negative and the $y$ value is positive. Therefore, $\theta$ will be between $\pi/2$ and $\pi$. Solving the equations $-\sqrt{3}= 2 \cos \left(\theta\right) $ $1 = 2 \sin \left( \theta\right) $ we find that $\theta = 5\pi /6.$ Hence the polar coordinates for this point are $\left(2, 5\pi /6 \right)$.

Consider this example. Suppose we used $r=-2$ and $\theta =2\pi -\left( \pi /6\right) = 11\pi /6$. These coordinates specify the same point as above. Observe that there are infinitely many ways to identify this particular point with polar coordinates. In fact, every point can be represented with polar coordinates in infinitely many ways. Because of this, it will usually be the case that $\theta$ is confined to lie in some interval of length $2\pi$ and $r>0$, for real numbers $r$ and $\theta$.

Just as with Cartesian coordinates, it is possible to use relations between the polar coordinates to specify points in the plane. The process of sketching the graphs of these relations is very similar to that used to sketch graphs of functions in Cartesian coordinates. Consider a relation between polar coordinates of the form, $r=f\left( \theta \right)$. To graph such a relation, first make a table of the form

Graph the resulting points and connect them with a curve. The following picture illustrates how to begin this process.

To find the point in the plane corresponding to the ordered pair $\left( f\left( \theta \right) ,\theta \right)$, we follow the same process as when finding the point corresponding to $\left( r, \theta \right)$.

Consider the following example of this procedure, incorporating computer software.

Example 5: Graphing a Polar Equation 例 5：绘制极坐标方程图像

Graph the polar equation $r=1+\cos \theta$.

Solution. We will use the computer software Maple to complete this example. The command which produces the polar graph of the above equation is plot(1+cos(t), t=0..2*Pi, coords=polar). Here we use $t$ to represent the variable $\theta$ for convenience. The command tells Maple that $r$ is given by $1+\cos \left( t\right)$ and that $t\in \left[ 0,2\pi \right]$.

The above graph makes sense when considered in terms of trigonometric functions. Suppose $\theta =0,r=2$ and let $\theta$ increase to $\pi /2$. As $\theta$ increases, $\cos \theta$ decreases to 0. Thus the line from the origin to the point on the curve should get shorter as $\theta$ goes from $0$ to $\pi /2$. As $\theta$ goes from $\pi /2$ to $\pi$, $\cos \theta$ decreases, eventually equaling $-1$ at $\theta =\pi$. Thus $r=0$ at this point. This scenario is depicted in the above graph, which shows a function called a cardioid.

The following picture illustrates the above procedure for obtaining the polar graph of $r=1+\cos(\theta)$. In this picture, the concentric circles correspond to values of $r$ while the rays from the origin correspond to the angles which are shown on the picture. The dot on the ray corresponding to the angle $\pi/6$ is located at a distance of $r = 1+\cos(\pi/6)$ from the origin. The dot on the ray corresponding to the angle $\pi/3$ is located at a distance of $r = 1+\cos(\pi/3)$ from the origin and so forth. The polar graph is obtained by connecting such points with a smooth curve, with the result being the figure shown above.

Consider another example of constructing a polar graph.

Example 6: A Polar Graph 例 6：一个极坐标图像

Graph $r=1+2\cos \theta$ for $\theta \in \left[ 0,2\pi \right]$.

Solution. The graph of the polar equation $r=1+2\cos \theta$ for $\theta \in \left[ 0,2\pi \right]$ is given as follows.

To see the way this is graphed, consider the following picture. First the indicated points were graphed and then the curve was drawn to connect the points. When done by a computer, many more points are used to create a more accurate picture.

Consider first the following table of points.

Note how some entries in the table have $r<0.$ To graph these points, simply move in the opposite direction. These types of points are responsible for the small loop on the inside of the larger loop in the graph.

The process of constructing these graphs can be greatly facilitated by computer software. However, the use of such software should not replace understanding the steps involved.

The next example shows the graph for the equation $r=3+\sin \left( \displaystyle \frac{7\theta }{6}\right)$. For complicated polar graphs, computer software is used to facilitate the process.

Example 7: A Polar Graph 例 7：一个极坐标图像

Graph $r=3+\sin \left( \displaystyle \frac{7\theta }{6} \right)$ for $\theta \in \left[ 0,14\pi \right]$.

Solution.

The next example shows another situation in which $r$ can be negative.

Example 8: A Polar Graph: Negative $r$ 例 8：一个极坐标图像（$r$ 为负）

Graph $r=3\sin(4\theta)$ for $\theta \in \left[ 0,2\pi \right]$.

Solution.

We conclude this section with an interesting graph of a simple polar equation.

Example 9: The Graph of a Spiral 例 9：螺线图像

Graph $r=\theta$ for $\theta \in [0,2\pi]$.

Solution. The graph of this polar equation is a spiral. This is the case because as $\theta$ increases, so does $r$.

In the next section, we will look at two ways of generalizing polar coordinates to three dimensions.

8.2 Spherical and Cylindrical Coordinates 球面坐标与柱面坐标

Outcomes.

1. Understand cylindrical and spherical coordinates.
2. Convert points between Cartesian, cylindrical, and spherical coordinates.

1. 理解柱面坐标与球面坐标。
2. 在直角坐标、柱面坐标与球面坐标之间转换点。

Spherical and cylindrical coordinates are two generalizations of polar coordinates to three dimensions. We will first look at cylindrical coordinates.

When moving from polar coordinates in two dimensions to cylindrical coordinates in three dimensions, we use the polar coordinates in the $xy$ plane and add a $z$ coordinate. For this reason, we use the notation $(r, \theta, z)$ to express cylindrical coordinates. The relationship between Cartesian coordinates $(x,y,z)$ and cylindrical coordinates $(r, \theta, z)$ is given by $$ \begin{aligned} x& =r\cos \left( \theta \right) \\ y& =r\sin \left( \theta \right) \\ z& =z\end{aligned} $$ where $r\geq 0$, $\theta \in \lbrack 0,2\pi ),$ and $z$ is simply the Cartesian coordinate. Notice that $x$ and $y$ are defined as the usual polar coordinates in the $xy$-plane. Recall that $r$ is defined as the length of the ray from the origin to the point $(x,y,0)$, while $\theta$ is the angle between the positive $x$-axis and this same ray.

To illustrate this coordinate system, consider the following two pictures. In the first of these, both $r$ and $z$ are known. The cylinder corresponds to a given value for $r$. A useful way to think of $r$ is as the distance between a point in three dimensions and the $z$-axis. Every point on the cylinder shown is at the same distance from the $z$-axis. Giving a value for $z$ results in a horizontal circle, or cross section of the cylinder at the given height on the $z$ axis (shown below as a black line on the cylinder). In the second picture, the point is specified completely by also knowing $\theta$ as shown.

Every point of three dimensional space other than the $z$ axis has unique cylindrical coordinates. Of course there are infinitely many cylindrical coordinates for the origin and for the $z$-axis. Any $\theta$ will work if $r=0$ and $z$ is given.

Consider now spherical coordinates, the second generalization of polar form in three dimensions. For a point $(x,y,z)$ in three dimensional space, the spherical coordinates are defined as follows. $$ \begin{array}{l} \rho:\ \mbox{the length of the ray from the origin to the point}\\ \theta:\ \mbox{the angle between the positive $x$-axis and the ray from the origin to the point $(x,y,0)$}\\ \phi:\ \mbox{the angle between the positive $z$-axis and the ray from the origin to the point of interest} \end{array} $$ The spherical coordinates are determined by $\left( \rho ,\phi ,\theta \right)$. The relation between these and the Cartesian coordinates $\left( x,y,z \right)$ for a point are as follows. $$ \begin{aligned} x& =\rho \sin \left( \phi \right) \cos \left( \theta \right) ,\ \phi \in \left[ 0,\pi \right] \\ y& =\rho \sin \left( \phi \right) \sin \left( \theta \right) ,\ \theta \in \lbrack 0,2\pi ) \\ z& =\rho \cos \phi \text{, }\rho \geq 0. \end{aligned} $$

Consider the pictures below. The first illustrates the surface when $\rho$ is known, which is a sphere of radius $\rho$. The second picture corresponds to knowing both $\rho$ and $\phi$, which results in a circle about the $z$-axis. Suppose the first picture demonstrates a graph of the Earth. Then the circle in the second picture would correspond to a particular latitude.

Giving the third coordinate, $\theta$ completely specifies the point of interest. This is demonstrated in the following picture. If the latitude corresponds to $\phi$, then we can think of $\theta$ as the longitude.

The following picture summarizes the geometric meaning of the three coordinate systems.

Therefore, we can represent the same point in three ways, using Cartesian coordinates, $\left(x,y,z\right)$, cylindrical coordinates, $\left( r, \theta, z \right)$, and spherical coordinates $\left( \rho, \phi, \theta \right)$.

Using this picture to review, call the point of interest $P$ for convenience. The Cartesian coordinates for $P$ are $(x,y,z)$. Then $\rho$ is the distance between the origin and the point $P$. The angle between the positive $z$ axis and the line between the origin and $P$ is denoted by $\phi$. Then $\theta$ is the angle between the positive $x$ axis and the line joining the origin to the point $(x,y,0)$ as shown. This gives the spherical coordinates, $(\rho, \phi, \theta)$. Given the line from the origin to $(x,y,0)$, $r=\rho \sin(\phi)$ is the length of this line. Thus $r$ and $\theta$ determine a point in the $xy$-plane. In other words, $r$ and $\theta$ are the usual polar coordinates and $r\geq 0$ and $\theta \in \lbrack 0,2\pi )$. Letting $z$ denote the usual $z$ coordinate of a point in three dimensions, $(r,\theta ,z)$ are the cylindrical coordinates of $P$.

The relation between spherical and cylindrical coordinates is that $r=\rho \sin(\phi)$ and the $\theta$ is the same as the $\theta$ of cylindrical and polar coordinates.

We will now consider some examples.

Example 1: Describing a Surface in Spherical Coordinates 例 1：用球面坐标描述曲面

Express the surface $z=\frac{1}{\sqrt{3}}\sqrt{x^{2}+y^{2}}$ in spherical coordinates.

Solution. We will use the equations from above: $$ \begin{array}{l} x =\rho \sin \left( \phi \right) \cos \left( \theta \right), \phi \in \left[ 0,\pi \right]\\ y =\rho \sin \left( \phi \right) \sin \left( \theta \right) ,\text{ }\theta \in \lbrack 0,2\pi ) \\ z =\rho \cos \phi \text{, }\rho \geq 0 \end{array} $$ To express the surface in spherical coordinates, we substitute these expressions into the equation. This is done as follows: $$ \rho \cos \left( \phi \right) =\frac{1}{\sqrt{3}}\sqrt{\left( \rho \sin \left( \phi \right) \cos \left( \theta \right) \right) ^{2}+\left( \rho \sin \left( \phi \right) \sin \left( \theta \right) \right) ^{2}}= \frac{1}{3}\sqrt{3}\rho \sin \left( \phi \right). $$ This reduces to $\tan \left( \phi \right)=\sqrt{3}$ and so $\phi =\pi /3$.

Example 2: Describing a Surface in Spherical Coordinates 例 2：用球面坐标描述曲面

Express the surface $y=x$ in terms of spherical coordinates.

Solution. Using the same procedure as the previous example, this says $\rho \sin \left( \phi \right) \sin \left( \theta \right) =\rho \sin \left( \phi \right) \cos \left( \theta \right)$. Simplifying, $\sin \left( \theta \right) =\cos \left( \theta \right)$, which you could also write $\tan \left( \theta \right)=1$.

We conclude this section with an example of how to describe a surface using cylindrical coordinates.

Example 3: Describing a Surface in Cylindrical Coordinates 例 3：用柱面坐标描述曲面

Express the surface $x^{2}+y^{2}=4$ in cylindrical coordinates.

Solution. Recall that to convert from Cartesian to cylindrical coordinates, we can use the following equations: $x =r\cos \left( \theta \right) , y=r\sin \left( \theta \right) , z =z$. Substituting these equations in for $x,y,z$ in the equation for the surface, we have $r^{2}\cos ^{2} \left( \theta \right) +r^{2}\sin ^{2} \left( \theta \right)=4 $. This can be written as $r^2 ( \cos^{2} \left( \theta \right)+ \sin^{2} \left(\theta\right) ) = 4$. Recall that $\cos^{2} \left( \theta \right)+ \sin^{2} \left( \theta \right)=1$. Thus $r^{2} = 4$ or $r=2$.

8.E Exercises 习题

Exercise 1 习题 1

In the following, polar coordinates $(r,\theta)$ for a point in the plane are given. Find the corresponding Cartesian coordinates.

1. $(2,\pi /4)$
2. $(-2, \pi/4)$
3. $(3, \pi/3)$
4. $(-3, \pi/3)$
5. $(2,5\pi /6)$
6. $(-2, 11\pi /6)$
7. $(2,\pi /2)$
8. $(1,3\pi /2)$
9. $(-3, 3\pi/4)$
10. $(3, 5\pi/4)$
11. $(-2, \pi /6)$

1. $(2,\pi /4)$
2. $(-2, \pi/4)$
3. $(3, \pi/3)$
4. $(-3, \pi/3)$
5. $(2,5\pi /6)$
6. $(-2, 11\pi /6)$
7. $(2,\pi /2)$
8. $(1,3\pi /2)$
9. $(-3, 3\pi/4)$
10. $(3, 5\pi/4)$
11. $(-2, \pi /6)$

Exercise 2 习题 2

Consider the following Cartesian coordinates $(x, y)$. Find polar coordinates corresponding to these points.

1. $(-1,1)$
2. $(\sqrt{3},-1)$
3. $(0,2)$
4. $(-5,0)$
5. $(-2\sqrt{3},2)$
6. $(2,-2)$
7. $(-1,\sqrt{3})$
8. $(-1,-\sqrt{3})$

1. $(-1,1)$
2. $(\sqrt{3},-1)$
3. $(0,2)$
4. $(-5,0)$
5. $(-2\sqrt{3},2)$
6. $(2,-2)$
7. $(-1,\sqrt{3})$
8. $(-1,-\sqrt{3})$

Exercise 3 习题 3

The following relations are written in terms of Cartesian coordinates $(x, y)$. Rewrite them in terms of polar coordinates, $(r,\theta )$.

1. $y=x^2$
2. $y=2x+6$
3. $x^2+y^2=4$
4. $x^2-y^2=1$

1. $y=x^2$
2. $y=2x+6$
3. $x^2+y^2=4$
4. $x^2-y^2=1$

Exercise 4 习题 4

Use a calculator or computer algebra system to graph the following polar relations.

1. $r=1-\sin (2\theta ),\ \theta\in [0,2\pi ]$
2. $r=\sin (4\theta ),\ \theta\in [0,2\pi ]$
3. $r=\cos (3\theta )+\sin (2\theta ),\ \theta\in [0,2\pi ]$
4. $r=\theta,\ \theta\in [0,15]$

1. $r=1-\sin (2\theta ),\ \theta\in [0,2\pi ]$
2. $r=\sin (4\theta ),\ \theta\in [0,2\pi ]$
3. $r=\cos (3\theta )+\sin (2\theta ),\ \theta\in [0,2\pi ]$
4. $r=\theta,\ \theta\in [0,15]$

Exercise 5 习题 5

Graph the polar equation $r = 1+\sin\theta$ for $\theta \in [0, 2\pi]$.

Exercise 6 习题 6

Graph the polar equation $r = 2+\sin\theta$ for $\theta \in [0, 2\pi]$.

Exercise 7 习题 7

Graph the polar equation $r = 1+2 \sin\theta$ for $\theta \in [0, 2\pi]$.

Exercise 8 习题 8

Graph the polar equation $r = 2+\sin(2\theta)$ for $\theta \in [0, 2\pi]$.

Exercise 9 习题 9

Graph the polar equation $r = 1+\sin(2\theta)$ for $\theta \in [0, 2\pi]$.

Exercise 10 习题 10

Graph the polar equation $r = 1+\sin(3\theta)$ for $\theta \in [0, 2\pi]$.

Exercise 11 习题 11

Describe how to solve for $r$ and $\theta$ in terms of $x$ and $y$ in polar coordinates.

Exercise 12 习题 12

This problem deals with parabolas, ellipses, and hyperbolas and their equations. Let $l$, $e > 0$ and consider $r=\frac{l}{1\pm e\cos\theta}$. Show that if $e = 0$, the graph of this equation gives a circle. Show that if $0 < e < 1$, the graph is an ellipse, if $e = 1$ it is a parabola and if $e > 1$, it is a hyperbola.

Exercise 13 习题 13

The following are the cylindrical coordinates of points, $(r,\theta,z)$. Find the Cartesian and spherical coordinates of each point.

1. $(5,\frac{5\pi}{6},-3)$
2. $(3,\frac{\pi}{3},4)$
3. $(4,\frac{2\pi}{3},1)$
4. $(2,\frac{3\pi}{4},-2)$
5. $(3,\frac{3\pi}{2},-1)$
6. $(8,\frac{11\pi}{6},-11)$

1. $(5,\frac{5\pi}{6},-3)$
2. $(3,\frac{\pi}{3},4)$
3. $(4,\frac{2\pi}{3},1)$
4. $(2,\frac{3\pi}{4},-2)$
5. $(3,\frac{3\pi}{2},-1)$
6. $(8,\frac{11\pi}{6},-11)$

Exercise 14 习题 14

The following are the Cartesian coordinates of points, $(x, y,z)$. Find the cylindrical and spherical coordinates of these points.

1. $(\frac{5}{2}\sqrt{2},\frac{5}{2}\sqrt{2},-3)$
2. $(\frac{3}{2},\frac{3}{2}\sqrt{3},2)$
3. $(-\frac{5}{2}\sqrt{2},\frac{5}{2}\sqrt{2},11)$
4. $(-\frac{5}{2},\frac{5}{2}\sqrt{3},23)$
5. $(-\sqrt{3},-1,-5)$
6. $(\frac{3}{2},-\frac{3}{2}\sqrt{3},-7)$
7. $(\sqrt{2},\sqrt{6},2\sqrt{2})$
8. $(-\frac{1}{2}\sqrt{3},\frac{3}{2},1)$
9. $(-\frac{3}{4}\sqrt{2},\frac{3}{4}\sqrt{2},-\frac{3}{2}\sqrt{3})$
10. $(-\sqrt{3},1,2\sqrt{3})$
11. $(-\frac{1}{4}\sqrt{2},\frac{1}{4}\sqrt{6},-\frac{1}{2}\sqrt{2})$

1. $(\frac{5}{2}\sqrt{2},\frac{5}{2}\sqrt{2},-3)$
2. $(\frac{3}{2},\frac{3}{2}\sqrt{3},2)$
3. $(-\frac{5}{2}\sqrt{2},\frac{5}{2}\sqrt{2},11)$
4. $(-\frac{5}{2},\frac{5}{2}\sqrt{3},23)$
5. $(-\sqrt{3},-1,-5)$
6. $(\frac{3}{2},-\frac{3}{2}\sqrt{3},-7)$
7. $(\sqrt{2},\sqrt{6},2\sqrt{2})$
8. $(-\frac{1}{2}\sqrt{3},\frac{3}{2},1)$
9. $(-\frac{3}{4}\sqrt{2},\frac{3}{4}\sqrt{2},-\frac{3}{2}\sqrt{3})$
10. $(-\sqrt{3},1,2\sqrt{3})$
11. $(-\frac{1}{4}\sqrt{2},\frac{1}{4}\sqrt{6},-\frac{1}{2}\sqrt{2})$

Exercise 15 习题 15

The following are spherical coordinates of points in the form $(\rho,\phi,\theta)$. Find the Cartesian and cylindrical coordinates of each point.

1. $(4,\frac{\pi}{4},\frac{5\pi}{6})$
2. $(2,\frac{\pi}{3},\frac{2\pi}{3})$
3. $(3,\frac{5\pi}{6},\frac{3\pi}{2})$
4. $(4,\frac{\pi}{2},\frac{7\pi}{4})$
5. $(4,\frac{2\pi}{3},\frac{\pi}{6})$
6. $(4,\frac{3\pi}{4},\frac{5\pi}{3})$

1. $(4,\frac{\pi}{4},\frac{5\pi}{6})$
2. $(2,\frac{\pi}{3},\frac{2\pi}{3})$
3. $(3,\frac{5\pi}{6},\frac{3\pi}{2})$
4. $(4,\frac{\pi}{2},\frac{7\pi}{4})$
5. $(4,\frac{2\pi}{3},\frac{\pi}{6})$
6. $(4,\frac{3\pi}{4},\frac{5\pi}{3})$

Exercise 16 习题 16

Describe the surface $\phi = \pi/4$ in Cartesian coordinates, where $\phi$ is the polar angle in spherical coordinates.

Exercise 17 习题 17

Describe the surface $\theta = \pi/4$ in spherical coordinates, where $\theta$ is the angle measured from the positive $x$ axis.

Exercise 18 习题 18

Describe the surface $r=5$ in Cartesian coordinates, where $r$ is one of the cylindrical coordinates.

Exercise 19 习题 19

Describe the surface $\rho =4$ in Cartesian coordinates, where $\rho$ is the distance to the origin.

Exercise 20 习题 20

Give the cone described by $z=\sqrt{x^2+y^2}$ in cylindrical coordinates and in spherical coordinates.

Exercise 21 习题 21

The following are described in Cartesian coordinates. Rewrite them in terms of spherical coordinates.

1. $z=x^2+y^2$
2. $x^2-y^2=1$
3. $z^2+x^2+y^2=6$
4. $z=\sqrt{x^2+y^2}$
5. $y=x$
6. $z=x$

1. $z=x^2+y^2$
2. $x^2-y^2=1$
3. $z^2+x^2+y^2=6$
4. $z=\sqrt{x^2+y^2}$
5. $y=x$
6. $z=x$

Exercise 22 习题 22

The following are described in Cartesian coordinates. Rewrite them in terms of cylindrical coordinates.

1. $z=x^2+y^2$
2. $x^2-y^2=1$
3. $z^2+x^2+y^2=6$
4. $z=\sqrt{x^2+y^2}$
5. $y=x$
6. $z=x$

1. $z=x^2+y^2$
2. $x^2-y^2=1$
3. $z^2+x^2+y^2=6$
4. $z=\sqrt{x^2+y^2}$
5. $y=x$
6. $z=x$