Coordinates in 3D

Understanding graphs and surfaces requires us to delve a little deeper into the place where they live: $\mathbb{R}^3.$

This lesson covers the essentials of three-dimensional coordinate systems. We'll need some right triangle trigonometry in order to construct the polar, cylindrical, and spherical coordinate systems.

We'll start with the $3$D Cartesian system, which extends the familiar $2$D $xy$-coordinate system into the full three dimensions of our day-to-day experience by adding a new axis. (Check out the animation below!)

The new axis is labeled $z,$ and we have three coordinates $( x,y,z)$ for every point in space $\mathbb{R}^3.$

To get to the point $(1, 2 , 2)$ pictured (the red dot), we start at the origin where all axes meet, move $1$ unit in the positive $x$ direction, $2$ units in the positive $y$ direction, and then $2$ units up in the positive $z$ direction.

Not all creatures prefer Cartesian coordinates.
并非所有生物都喜欢笛卡尔坐标。

Bees do a waggle-dance to communicate distance and direction information, in effect using a coordinate system based on circles instead of perpendicular lines.
蜜蜂通过摇摆舞来传达距离和方向信息，实际上使用基于圆形而不是垂直线的坐标系。

If we're to be at least as clever as bees, we should develop alternative coordinate systems, too!
如果我们至少要像蜜蜂一样聪明，我们也应该开发替代坐标系！

Cylindrical and spherical coordinate systems for $\mathbb{R}^3$ are just such coordinate systems, and both are built up from polar coordinates $(r,\theta)$ in $\mathbb{R}^2,$ the system of choice for talkative bees.
$\mathbb{R}^3$ 的柱坐标系和球坐标系就是这样的坐标系，并且两者都是从 $\mathbb{R}^2,$ 系统中的极坐标 $(r,\theta)$ 建立起来的，这是健谈的蜜蜂选择的系统。

The rest of the lesson will develop all three coordinate systems from scratch.
本课程的其余部分将从头开始开发所有三个坐标系。

Let's start with polar coordinates.
让我们从极坐标开始。

We arrive at the planar point $P =(x,y)$ by following a ray starting at the origin and making an angle of $\theta$ with respect to the positive $x$-axis for a distance of $r.$ Trigonometry then tells us that $x = r \cos(\theta ).$
我们通过沿着从原点开始的射线并相对于正 $x$ 轴形成 $\theta$ 角度一段距离来到达平面点 $P =(x,y)$ $r.$ 三角函数告诉我们 $x = r \cos(\theta ).$

Polar coordinates and Cartesian coordinates are related through
极坐标和笛卡尔坐标通过以下关系相关

Suppose we're given the Cartesian coordinates for $P$ but want to know the polar form. Since $\sin^2(\theta)+ \cos^2(\theta) =1,$
假设我们给出了 $P$ 的笛卡尔坐标，但想知道极坐标形式。自 $\sin^2(\theta)+ \cos^2(\theta) =1,$

For $3$D problems, we can build a hybrid of the $2$D polar system and Cartesian coordinates called polar cylindrical (or just cylindrical) coordinates.
对于 $3$ D 问题，我们可以构建 $2$ D 极坐标系和笛卡尔坐标的混合体，称为极柱坐标（或简称柱坐标）。

Suppose $P = (x,y,z) \in \mathbb{R}^3.$ The first two numbers represent a point in the plane, which we can describe using $( r, \theta).$ The cylindrical coordinates of $P$ are then $(r,\theta,z).$
假设 $P = (x,y,z) \in \mathbb{R}^3.$ 前两个数字代表平面上的一个点，我们可以用 $( r, \theta).$ 来描述，那么 $P$ 的柱坐标就是 $(r,\theta,z).$

Cylindrical coordinates are ideally suited for problems in $\mathbb{R}^3$ symmetric about the $z$-axis, like describing $C_{R},$ the cylinder of radius $R$ about the $z$-axis, which is made up of all points of distance $R$ from this line.
圆柱坐标非常适合解决 $\mathbb{R}^3$ 关于 $z$ 轴对称的问题，例如描述 $C_{R},$ 半径为 $R$ 的圆柱体 $z$ - 轴，由距该线 $R$ 距离的所有点组成。

For this diagram, we use dashed lines to indicate portions of the picture that continue out to infinity. The cylinder above continues parallel to the $z$-axis in both the positive and negative directions.
对于此图，我们使用虚线来表示图片中一直延伸到无穷大的部分。上面的圆柱体在正方向和负方向上都继续平行于 $z$ 轴。

Of the equations presented, which one best describes $C_{R}$ in cylindrical coordinates?
在所提出的方程中，哪一个最能描述柱坐标系中的 $C_{R}$ ？

$r = R$$\theta = \dfrac{\pi}{2}$$z = R$

Show explanation

Explanation

Setting $r = R$ is equivalent to setting $\sqrt{x^2+y^2} = R,$ fixing the distance of the point $(x,y,z)$ from the $z$-axis to be exactly $R$ units. This perfectly describes $C_{R}.$

The other options have interesting geometric interpretations as well. If $\theta$ is held fixed, we have part of a plane containing the $z$ -axis.

It's probably easiest to see why this is true by taking a bird's eye view. Looking down the $z$-axis, we see that holding $\theta$ fixed gives us a line in the plane.

Since $r \geq 0$ always, the plane is actually formed from only half this line. A copy of this half line sits at each position along the $z$-axis. Putting all the lines together gives us a half plane.

The interactive below shows a $\theta = \text{constant}$ line segment (blue) in the $z = 0$ plane. Adjust the slider to see the result of stacking $\theta = \text{constant}$ line segments for all $z$-values from $0$ up to height $h > 0.$

$z = R$ also defines a plane, but one that cuts through the $z$-axis perpendicularly.

Since $\mathbb{R}^3$ is the set of all ordered triples, setting $z = \text{const.} = R$ fixes the third coordinate to some value. This leaves the first two coordinates free to be anything, which is virtually the same thing as the Cartesian plane $\mathbb{R}^2.$ So $z = R$ is a plane parallel to the $xy$-plane (perpendicular to the $z$-axis) sitting at the $R$ tick mark on the $z$-axis.

Back to problem

When a problem has complete symmetry around the origin, spherical coordinates are usually better than cylindrical coordinates.

Suppose $P = (x,y,z) \in \mathbb{R}^3.$ We know

where $r$ is the distance between the point $( 0,0)$ and $(x,y)$ in the plane.

$\theta$ is taken to be one of the new spherical coordinates; the other two are $\rho$ $\big($the distance between $P$ and $(0,0,0)\big)$ and $\phi,$ the angle between the positive $z$-axis and the ray joining $P$ with the origin.

Since $r = \rho \sin(\phi),$

We need to find a formula for $z$ to complete the relationship between Cartesian and spherical coordinates.

Finally, let's understand why the coordinates $( \rho, \theta, \phi)$ given by

are called spherical.

Of the options presented, which one correctly describes the sphere $S_{R}$ of radius $R$ centered at the origin in spherical coordinates?

$\rho = R$$\theta = \dfrac{\pi}{2}$$\phi = \pi$

Show explanation

Explanation

A direct calculation shows that $\rho^2 = x^2 + y^2 + z^2.$ Alternatively, $\rho$ is defined to be the distance from $(0,0,0)$ to $(x,y,z).$ Either way we see that the sphere $S_{R}$ is perfectly represented by $\rho = R.$

Fixing $\theta$ to be constant gives us a plane, just as we argued in cylindrical coordinates. The result of fixing $\phi,$ however, depends on the value.

First, note that $\phi \in [0, \pi].$ This makes sure that every point in $\mathbb{R}^3$ has one and only one representation in spherical coordinates. If $\phi = 0$ or $\phi = \pi,$ the result is a ray starting at the origin.

If $\phi \neq \frac{\pi}{2}$ is held fixed somewhere between $0$ and $\pi,$ the result is a cone:

Looking down the $x$ -axis when $\phi$ is held fixed, we see a ray starting at the origin having slope $\cot(\phi).$

This is because the line segment sits in the $yz$-plane (which corresponds to $\theta = \frac{\pi}{2}$), and is described by

$$\frac{z}{y} = \frac{\rho \cos( \phi)}{\rho \sin(\phi) \sin(\theta)} = \frac{\cot(\phi)}{\sin \left( \frac{\pi}{2} \right) } = \cot(\phi)$$

or $z = \cot(\phi)y.$

Revolving this shape around the $z$-axis gives us $\phi = \text{constant},$ which is a cone.

The set corresponding to $\phi = \frac{\pi}{2}$ is just the $xy$-plane.

Back to problem

Visualizing mathematical objects like surfaces in $\mathbb{R}^3$ can be very helpful in solving many multivariable calculus problems.
可视化数学对象（例如 $\mathbb{R}^3$ 中的曲面）对于解决许多多变量微积分问题非常有帮助。

We saw one example of this already when we visualized the graph of the depth function at the end of the optimization lesson. At a glance we were able to see where the minimum and maximum values of the depth function occur.
当我们在优化课程结束时可视化深度函数图时，我们已经看到了这样的一个例子。我们一眼就能看出深度函数的最小值和最大值出现在哪里。

The Cartesian, spherical, and cylindrical systems provide the means for visualizing a large variety of useful objects in multivariable calculus.
笛卡尔、球面和柱面系统提供了可视化多元微积分中各种有用对象的方法。

The final lesson of this intro chapter shows one particular and very important example.
本章的最后一课展示了一个特殊且非常重要的示例。

There, we'll use the $3$D coordinate system to understand what place integrals have in the world of multivariable calculus.
在那里，我们将使用 $3$ D 坐标系来了解积分在多变量微积分世界中的地位。