Math 243: Calculus II

12.1 Coordinate Systems in $\mathbb{R}^n$

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12.1.1 Coordinate Systems

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A point in three-dimensional space requires three coordinates to uniquely describe each point in the space.

Definition

A space in mathematics is set of mathematical objects called the elements of the space (set) and definitions for relationships between the elements of the space.

Elements of a space are the mathematical objects in the space. These elements are often referred to as points in the space. However there are many (many, many, many) examples of spaces of mathematical objects and the relationships between them.

Note:

There are two parts to the definition (axioms, or postulates) of a space.

1. The definition of the elements or points in the space. Often the notation used to describe the elements of a space is included with this part of the definition.

2. The relationships or structure of a space describe relationships between elements of the space.

Euclidean Space

Euclidean Space is designed to describe points in our physical world. Greek mathematician Euclid wrote 13 books to describe our physical world using the most efficient definition. Euclid believed that the most efficient definition contains the smallest number of axioms or postulates to define the space. This is followed by a much larger number of theorems whose validity can be derived from the axioms utilizing Aristotelian logic.

The elements or points in 3-dimensional space are described using coordinates. Coordinates of a point in Euclidean space are a tuple or ordered list of numbers. Often one denotes the coordinates of a point using the same lower case letter in normal font to indicate it is a scalar, and with a subscript to indicate which coordinate it is.

$$ P = \left( p_1, p_2, p_3 \right) $$

A point in $n$-dimensional Euclidean space requires an $n$-tuple to describe it.

$$ Q = \left( q_1,\ \dots,\ q_n \right) $$

The coordinates of a point in 3-dimensional space describe a distance along a number line or axis. Geometrically, three dimensional space is pictured using three perpendicular axes. These axes are often described as the $x$-axis, $y$-axis and $z$-axis. However, these labels are arbitrary. In our calculus one will often want to distinguish between two copies of three dimensional space. In these cases, the axes will have different labels.

Figure 1

The positive and negative direction of each axes is very important.

The three perpendicular axes in three dimensional Euclidean space meet at a single point called the origin, $O$. The coordinates of the origin are $O = (0,0,0)$. In figure 1 we see several line segments $\overline{OQ}$, $\overline{OR}$, and $\overline{OS}$. We also see several vectors

$$ \overrightarrow{OQ},\ \overrightarrow{OR},\ \overrightarrow{OS} $$

Notice the difference between the notation for points using round brackets, line segments using an over line with labels for the end points, and vectors using an over arrow with labels for the end points. Euclidean $n$-dimensional space is a vector space. We will introduce vectors in the next section. Here we need the definition and notation for points in our Euclidean space, and the algebraic structure given to relationships between these points.

Points in Euclidean space and their coordinates are tied to the relationship or distance between them. This distance between points is defined in the next section using dot product. However we can discuss the Aristotelian consequence of the definition of dot product.

Definition

The Cartesian coordinates, or rectangular coordinates of a point in Euclidean space are a tuple of signed real numbers. Each number represents a distance parallel to the corresponding axis.

In figure 1 we represent the point $(2,3,4)$ using Cartesian coordinates or three dimensional rectangular coordinates:

• The first number in the 3-tuple specifies a distance of 2 parallel to the first axis in the positive direction.

• The second number in the 3-tuple specifies a distance of 3 parallel to the second axis in the positive direction.

• The third number in the 3-tuple specifies a distance of 4 parallel to the third axis in the positive direction.

In this way every point in three dimensional space has a unique tuple or set of coordinates, and every three-tuple describes a unique point in three dimensional space. Because every point is described using three mutually perpendicular number lines $\mathbb{R}$. Mathematicians think the number line as a one dimensional Euclidean space, $\mathbb{R}$.

Thus three dimensional Euclidean space is defined to be the Cartesian Product of three copies of the real numbers,

$$ \mathbb{R}^3 := \mathbb{R}\times\mathbb{R}\times\mathbb{R} = \left\{\,(x,y,z)\,:\,x,y,z\in\mathbb{R}\,\right\} $$

Likewise $n$-dimensional Euclidean space is defined to be the Cartesian product of $n$ copies of the real numbers,

$$ \begin{align*} \mathbb{R}^n &= \mathbb{R}\times\dots\times\mathbb{R} = {\displaystyle\Large{\times}}_{k=1}^n \mathbb{R} \\ \\ &= \left\{\,(x_1, \dots, x_n)\,:\,x_k\in\mathbb{R}\ \text{for all }1\le k\le n\,\right\} \end{align*} $$

The points $(2,3,1)$ and $(-1,-3,-2)$ are plotted as follows:

Figure 3

Here the points are plotted using an isometric angle. The viewing angle here is useful because a unit of travel in each of the $x$, $y$, and $z$ directions is visually equivalent. This is by no means the only choice of orientation for the axes, and in 3D there is not a clear-cut standard orientation like there is in 2D. It is usual for the axes to be in this rough configuration with $x$ pointed at the viewer, $y$ to the right and $z$ upwards, but the precise viewing angle is not fixed. For instance, we may choose to have the $xy$-plane closer to being parallel with the floor (relative to the viewer) than in the isometric case

Figure 4

The reason we have difficulty here is that there are fundamental issues with using a two-dimensional medium, such a printed page or computer screen, to represent a three-dimensional idea. Do not allow yourself to be limited by this. It is important to look at these images and other representations of 3D objects that you are familiar with and begin to train yourself to visualize and rotate these objects in your mind. This will be extremely useful in both your studies for this course, and for other activities you might be involved in in the future such as computer aided drafting/design.

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12.1.2 Plotting Functions in 2D and 3D

Example 1

If you were told to plot the function $y = \frac{1}{8}x^2$ in a previous course, the procedure is pretty straightforward. You are familiar enough with that function to know that it is a parabola, and you would plot it over some subset of values, say $[-4,4]$, to show the shape of the curve.

Figure 5

However, the construction of this plot has many elements that were not explicitly mentioned. More formally, this is a function

$$ f:\,\mathbb{R}\rightarrow\mathbb{R} $$

given by $f(x) = y = \frac{1}{8}x^2$.

Definition

A function is a relation between to sets called the domain and the codomain, $f\,:\,D\rightarrow R$, where every element of the domain set $D$ is associated with exactly one element of the codomain $R$.

This definition of a function allows us to draw the graph of a function in Euclidean $2$-space

Definition

The graph of a real-valued function $f\,:\,D\rightarrow R$ where $D$ and $R$ are subsets of the real numbers $R$ is the locus of all points

$$ \left\{\,(x,f(x))\,:\,x\in D\,\right\} $$

The graph of a function in Euclidean $2$-space is one-dimensional because the domain is one-dimensional and every point in the domain appears only once in the graph. This occurs because the definition of function allows for only one element of the co-domain to be associated with every element of the domain. This graph will look like we drew it on the plane using a pencil. We call these graphs a curve.

Definition

A curve in Euclidean $2$-space is the graph of a function $f\,:\,D\rightarrow R$, where the domain $D$ and the co-domain $R$ are subsets of $\mathbb{R}$.

Typically, when a function is discussed, it is presented with just the formula. It is assumed that the reader will interpret this function as being defined on the largest possible domain where that formula is valid and that the codomain is implied by the formula. For instance,

$$ y = \sqrt{x} $$

would only be defined for $x\geq 0$ because taking the square root of a negative number produces a complex result.

What is not given by that formula is the domain and codomain. The most common way to consider that formula would be as a 2D graph with the $x$ values for the domain and the $y$ values as the codomain as seen on the left.

However, there is nothing preventing us from interpreting this as a 3D plot. The $z$ value is not specified by the formula, so any possible value for $z$ is allowed so long as $y = \sqrt{x}$ is satisfied as seen in the right figure. At each $z$ value, the curve is exactly the graph of the square root.

What matters here is context. You need to know that the same formula in 2D and 3D lead to different graphs.

Example 2

As a simpler example, we can look at $y = 1$. In 2D, this is a line and in 3D this is a plane:

Figure 8

Be careful about the context of the graph. Make sure that you are interpreting the equation correctly. What you are plotting is called the locus of solutions to the equation in 2D or 3D. This is the collection of points that satisfy the equation, and could be a curve or surface depending on the circumstance. It is represented symbolically as either

$$\{ (x,1)\ \vert\ x\in\mathbb{R}\}\qquad\text{or}\qquad\{ (x,y)\in\mathbb{R}^2\ \vert\ y=1\}$$

in 2D or

$$\{ (x,1,z)\ \vert\ x,z\in\mathbb{R}\}\qquad\text{or}\qquad\{ (x,y,z)\in\mathbb{R}^3\ \vert\ y=1\}$$

in 3D.

Exercise 1

Sketch a plot of each of the following:

1. $x^2 + y^2 = 1$ in $\mathbb{R}^2$
2. $x^2 + y^2 = 1$ and $z = 1$ in $\mathbb{R}^3$
3. $x^2 + y^2 = 1$ in $\mathbb{R}^3$

Check Your Work

Figure 9

The cylinder in 3D is a "true" cylinder of infinite height in both directions. For drawing the image it is necessary to truncate the surface. Also, only the surface on the sides of the cylinder are plotted; it is hollow and has no top or bottom. Imagine the circle in the middle plot being copied for every possible height $z$.

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12.1.3 Surfaces

Definition

The graph of a function in three dimensional space where one of the elements of the 3-tuple is a function of the other two elements is called a surface. In this case function $f$ has domain $D\subset\mathbb{R}^2$ and co-domain $R\subset\mathbb{R}$.

Example 3

For example, the graph of the function $f\,:\,\mathbb{R}^2\rightarrow\mathbb{R}$ defined by $z=\pi$ is a constant function; for every input $(x,y)$ in $\mathbb{R}^2$, the output is $\pi$. Hence the graph of this function is given by

$$ \left\{\,(x,y,z)\,:\,(x,y)\in\mathbb{R}^2,\ \text{and}\ z=f(x,y)=\pi\,\right\} $$

Figure 9

Example 4

How would one define the $yz$-plane? While there are infinitely many functions whose graph is the $yz$-plane, consider the function $g\,:\,\mathbb{R}^2\rightarrow\mathbb{R}$ defined by $x=0$. In this case, the first coordinate is a function of the last two coordinates. This is also a constant function since for every element of the domain $(y,z)\in\mathbb{R}^2$ the output is $x=0$.

$$ \left\{\,(x,y,z)\,:\,(y,z)\in\mathbb{R}^2,\ \text{and}\ x=g(y,z)=0\,\right\} $$

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Figure 10

Surfaces are two-dimensional because the domain is two dimensional and there is only one real output for every element of the domain. A surface can be described as the graph of a system of equations.

Example 5

Consider the graph of the system of equations

$$ \begin{align*} y^2 + z^2 = 4 \\ x = \pi \end{align*} $$

Figure 11

Example 6

Consider the graph of the right circular cylinder $y^2 + z^2 = 9$

Figure 12

In general, if one starts with the graph of a curve $v = h(u)$ in two of the planes in three dimensional space and leave the third coordinate unconstrained, then the graph is called a cylinder.

Example 7

The graph of the equation $z=y^2$ is a cylinder because exactly one of the coordinates is unconstrained. The curve $\left\{\,(0,y,z)\,:\,z=y^2,\ \text{and}\ y\in\mathbb{R}\,\right\}$ is called a base curve of the cylinder.

Figure 13

In this way the curve in Example 3 can be considered the intersection of the surfaces

$$ y^2 + z^2 = 4 $$

and

$$ x=\pi $$

Figure 14

A system of equations whose graph describes a curve or surface is called a parametrization#:~:text=Parametrization%20is%20a%20mathematical%20process,some%20independent%20quantities%20called%20parameters.) of the curve or surface. The curve in Example 3 can also be parameterized using three coordinate functions

$$ \begin{align*} x(t) &= \pi \\ y(t) &= 3\cos(t) \\ z(t) &= 3\sin(t) \end{align*} $$

This system of equations can also be described as a function $F\,:\,[0,2\pi]\rightarrow\mathbb{R}^3$ defined by

$$ F(t) = \left( x(t), y(t), z(t) \right) $$

Exercise 2

Create another parametrization of the curve in Example 3

View Solution

Define $F\,:\,[2\pi,4\pi]\rightarrow\mathbb{R}^3$ by the same set of equations $G(t) = \left(x(t),y(t),z(t)\right)$, where
$$ \begin{align*} x(t) &= \pi \\ y(t) &= 3\cos(t) \\ z(t) &= 3\sin(t) \\ \end{align*} $$
Since the domain of $G$ is different from $F$, the functions $F$ and $G$ are different. Therefore they are different parametrizations of the curve.

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12.1.4 Distance and Spheres

The distance formula in Euclidean space $\mathbb{R}^3$ is a consequence of dot product on $\mathbb{R}^3$ as we shall see in the next section.

Distance formula in $\mathbb{R}^3$

The distance $d(P_1,P_2) = \left|\overline{P_1P_2}\right|$ between two points $P_1 = (x_1,y_1,z_1)$ and $P_2 = (x_2,y_2,z_2)$ in $\mathbb{R}^3$ is given by

$$ d(P_1,P_2) = \left|\overline{P_1P_2}\right| = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2} $$

Figure 15

Consider the faces of the solid prism or rectangular solid in Figure 9 with diagonal $\overline{P_1P_2}$, as well as the right triangles $\triangle ABP_2$ and $\triangle P_1AP_2$. To determine the length of the hypotenuse $\overline{P_1P_2}$, one must first determine the length of hypotenuse $\overline{AP_2}$.

The length of hypotenuse $\overline{AP_2}$ can be computed using the Pythagorean theorem and the lengths of line segments $\overline{AB}$ and $\overline{BP_2}$. The coordinates of the relevant points are

$$ \begin{align*} P_1 = \left( x_1, y_1, z_1 \right) \\ A = \left( x_1, y_1, z_2 \right) \\ B = \left( x_2, y_1, z_2 \right) \\ P_2 = \left( x_2, y_2, z_2 \right) \end{align*} $$

Hence the line segments

$$ \begin{align*} \left|\overline{AB}\right| &= x_2-x_1 \text{ because the line segment is parallel to the }x\text{-axis} \\ \left|\overline{BP_2}\right| &= y_2-y_1 \text{ because the line segment is parallel to the }y\text{-axis} \end{align*} $$

Using the Pythagorean theorem

$$ \left|\overline{AP_2}\right| = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} $$

Similarly

$$ \begin{align*} \left|\overline{P_1A}\right| &= z_2-z_1 \text{ because the line segment is parallel to the }z\text{-axis} \\ \left|\overline{AP_2}\right| &= \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \\ \end{align*} $$

Using the Pythagorean theorem again

$$ \begin{align*} \left|\overline{P_1P_2}\right| &= \sqrt{ \left|AP_2\right|^2 + (z_2-z_1)^2} \\ \\ &= \sqrt{ (x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2} \end{align*} $$

A sphere in $\mathbb{R}^3$ is the locus of all points that are equi-distant from a single point called the center of the circle. This is a surface in $\mathbb{R}^3$

Figure 16

Any point $P=(x,y,z)$ on the sphere must be a constant distance from the center $C=(h,k,l)$. The constant distance $r$ called the radius of the sphere. Hence

$$ r = \left|\overline{CP}\right| = \sqrt{ (x-h)^2 + (y-k)^2 + (z-l)^2 } $$

Squaring both sides of this equation yields the standard equation for a sphere.

$$ (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2 $$

Dividing both sides of this equation by $r^2$ provides

$$ \frac{(x-h)^2}{r^2} + \frac{(y-k)^2}{r^2} + \frac{(z-l)^2}{r^2} = 1 $$

For an elipsoid, each of the axes may have a different radius. Hence the standard equation of an ellipsoid becomes

$$ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} + \frac{(z-l)^2}{c^2} = 1 $$

Recall the equations for the distance, circle and ellipse in the plane, $\mathbb{R}^2$. Notice how similar they are. We can extend this definition to Euclidean space $\mathbb{R}^n$

Distance formula in $\mathbb{R}^n$

The distance $d(P_1,P_2) = \left|\overline{P_1P_2}\right|$ between two points $P_1 = (x_1,\dots,x_n)$ and $P_2 = (y_1,\dots,y_n)$ in $\mathbb{R}^n$ is given by

$$ d(P_1,P_2) = \left|\overline{P_1P_2}\right| = \sqrt{(y_1-x_1)^2 + \dots + (x_n-y_n)^2} $$

Similarly the equation for an $n-1$ dimensional hypersphere in $\mathbb{R}^n$ with center $C=(c_1,\dots,c_n)$ and radius $r$ is given by

$$ (x_1-c_1)^2 + \dots + (x_n-c_n)^2 = r^2 $$

This is the $n-1$ dimensional hypersurface of points equi-distant from the center $C$.

Exercise 3

Given the equation

$$ x^2 + 4x + y^2 - 2y + z^2 = -1 $$
determine the solution set and sketch its graph. ( Hint: Complete the square! )

Check Your Work

$$ \begin{align*} x^2 + 2\cdot\color{#0066CC}{2}x \color{#0066CC}{+2^2}\color{#CC0099}{-2^2} + y^2 - 2\cdot\color{#006666}{1}y \color{#006666}{+1^2} \color{#9900FF}{-1^2} + z^2 &= -1 \\ \\ x^2 + 2\cdot\color{#0066CC}{2}x \color{#0066CC}{+2^2} + y^2 - 2\cdot\color{#006666}{1}y \color{#006666}{+1^2} + z^2 &= -1 \color{#CC0099}{+2^2}\color{#9900FF}{+1^2}\\ \\ (x\color{#0066CC}{+2})^2 + (y\color{#006666}{-1})^2 + z^2 &= -1+4+1 \\ \\ (x+2)^2 + (y-1)^2 + z^2 &= 2^2 \end{align*}$$
After completing the square, the equation tells us that this is sphere of radius $2$ centered at $(-2,1,0)$.

Figure 17

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Exercise 4

Given the equation

$$ x^2 + 4x + y^2 - 2y + z^2 = -10 $$
determine the solution set and sketch its graph.

Check Your Work

The left hand side is the same as before, so we have

$$ \begin{align*} x^2 + 2\cdot\color{#0066CC}{2}x \color{#0066CC}{+2^2}\color{#CC0099}{-2^2} + y^2 - 2\cdot\color{#006666}{1}y \color{#006666}{+1^2} \color{#9900FF}{-1^2} + z^2 &= -10 \\ \\ x^2 + 2\cdot\color{#0066CC}{2}x \color{#0066CC}{+2^2} + y^2 - 2\cdot\color{#006666}{1}y \color{#006666}{+1^2} + z^2 &= -10 \color{#CC0099}{+2^2}\color{#9900FF}{+1^2}\\ \\ (x\color{#0066CC}{+2})^2 + (y\color{#006666}{-1})^2 + z^2 &= -10+4+1 \\ \\ (x+2)^2 + (y-1)^2 + z^2 &= -5 \end{align*}$$
Which has no solutions, since the left hand side must be a positive quantity and there is a negative value on the right hand side. Since there is no valid solution set, there is no graph to sketch.

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12.1.5 Annular Cylinders

Remember that the area between two concentric circles in $\mathbb{R}^2$ is called an annulus. A cylinder in $\mathbb{R}^3$ with an annular base is called an annular cylinder, an annular region a ring, a cylindrical shell. It is the solid region between two concentric circular cylinders.

Figure 18

An annular region with a small height compared to its radius is called a washer. This region is described using inequalities

$$ \begin{align*} 4 &\le x^2 + y^2 \le 16 \\ 0 &\le z \le 4 \end{align*} $$

In general, the system of equations for an annular region in Figure 11 consists of

$$ \begin{align*} r^2 &\le (x-h)^2 + (y-k)^2 \le R^2 \\ 0 &\le z \le h \end{align*} $$

where the radius of the inner cylinder is $r$, the radius of the outer cylinder is $R$, the center of the concentric circles is the point $(h,k,0)$, and the height of the region is $h$. One will need to make adjustments to this system of equations if the orientation is changed.

Figure 19

One obtains an annular sector by limiting the beginning and end angles for the annular base. We will need polar coordinates in a future chapter to efficiently describe this region with a system of equations. In rectangular coordinates the function

$$ F(r,\theta,z) = \left( r\cos(\theta), r\sin(\theta), z \right), $$

where $2\le r\le 4$, $0 \le \theta \le \frac{3\pi}{2}$, and $0 \le z \le 4$ describes the annular section in Figure 12.

Exercise 5

Plot the solution set for the half-sphere
$$ 1 \leq (x+2)^2 + (y-1)^2 + z^2 \leq 4 \quad\text{and}\quad z\leq 0 $$

Check Your Work

Figure 20

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