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Functions in higher dimensions

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1 Multiple variables, multiple dimensions

Why do we need to study multidimensional spaces?

These are the two main sources of a multiple dimension:

  • the physical space, dimension $3$,
  • abstract spaces designed to accommodate multiple variables (such as the stock and the commodity prices), and
  • multiple spaces of single dimension interconnected via functional relations.

We will start with the last item. Even though in the early calculus (Part I) we deal with only numbers, the graph of a function of one variable lies in the $xy$-plane, a space of dimension $2$.

Let's furthermore recall the following problem from Part I. A car is driven through a mountain terrain. Its location and its speed, as seen on the map, are known. The grade of the road is also known. How fast is the car climbing?

Time-location-altitude.png

The first variable is time, $t$. We also have two spacial variables: the horizontal location $x$ and the elevation (the vertical location) $z$. Then $z$ depends on $x$ and $x$ depends on $t$. Therefore, $z$ depends on $t$ via the composition:

Composition -- time-location-altitude.png

As we can see, we have to consider the Cartesian space of dimension $3$ in order to understand what is going on.

Furthermore, these are the two observations that we made:

  • 1. your vertical speed, $\frac{\Delta z}{\Delta t}$ or $\frac{dz}{dt}$, is proportional to your horizontal speed, $\frac{\Delta x}{\Delta t}$ or $\frac{dx}{dt}$;
  • 2. your vertical speed, $\frac{\Delta z}{\Delta t}$ or $\frac{dz}{dt}$, is proportional to the slope of the road, $\frac{\Delta z}{\Delta x}$ or $\frac{dz}{dx}$.

This lead us to discover the Chain Rule: $$\frac{\Delta z}{\Delta t}=\frac{\Delta z}{\Delta x}\frac{\Delta x}{\Delta t} \text{ or } \frac{dz}{dt}=\frac{dz}{dx}\frac{dx}{dt}.$$ These aren't locations anymore but vectors.

Now, the road in this example is straight! More realistically, it should have turns and curves...

Planning a road trip, we create a trip plan: the times and the places put on a simple automotive map:

Trip map.png

This is a parametric curve: $$x=f(t),\ y=g(t),$$ with $x$ and $y$ providing the coordinates of your location. Conversely, motion in time is a go-to metaphor for parametric curves!

We then bring the terrain map of the area:

Function of two variables -- heat map.png

Such a topographic map has the colors indicating the elevation:

Function of two variables -- surface 1.png

This is a function of two variables: $$z=f(x,y).$$ Conversely, a terrain map is a go-to metaphor for functions of two variables!

Now, back to the same question: how fast will we be climbing?

Map and terrain.png

We realize that we face new kinds of functions and a new kind of composition: $$\begin{array}{|ccccc|} \hline &\text{trip map} & & \bigg|\\ \hline t&\longrightarrow & (x,y) &\longrightarrow & z\\ \hline &\bigg| & &\text{terrain map}\\ \hline \end{array}$$ Both functions deal with $3$ variables at the same time with a total of $4$! Furthermore, planning a flight of a plane would require $3$ spacial variables in addition to time. The number increases to $6$ if we are to consider the orientation of the plane in the air: the roll, the pitch, and the yaw. In the meantime, there are many functions of the $3$ variables of location: the temperature and the pressure of the air or water, the humidity, the concentration of a particular chemical, etc.

The observations about the rates of change are still applicable:

  • If we double our horizontal speed (with the same terrain), the climb will be twice as fast.
  • If we double steepness of the terrain (with the horizontal speed), the climb will be twice as fast.

It follows that the speed of the climb is proportional to both our horizontal speed and the steepness of the terrain. That's the Chain Rule. What is it in this new setting? Both rates of change of and with respect to $x$ and $y$ will have to be involved. We will show that it is the sum of those: $$\frac{\Delta z}{\Delta t}=\frac{\Delta z}{\Delta x}\frac{\Delta x}{\Delta t} + \frac{\Delta z}{\Delta y}\frac{\Delta y}{\Delta t} \text{ or }\frac{dz}{dt}=\frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt}.$$ This number is then computed from these two vectors:

  • the rate of change of the parametric curve of the trip, i.e., the horizontal velocity, $\left< \frac{\Delta x}{\Delta t}, \frac{\Delta y}{\Delta t} \right>$ or $\left< \frac{dx}{dt}, \frac{dy}{dt} \right>$, and
  • the rates of change of the terrain function in the two directions, $\left< \frac{\Delta z}{\Delta x}, \frac{\Delta z}{\Delta x} \right> $ or $\left< \frac{\partial z}{\partial x}, \frac{\partial z}{\partial x} \right> $.

We saw in the last chapter an example of an abstract space: the space of prices. The space's dimension was $2$ with only two prices of two ingredients of bread. The dependence is as follows: $$t\ \longrightarrow\ (x,y) \ \longrightarrow \ z.$$ Multiple variables lead to high-dimensional abstract spaces such as in the case of the price of a car dependent on the prices of $1000$ of its parts: $$t\ \longrightarrow\ (x_1,x_2,...,x_{1000}) \ \longrightarrow \ z.$$ We will be developing algebra, geometry, and calculus that will be applicable to a space any dimension. We replace multiple single variables with a single variable in a space of multiple dimension. For example, $P$ below is such a variable, i.e., a location in a $1000$-dimensional space: $$t\ \longrightarrow\ P \ \longrightarrow \ z.$$ Initially however we will limit ourselves to dimensions that we can visualize!

Convention. We will use upper case letters for the entities that are (or may be) multidimensional, such as points and vectors: $$A,\ B,\ C,\ ...P,\ Q,\ ...,$$ and lower case letters for numbers: $$a,\ b,\ c,\ ..., x,\ y,\ z,\ ...$$

2 Euclidean spaces and Cartesian systems of dimensions $1$, $2$, $3$,...

We start with the Cartesian system for dimension $1$. It is a lie with a certain collection of features -- the origin, the positive direction, and the unit -- added:

Coordinate system dim 1.png

The main idea is this correspondence (i.e., a function that is one-to-one and onto):

  • a location $P\ \longleftrightarrow\ $ a real number $x$.

We can have such “Cartesian lines” as many as we like and we can arrange them in any way we like. Then the Cartesian system for dimension $2$ is made of two copies of the Cartesian system of dimension $1$ aligned at $90$ degrees (of rotation) from positive $x$ to positive $y$:

Coordinate system dim 2.png

Warning: the units of the two systems don't have to match. The most common $xy$-plane that we have seen has $x$ is time measured in seconds while $y$ is distance measured in feet. Sometimes even the spacial coordinates use different units such as in the case of a location of an aircraft: altitude in feet and distance in miles.

The Cartesian system for dimension $3$ is made of three copies of the Cartesian system of dimension $1$ as follows:

  • 1. three coordinate axes are chosen, the $x$-axis, the $y$-axis, and the $z$-axis;
  • 2. the two axes are put together at their origins so that it is a $90$-degree turn from the positive direction of one axis to the positive direction of the next -- from $x$ to $y$ to $z$ to $x$;
  • 3. use the marks on the axis to draw a grid.
Coordinate system dim 3.png

The second requirement is called the right-hand rule. Indeed, if we curl our fingers from the $x$-axis to the $y$-axis, our thumb will point in the direction of the $z$-axis.

Right-hand rule.png

The main idea is this correspondence:

  • a location $P\ \longleftrightarrow\ $ a triple of numbers $(x,y,z)$.

Remember, the three variables or quantities represented by the three axes may be unrelated. Then our visualization will remain valid with rectangles instead of the squares and boxes instead of cubes.

What about dimension $4$ and higher?

We cannot use our physical space as a reference anymore! We can't use it for visualization either. The space is abstract.

The idea of the $n$-dimensional space remains the same; it is the correspondence:

  • a location $P\ \longleftrightarrow\ $ a string of $n$ real numbers $(x_1,x_2,x_3,...,x_n)$.

Using the same letter with subscripts is preferable even for dimension $3$ as the symmetries between the axes and variables are easier to detect and utilize. However, using just $P$ is often even better!

Because of the difficulty or even impossibility of visualization of these “locations” in dimension $4$ this correspondence become much more than just a way to go back and forth whenever convenient. This time we just say: “it's the same thing”.

Example. This may be the data continuously collected by a weather center: $$\begin{array}{llll} 1&2&3&4&5&...\\ \hline \text{temperature}&\text{pressure}&\text{precipitation}&\text{humidity}&\text{sunlight}&...\\ \end{array}$$ They are all measured in different units and cannot be seen as an analog of our physical space. $\square$

How do we visualize an $n$-dimensional space? Let's first realize that, in a sense, we have failed even with the three-dimensional space! We had to squeeze these three dimensions on a two-dimensional piece of paper.

Axes as shadows.png

At best, we can see the shadows of the axes...

These are the spaces we will study and the notation for them:

  • ${\bf R}$, all real numbers;
  • ${\bf R}^2$, all pairs of real numbers (plane);
  • ${\bf R}^3$, all triples of real numbers (space);
  • ${\bf R}^4$, all quadruples of real numbers;
  • ...
  • ${\bf R}^n$, all strings of $n$ real numbers;
  • ...

Each of them is supplied with its own algebra and geometry.

We can build these consecutively adding one dimension at a time.

  • if ${\bf R}$ is given, we treat it as the $x$-axis and then add another axis, the $y$-axis, perpendicular to the first;
  • the result is ${\bf R}^2$, which we treat the $xy$-plane and then add another axis, the $z$-axis, perpendicular to the the first two;
  • the result is ${\bf R}^3$, which we treat the $xyz$-space and then add another axis perpendicular to the the first three; and so on.

Let's consider ${\bf R}^4$. This space is abstract but still made of copies of ${\bf R}$: the four coordinate axes. We just don't have a place to put them in so that we could see them... It is also seen to be made of copies of ${\bf R}^2$: the six coordinate planes. There are also six copies of ${\bf R}^3$:

Coordinate system dim 4.png

So, these spaces aren't unrelated!

In fact, we can see many copies of ${\bf R}^m$ in ${\bf R}^n$ with $n>m$. For example, the plane can be seen as a row of lines parallel to either of the coordinate axes:

Xy-plane as a stack.png

These lines are given by the equations: $x=a$ and $y=b$ over all real $a,b$. They are copies of ${\bf R}$ and will have the same algebra and geometry. Now, ${\bf R}^3$ is a stack of planes parallel to one of the coordinate planes:

Xyz-space as a stack.png

They are given by the equations: $x=a$, $y=b$, and $z=c$. These are copies of ${\bf R}^2$. Now, ${\bf R}^4$ is a “stack” of ${\bf R}^3$s... How they fit together is hard to visualize but they are still copies of ${\bf R}^3$ given by equations: $x_1=a_1$, $x_2=a_2$, etc.

3 Geometry of distances

The axes of the Cartesian system ${\bf R}^3$ for our physical space refer to the same: distances to the coordinate planes. They are (or should be) measured in the same unit. Even though, in general, the axes of ${\bf R}^n$ refer to unrelated quantities, they may be measured in the same unit, such as the prices of $n$ commodities being traded. When this is the case, doing geometry in ${\bf R}^n$ based entirely on the coordinates of points is possible.

A Cartesian system has everything in the space pre-measured.

Ruler.png

In particular, we compute (rather than measure) the distances between locations because the distance can be expressed in terms of the coordinates of the locations.

Coordinate system dim 1 -- distance formula.png

Theorem (Distance Formula for dimension $1$). The distance from point $P$ to point $Q$ in ${\bf R}$ given by real numbers $x$ and $x'$ respectively is $$d(P,Q)=|x-x'|.$$

Here, the geometry problem of finding distances relies on the algebra of real numbers (the subtraction).

The idea of how to compute how close two locations are to each other comes up very early in calculus. Recall the definition of the limit of a sequence:

  • $a_n\to a$ if for any $\varepsilon >0$ there is $N$ such that $|a_n-a|<\varepsilon$ whenever $n>N$.

Moreover, in the definition of the limit of a function this expression appears twice, for $x$- and $y$-axes:

  • $\lim_{x\to a}f(x)=L$ if for any $\varepsilon >0$ there is $\delta >0$ such that $|f(x)-L|<\varepsilon$ whenever $0<|x-a|<\delta$.

Both sequences (and their limits) and functions (and their limits) appear in multidimensional spaces and will be treated in this manner.

Now the coordinate system for dimension $2$. The formula for the distance between locations $P$ and $Q$ in terms of their coordinates $(x,y)$ and $(x',y')$ is found by using the distance formula from the $1$-dimensional case for either of the two axis in order to find

  • the distance between $x$ and $x'$, which is $|x-x'|=|x'-x|$, and
  • the distance between $y$ and $y'$, which is $|y-y'|=|y'-y|$,

respectively.

Coordinate system dim 2 -- distance formula.png

Then the two number are put together by the Pythagorean Theorem taking into account this simplification: $$|x-x'|^2=(x-x')^2,\ |y-y'|^2=(y-y')^2.$$

Theorem (Distance Formula for dimension $2$). The distance between points $P$ and $Q$ in ${\bf R}^2$ with coordinates $(x,y)$ and $(x',y')$ respectively is $$d(P,Q)=\sqrt{(x-x')^2+(y-y')^2}.$$

The two exceptional cases when $P$ and $Q$ lie on the same vertical or the same horizontal line (and the triangle “degenerate” into a segment) are treated separately.

Now the coordinate system for dimension $3$. We can guess that there will be another term in the sum of the Distance Formula (the proof was provided in Part II).

Theorem (Distance Formula for dimension $3$). The distance between points $P$ and $Q$ in ${\bf R}^3$ with coordinates $(x,y,z)$ and $(x',y',z')$ respectively is $$d(P,Q)=\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}.$$

A pattern starts to appear: there is the square root of the sum of the squares of the distances for each of the coordinates. Thinking by analogy we continue to include the case of dimension $4$: $$\begin{array}{l|ll|ll} \text{dimension}&\text{points}&\text{coordinates}&\text{distance}\\ \hline 1&P&x&\\ &Q&x'&d(P,Q)=\sqrt{(x-x')^2}\\ \hline 2&P&(x,y)&\\ &Q&(x',y')&d(P,Q)=\sqrt{(x-x')^2+(y-y')^2}\\ \hline 3&P&(x,y,z)&\\ &Q&(x',y',z')&d(P,Q)=\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}\\ \hline 4&P&(x_1,x_2,x_3,x_4)&\\ &Q&(x'_1,x'_2,x'_3,x'_4)&d(P,Q)=\sqrt{(x_1-x'_1)^2+(x_2-x'_2)^2+(x_3-x'_3)^2+(x_4-x'_4)^2}\\ \hline ...&... \end{array}$$ There are $n$ terms in dimension $n$: $$\begin{array}{l|ll|lll} \text{dimension}&\text{points}&\text{coordinates}&\text{distance}\\ \hline n&P&(x_1,x_2,...,x_n)&\\ &Q&(x'_1,x'_2,...,x'_n)&d(P,Q)=\sqrt{(x_1-x'_1)^2+(x_2-x'_2)^2+...+(x_n-x'_n)^2}\\ \hline \end{array}$$

The formula for $n=1,2,3$ is justified by what we know about the physical space. What about $n=4$ and above? Let's take a look at the copies of ${\bf R}^3$ that make up ${\bf R}^4$. One of them is given by $x_4=a_4$ for some real number $a_4$. If we take any two points $P,Q$ within it, the formula becomes: $$\begin{array}{ll} d(P,Q)&=\sqrt{(x_1-x'_1)^2+(x_2-x'_2)^2+(x_3-x'_3)^2+(a_4-a_4)^2}\\ &=\sqrt{(x_1-x'_1)^2+(x_2-x'_2)^2+(x_3-x'_3)^2}. \end{array}$$ In other words, the distance is the same as the one for dimension $3$. We conclude that the geometry of such a copy of ${\bf R}^3$ is the same as the “original”!

Exercise. Show that the geometry of any plane in ${\bf R}^3$ and ${\bf R}^4$ is the same as that of ${\bf R}^2$.

In general, we progress from understanding the geometry of $X={\bf R}^n$ to that of $Y={\bf R}^{n+1}$ by adding an extra axis -- perpendicular to the rest -- to ${\bf R}^n$. Then, the Pythagorean Theorem is applied:

Distance formula Rn proof.png

We thus replace the study the complex geometry of locations in a multi-dimensional space with a study of distances, i.e., numbers.

Exercise. State the definition of the limits of (a) a parametric curve, (b) a function of two variables.

Next, we would like to formulate three very simple properties of the distances that apply equally to all dimensions, without reference to the formulas. First, the distances can't be negative and, moreover, for the distance to be zero, the two points have to be the same. Second, the distance from $P$ to $Q$ is the same as the distance from $Q$ to $P$. And so on.

Theorem (Properties of Distance). Suppose $P,Q,S$ are points in ${\bf R}^n$. Then the following properties are satisfied:

  • Positivity: $d(P,Q)\ge 0$; and $d(P,Q)=0$ if and only of $P=Q$;
  • Symmetry: $d(P,Q)=d(Q,P)$;
  • Triangle Inequality: $d(P,Q)+d(Q,S)\ge d(P,S)$.

Proof. $\blacksquare$

Triangle Inequality.png

We know the last property from Euclidean geometry. We can justify it for dimension $n\ge 4$ by referring to the following fact: any three points lie within a single plane. This fact brings us back to Euclidean geometry... if that's what we want.

Example (city blocks). The Distance Formula for the plane gives us the distance measured along a straight line as if we are walking through a field. But what if we are walking through a city? We then cannot go diagonally and have to follow the streets. This fact dictates how we measure distances. To find the distance between two locations $P=(x,y)$ and $Q=(u,v)$, we go along the grid: $$d_T(P,Q)= |x-u|+|y-v|.$$

Euclidean vs taxicab metric.png

It is called the taxicab metric. $\square$

The example helps us realize that the Distance Formula is in fact a definition. In other words, for a given space of locations -- one of ${\bf R}^n$s -- we choose a specific way to compute distances from coordinates.

Definition. The Euclidean distance between points $P$ and $Q$ in ${\bf R}^n$ with coordinates $(x_1,x_2,...,x_n)$ and $(x'_1,x'_2,...,x'_n)$ respectively (or simply the distance) is defined to be $$d(P,Q)=\sqrt{\sum_{k=1}^{n}(x_n-x_n')^2}.$$ We refer to the formula as the Euclidean metric. The space ${\bf R}^n$ equipped with the Euclidean metric is called the $n$-dimensional Euclidean space.

When we deal with the “physical space” ($n=1,2,3$) as in the above theorems, the Euclidean metric is implied. For the “abstract spaces” ($n=1,2,...$), the Euclidean metric is the default choice; however, there are many examples when the Euclidean geometry and, therefore, the Euclidean metric (aka the Distance Formula) don't apply.

Example (prices). Let's consider the prices of wheat and sugar again. The space of prices is the same, ${\bf R}^2$. However, measuring the distance between two combinations of prices with the Euclidean metric leads to undesirable effects. For example, such a trivial step as changing the latter from “per ton” to “per kilogram” will change the geometry of the whole space. It is as if the space is stretched vertically. As a result, in particular, point $P$ that used to be closer to point $A$ than to $B$ might now satisfy the opposite condition. $\square$

Example (attributes). The problem is that the two (or more) measurements or other attributes might have nothing to do with each other. In extreme cases, they even have different units! For example, we might compare two persons built based to the two main measurements: weight and height. Unfortunately, if we substitute such numbers into our formula, we will be adding pounds to feet! $\square$

Compare:

  • a circle on the plane is defined to be the set of all point a given distance away from its center;
  • a sphere in the space is defined to be the set of all point a given distance away from its center.

What about higher dimensions? The pattern is clear:

  • a hypersphere in ${\bf R}^n$ is defined to be the set of all point a given distance away from its center.

In other words, each point $P$ on the hypersphere satisfies: $$d(P,Q)=R,$$ where $Q$ is its center and $R$ is its radius.

Example (Newton's Law of Gravity). According to the law, the force of gravity between two objects is

  • 1. proportional to either of their masses,
  • 2. inversely proportional to the square of the distance between their centers.

In other words, the force is given by the formula: $$F = G \frac{mM}{r^2},$$ where:

  • $F$ is the force between the objects;
  • $G$ is the gravitational constant;
  • $m$ is the mass of the first object;
  • $M$ is the mass of the second object;
  • $r$ is the distance between the centers of the mass of the two.

We can treat this formulas as a function. This may be seen as a function of the three variables listed above, like this: $$F(m,M,r) = G \frac{mM}{r^2}.$$ However, the dependence of $F$ on $n$ and $M$ is very simple and, furthermore, we can assume that the masses of planets are remain the same. The third variable is more interesting especially because it depends on the location $P$ of the second object in the $3$-dimensional space: $$r=d(O,P),$$ if, for simplicity, we assume that the first object is located at the origin. Note that this force is constant along any of the spheres centered at $O$. Now, we can re-write the law as a function of three variables: $$F(x,y,z) =\frac{G mM}{d(O,P)^2}=\frac{G mM}{\left(\sqrt{x^2+y^2+z^2}\right)^2}=\frac{G mM}{x^2+y^2+z^2},$$ where the three spacial variables $x,y,z$ are the coordinates of $P$. If we ignore the third variable ($z=0$), we can plot the graph of the resulting function of two variables:

Gravity graph.png

The information about the asymptotic behavior of the original function, $$F(x)\to +\infty \text{ as } r\to 0\text{ and }F(r)\to 0 \text{ as } r\to +\infty,$$ can now be restated: $$F(P)\to +\infty \text{ as } d(O,P)\to 0\text{ and }F(P)\to 0 \text{ as } d(O,P)\to +\infty.$$ This idea is developed in the next section. $\square$

4 Sequences and topology in ${\bf R}^n$

Recall from Part I the image of a a ping-pong ball bouncing off the floor. Recording its height every time gives us a sequence:

TT ball bounce.png

It is a sequence of numbers!

Imagine now watching a ball bouncing on an uneven surface with its locations recorded at equal periods of time. The result is a sequence of locations on the plane:

Drain.png

We will study infinite sequences of locations and especially their trends. An infinite sequence will be sometimes “accumulating” around a single location. The gap between the ball and the drain becomes invisible!

Every function $X=f(t)$ with an appropriate domain creates a sequence: $$A_k=f(k).$$ A function defined on a ray in the set of integers, $\{p,p+1,...\}$, is called an infinite sequence, or simply sequence with the abbreviated notation: $$A_k=\{A_k\}.$$ We visualize sequences as sequences of points on the $xy$-plane.

Example. The go-to example is that of the sequence made of the reciprocals: $$A_k=\left( \frac{\cos k}{k},\ \frac{\sin k}{k} \right).$$ It tends to $0$ while spiraling around it.

Reciprocal spiral.png

This fact is easily confirmed numerically. $\square$

Unfortunately, not all sequences are as simple as that. They may approach their respective limits in an infinite variety of ways. And then there are sequences with no limits. We need a more general approach.

Let's recall from Part I the idea of the convergence of (infinite) sequences of real numbers:

Sequence accumulation.png

The idea is that the distance from the $k$th point to the limit is getting smaller and smaller: $$|x_k-a|\to 0 \text{ as }k\to \infty.$$ In the $n$-dimensional case, the idea remains the same: a sequence of points in ${\bf R}^n$ is getting closer and closer to its limit, which is also a point in ${\bf R}^n$:

Convergence of sequence.png

We re-write what we want to say about the meaning of the limits in progressively more and more precise terms. $$\begin{array}{l|ll} k&X=A_k\\ \hline \text{As } k\to \infty, & \text{we have } X\to A.\\ \text{As } k\text{ approaches } \infty, & X\text{ approaches } A. \\ \text{As } k \text{ is getting larger and larger}, & \text{the distance from }X \text{ to } A \text{ approaches } 0. \\ \text{By making } k \text{ larger and larger},& \text{we make } d(X,A) \text{ as small as needed}.\\ \text{By making } k \text{ larger than some } N>0 ,& \text{we make } d(X,A) \text{ smaller than any given } \varepsilon>0. \end{array}$$ Then, the following condition holds:

  • for each real number $\varepsilon > 0$, there exists a number $N$ such that, for every natural number $k > N$, we have

$$d(A_k , A) < \varepsilon .$$

Our study become much easier once we realize that distances are numbers and $d(A_k , A)$ is just a sequence of numbers! Understanding distances in ${\bf R}^n$ and numerical sequences allows us easily sort this out.

Definition. Suppose $\{A_k:\ k=1,2,3...\}$ is a sequence of points in ${\bf R}^n$. We say that the sequence converges to another point $A$ in ${\bf R}^n$, called the limit of the sequence, if: $$d(A_n,A)\to 0\text{ as }k\to \infty,$$ denoted by: $$A_k\to A \text{ as }k\to \infty,$$ or $$A=\lim_{k\to \infty}A_k.$$ If a sequence has a limit, then we call the sequence convergent and say that it converges; otherwise it is divergent and we say it diverges.

In other words, the points start to accumulate in smaller and smaller circles around $A$. A way to visualize a trend in a convergent sequence is to enclose the tail of the sequence in a disk:

Definition of limit Rn.png

It should be, in fact, a narrower and narrower band; its width is $2\varepsilon$. Meanwhile, the starting point of the band moves to the right; that's $N$.

Examples of divergence are below.

Example. A sequence may tend to infinity, such as $A_k=(k,k)$ at the simplest:

Identity sequence Rn.png

Then no disk -- no matter how large -- will contain the sequence's tail. $\square$

This behavior however has a meaningful pattern.

Definition. We say that a sequence $A_n$ tends to infinity if the following condition holds:

  • for each real number $R$, there exists a natural number $N$ such that, for every natural number $k > N$, we have

$$d(0,A_n) >R.$$

We describe such a behavior with the following notation: $$A_k\to \infty \text{ as } k\to \infty .$$

We need to justify “the” in “the limit”.

Theorem (Uniqueness). A sequence can have only one limit (finite or infinite); i.e., if $A$ and $B$ are limits of the same sequence, then $A=B$.

Proof. The geometry of the proof is the following: we want to separate these two points by two non-overlapping disks. Then the tail of the sequence would have to fit one or the other, but not both. In order for them to be disjoint, their radii (that's $\varepsilon$!) should be less than half the distance between the two points.

Proof of uniqueness of limit of sequence Rn.png

The proof is by contradiction. Suppose $A$ and $B$ are two limits, i.e., either satisfies the definition. Let $$\varepsilon = \frac{d(A,B)}{2}.$$ Now, we write the definition twice:

  • there exists a number $L$ such that, for every natural number $k > L$, we have

$$d(A_k, A) < \varepsilon ,$$ and

  • there exists a number $M$ such that, for every natural number $k > M$, we have

$$d(A_k, B) < \varepsilon .$$ In order to combine the two statements we need them to be satisfied for the same values of $k$. Let $$N=\min\{ L,M\}.$$ Then,

  • for every number $k > N$, we have

$$d(A_k , A) < \varepsilon \text{ and }d(A_k, B) < \varepsilon .$$ In particular, for every $k > N$, we have by the Triangle Inequality: $$d(A,B)\le d(A , A_k)+d(A_k , B) < \varepsilon+\varepsilon<2\varepsilon.$$ A contradiction. $\blacksquare$

Exercise. Follow the proof and demonstrate that that it is impossible to for a sequence to have as limit: a point and infinity.

Thus, there can be no two limits and we are justified to speak of the limit.


Topology convergent sequence.png

What is the $n$-dimensional analog of a closed interval?

Open and closed.png

Compare the disk and the disk with its boundary (the circle) removed: $$\{(x,y):\ x^2+y^2\le 1\}\ \text{ vs. }\ \{(x,y):\ x^2+y^2< 1\}.$$ Or think of the ball and the ball with its boundary (the sphere) removed. The difference is that in the latter one can reach the boundary -- and the outside of the set -- by following a sequence that lies entirely inside the set!

Definition. A set in ${\bf R}^n$ is called closed if it contains the limits of all of its convergent sequences.

Closure is closed.png

Definition. A set $S$ in ${\bf R}^n$ is bounded if it fits in a sphere (or a box) of a large enough size: $$d(x,0) < Q \text{ for all } x \text{ in } S.$$

Bounded set.png

5 The coordinate-wise treatment of sequences

Here we go back from the treatment of the space to the spread-out coordinate-wise approach.

Suppose space ${\bf R}^n$ is supplied with a Cartesian system.

Let's first look at the definition of the limit of a sequence $X_k$ in ${\bf R}^n$. The limit is defined to be such a point $A$ in ${\bf R}^n$ that $$d(X_k,A)\to 0\text{ as } k\to \infty.$$

Suppose we use the Euclidean metrics is our space ${\bf R}^3$, what does the above condition mean? Suppose $$X_k=\big( x_k,y_k,z_k \big)\text{ and }A=(a,b,c).$$ Then we have: $$\sqrt{(x_k-a)^2+(y_k-b)^2+(z_k-c)^2}\to 0.$$

Convergence of sequence from various directions.png

This limit is equivalent to the following: $$(x_k-a)^2+(y_k-b)^2+(z_k-c)^2\to 0,$$ because the function $u^2$ is continuous at $0$. Since these three terms all non-negative, all three have to approach $0$! Then we have: $$(x_k-a)^2\to 0,\ (y_k-b)^2\to 0,\ (z_k-c)^2\to 0.$$ These limits are equivalent to the following: $$|x_k-a|\to 0,\ |y_k-b|\to 0,\ |z_k-c|\to 0,$$ because the function $\sqrt{u}$ is continuous from the right at $0$. Finally, we have: $$x_k\to a,\ y_k\to b,\ z_k\to c.$$ All coordinate sequences converge!

For the $n$-dimensional case, build a table: $$\begin{array}{c|ccccc} &1&2&...&n\\ \hline X_k&x_k^1&x_k^2& ...&x_k^n\\ \downarrow&\downarrow&\downarrow&...&\downarrow&\text{ as } k\to\infty\\ A&a^1&a^2& ...&a^n \end{array}$$ The following is a summary.

Theorem (Coordinate-wise convergence). A sequence of points $X_k$ in ${\bf R}^n$ converge to another point $A$ if and only if every coordinate of $X_k$ converges to the corresponding coordinate of $A$; i.e., $$X_k\to A \text{ as } k\to \infty\ \Longleftrightarrow\ x_k^i\to a^i \text{ as } k\to \infty \text{ for all } i=1,2,...,n,$$ where $$X_k=(x_k^1,\ x_k^2,\ ...,\ x_k^n) \text{ and } A=(a^1,\ a^2,\ ...,\ a^n).$$

Exercise. Prove the “if” part of the theorem.

Example. We compute using the Continuity Rule for numerical sequences: $$\begin{array}{lll} \lim_{k\to \infty}\left( \cos \frac{1}{k},\ \sin \frac{1}{k} \right)&=\left( \lim_{k\to \infty}\cos \frac{1}{k},\ \lim_{t\to \infty} \sin \frac{1}{k} \right)\\ &=(\cos 0,\ \sin 0)\\ &=(1,0), \end{array}$$ because $\sin t$ and $\cos t$ are continuous. $\square$

This theorem also makes it easy to prove the algebraic properties of limits from those for numerical functions.

6 Vectors

If $P$ and $Q$ are two locations then we can say that $PQ$ is the displacement (from $P$ to $Q$). The idea applies to any space ${\bf R}^n$ but we will start with the physical space devoid of a Cartesian system.

From this point of view, a vector is a pair, $PQ$, of locations $P$ and $Q$.

Two locations and the displacement.png

It is an ordered pair so that $$PQ\ne QP.$$

The locations and displacements and, therefore, points and vectors are subject to algebraic operations: $$P+PQ=Q.$$ As you can see, we add a vector to a point that is its initial point and the result is its terminal point.

We can continue from points to point as if following a parametric curve.

Vectors as displacements.png

This how we can understand addition of vectors as displacements: $$\begin{array}{ccll} \text{initial location}&\text{displacement}&\text{terminal location}\\ \hline P&PQ&Q=P+PQ\\ Q&QR&R=Q+QR&=P+PQ+QR\\ R&RS&S=R+RS&=P+PQ+QR+RS\\ S&ST&T=S+ST&=P+PQ+QR+RS+ST \end{array}$$ For the general case of $m$ steps, we have: $$\sum_{k=0}^mX_kX_{k+1}=X_m-X_0,$$ which, with the velocities denoted by $V_k,\ k=1,2,...,m$, respectively, takes the form: $$\sum_{k=0}^mV_k\Delta t=X_m-X_0.$$ The formula resembles -- and not by coincidence -- the Fundamental Theorem of Calculus!

The construction suggests the following “head-to-tail” alignment of vectors for addition: $$PR=PQ+QR.$$ However, we will follow rule that says:

  • two vector can only be added when they have the same initial point,

and use the above construction as one of the ways to find the sum.

We think about vectors as a line segment in a Euclidean space ${\bf R}^n$ with $n=1,2,3$. AS such, it has a direction and the length. It is possible to have both the direction and the length to be $0$; that's the zero vector.

To add two of such vectors, we follow either

  • the head-to-tail construction, or
  • the parallelogram construction,

illustrated below.

Adding vectors.png

For the former, we make a copy $B'$ of $B$, attach to the end of $A$, and then create a new vector with the initial point that of $A$ and terminal point that of $B'$. For the latter, we make a copy $B'$ of $B$, attach to the end of $A$, also make a copy $A'$ of $A$, attach to the end of $B$. Then the sum $A+B$ of two vectors $A$ and $B$ with the same initial point is the vector with the same initial point that is the diagonal of the parallelogram with sides $A$ and $B$. The result is the same according to what we know from Euclidean geometry.

Once we know who to add, we know subtraction is the inverse operation. Indeed, given vectors $A$ and $B$, we need to find the vector $C$ such that $B+C=A$.

Subtraction of vectors.png

In other words, $A-B$ is found by constructing the vector, $C$, from the end of $B$ to the end of $A$ and then making a copy of $C$ with the same starting point as $A$.

There is another algebraic operation on vectors. A force might keep its direction but change its magnitude! It might also change the direction to the opposite. The actual construction is nothing but stretching of the vector:

Scalar multiplication of vectors.png

Thus the scalar product $cQ$ of a vector $A$ and a real number $c$ is the vector with the same initial point as $A$, with the direction which is

  • same as that of $A$ when $c>0$,
  • opposite to that of $A$ when $c<0$, and
  • zero when $c=0$;

and the length equal to the length of $A$ multiplied by $|c|$.

Consider:

  • displacement $=$ time $\cdot$ velocity.

Both the displacement and the velocity are vectors while the time is a number (aka a scalar): $$\Delta X=\Delta t \cdot V.$$ This idea -- via limits -- will take us to the notion of the derivative of a parametric curve at a given point, a vector!

Warning: When units are involved, the vectors $\Delta X$ and $V$ belong to different spaces.

Note that both vector operations, even repeated many times, will always produce vectors with the same initial point!

What is the dimension of the space? As we know from Euclidean geometry, two lines and, therefore, two vectors, determine a plane. This is why the dimension doesn't matter because the situation is always $2$-dimensional.

Addition and scalar multiplication are 2-dim.png

Note that we can also represent the displacement from $P$ to $Q$ algebraically: $$PQ=Q-P.$$ Therefore, we can add a vector to a point as long as the point is the initial point of the vector: $$P+PQ=Q.$$ Furthermore, $QP=P-Q$, therefore, $QP=-PQ$. So, even though we don't do algebra with vectors when they don't have the same initial point, they aren't unrelated.

In the physical world, there are “metaphors” for vectors in addition to displacements.

First, we look at velocities as vectors. We can look at the velocity as the rate of change of the location, or independently. Velocities appear as the wind speed:

Wind as vector field.png

or as the velocities of particles in a stream:

Canal - boat.png

If one to swim or row across this stream, his velocity is combined with the velocity of the stream. That's vector addition! If we want to go faster, we row twice as hard. That's scalar multiplication!

Let's also look at forces as vectors. We can look at the force as a multiple of the acceleration ($F=ma$) or independently. For example, springs attached to an object will pull it in their respective directions:

Springs and ball.png

We use vector addition to find the combined force as if produced by a single spring. Furthermore, one can attach two springs in a consecutive manner to double the force or cut any portion of the spring to reduce the force proportionally with the result computed using scalar multiplication.

7 Algebra of vectors

In dimension $1$, the algebra of vector is the algebra of real numbers. However, even without a Cartesian system, we have the algebra of signed lengths. The addition is illustrated below:

Algebra of signed lengths.png

Scalar multiplication is just stretching.

Scalar multiplication of vectors (dimension above $1$) is similar to that.

First, we can apply two (or more) scalar multiplications in a row. Given a vector $A$ and real numbers $a$ and $b$, we create new vectors: $B=aA$ from $A$ and then $C=bB$ from $B$. Since scale multiplication is just stretching we can combine the two into one and create $C$ directly from $A$: $C=(ab)A$.

Scalar multiplication -- associativity.png

Therefore, we have the Associativity Property of Scalar Multiplication: $$b(aA)=(ba)A.$$

The following simple property, $$A+A=2A,$$ connects vector addition to scalar multiplication. Its generalization is the First Distributivity Property of Vector Algebra: $$(a+b)A=aA+bA.$$

Vector algebra -- distributivity 1.png

In other words, we distribute scalar multiplication over addition of real numbers.

Next, we distribute multiplication of real numbers over addition of vectors. The Second Distributivity Property of Vector Algebra is: $$a(A+B)=aA+aB.$$

Vector algebra -- distributivity 2.png

Recall that to find $A+B$ we make a copy $B'$ of $B$, attach to the end of $A$, and then create a new vector with the initial point that of $A$ and terminal point that of $B'$. Now, to find $B+A$ we make a copy $A'$ of $A$, attach to the end of $B$, and then create a new vector with the initial point that of $B$ and terminal point that of $A'$.

Commutativity of addition of vectors.png

From what we know from Euclidean geometry -- two triangles with two identical sides and identical angle between them are identical -- these two triangles form a parallelogram. It has sides $A,B,A',B'$. Therefore, we have the Commutativity Property of Vector Addition: $$A+B=B+A.$$

We have the Associativity Property of Vector Addition: $$A+(B+C)=(A+B)+C.$$

Associativity of addition of vectors.png

There are special numbers and there are special vectors...

There is $0$, the real number, and then there is $0$, the vector. The latter is called the zero vector and can mean no displacement, no motion (zero velocity), forces that cancel each other, etc. The two are related: $$0\cdot A=0.$$ We have here: $$\text{number }\cdot \text{ vector }=\text{ vector }.$$ Similarly, $$1\cdot A=A.$$ Of course, we have (all vectors): $$A+0=A.$$

Taken together, these properties of vectors match the property of numbers perfectly! This means that all manipulations of algebraic expressions that we have done with numbers are still allowed -- even when the expression includes vectors -- as long as the expression itself makes sense. In other words, we just need to avoid operations that haven't been defined: no multiplication (for now) of vectors, no division of vectors, no adding numbers to vectors, etc.

As a summary...

Theorem (Vector Operations). The two operations -- addition of two vectors and multiplication of a vector by a scalar -- satisfy the following properties:

  1. $X+Y=Y+X$ for all $X$ and $Y$;
  2. $X+(Y+Z)=(X+Y)+Z$ for all $X$, $Y$, and $Z$;
  3. $X+0=X=0+X$ for some vector $0$ and all $X$;
  4. $X+(-X)=0$ for any X and some vector $-X$;
  5. $a(bX)=(ab)X$ for all $X$ and all scalars $a, b$;
  6. $1X=X$ for all $X$;
  7. $a(X+Y)=aX+aY$ for all $X$ and $Y$;
  8. $(a+b)X=aX+bX$ for all $X$, $Y$ and all scalars $a, b$.

8 Components of vectors

We now understand vectors -- at least in the lower-dimensional setting. Now we add a Cartesian system to these spaces and also include the abstract spaces of arbitrary dimensions, ${\bf R}^n$.

A vector is a pair $PQ$ of points in ${\bf R}^n$, either of which corresponds to a string of $n$ numbers called coordinates.

On the line ${\bf R}^1$, points are numbers and the vectors are simply differences of these numbers: $$PQ=Q-P.$$

On the plane ${\bf R}^2$, we might have: $$P=(1,2) \text{ and } Q=(2,5).$$ How can we express vector $PQ$ in terms of these four numbers?

Components of vectors.png

We look at the change from $P$ to $Q$:

  • the change with respect to $x$, which is $2-1=1$, and
  • the change with respect to $y$, which is $5-2=3$.

We combine these to into a new pair of numbers (with triangular brackets to distinguish these from points): $$PQ=<1,3>.$$ Technically, however, for a complete information on a vector we have to mention its initial point $P=(1,2)$.

Now in ${\bf R}^3$, we might have: $$P=(x,y,z) \text{ and } Q=(x',y',z').$$ How can we express vector $PQ$ in terms of these six numbers?

Coordinate system dim 3 -- vectors.png

There are three changes (differences) along the three axes, i.e., a triple: $$PQ=<x'-x,y'-y,z'-z>.$$

Definition. A vector $PQ$ in ${\bf R}^n$ with its initial point $$P=(x_1,x_3,...,x_n)$$ and its terminal point $$Q=(x_1',x_2',...,x_n')$$ is given by the string of $n$ numbers called the components of the vector: $$x_1'-x_1,\ x_2'-x_3,\ ...,\ x_n'-x_n.$$

A vector may emerge from its initial and terminal points or independently. In either case, we assemble the components according to the following row notation: $$A=<a_1,a_2,...,a_n>,$$ or the column notation: $$A=\left[\begin{array}{ccc} a_1\\ a_2\\ ...\\ a_n\end{array}\right].$$

Once again we can only carry out vector addition on vectors with the same initial point. What happens if we change the initial point while leaving the components of the vectors intact? Not only each vector are “copied” but so are the results of the algebraic operations. They are the same just shifted to a new location:

A+B+C.png

It is then sufficient to provide results for the vectors that start at the origin $O$ only! Only these are allowed:

All vectors.png

In that case, the components of a vectors are simply the coordinates of its end: $$\begin{array}{rlccccccc} P&=(x_1,&x_3,&...,&x_n)& \Longrightarrow\\ OP&=<x_1,&x_2,&...,&x_n>. \end{array}$$

Next we consider the familiar algebraic operations but this time the vectors are represented by their components.

We carry out operations component-wise.

We demonstrate these operations for dimension $n=3$ and for both row and column style of notation. Vector addition: $$\begin{array}{rcccccc} A&=<&x,&y,&z&>\\ +\\ B&=<&u,&v,&w&>\\ \hline A+B&=<&x+u,&y+v,&z+w&> \end{array},\qquad \left[\begin{array}{c}x\\y\\z\end{array}\right]+ \left[\begin{array}{c}u\\v\\w\end{array}\right]= \left[\begin{array}{c}x+u\\y+v\\z+w\end{array}\right]. $$ Scalar multiplication: $$\begin{array}{rcccccc} A&=<&x,&y,&z&>\\ \times\\ c&\\ \hline cA&=<&cx,&cy,&cz&> \end{array},\qquad c\cdot\left[\begin{array}{c}x\\y\\z\end{array}\right]= \left[\begin{array}{c}cx\\cy\\cz\end{array}\right]. $$ In either case the components are aligned. Even though both seems equally convenient, the former will be seen as an abbreviation of the latter.

We thus have the algebra of vectors for a space of any dimension! These operations can be proven to satisfy the same properties as the vectors in ${\bf R}^3$ as discussed in the last section. The proof is straight-forward and relies on the corresponding property of real numbers. For example, to prove the commutativity of vector addition we use the commutativity of addition of numbers as follows: $$A+B=\left[\begin{array}{c}x\\y\\z\end{array}\right]+ \left[\begin{array}{c}u\\v\\w\end{array}\right]= \left[\begin{array}{c}x+u\\y+v\\z+w\end{array}\right]= \left[\begin{array}{c}u+x\\v+y\\w+z\end{array}\right]= \left[\begin{array}{c}u\\v\\w\end{array}\right]+ \left[\begin{array}{c}x\\y\\z\end{array}\right]=B+A.$$

The reason for the word “component” becomes clear: a vector $A$ is decomposed into the sum of vectors each of which is aligned with one of the axes; for example, $$A=<3,2>=<3,0>+<0,2>.$$ They are called the component vectors of $A$. We can take this analysis one step further with scalar multiplication: $$A=<3,2>=<3,0>+<0,2>=3<1,0>+2<0,1>.$$ Then, similarly, any vector can be represented in such a way: $$<a,b>=a<1,0>+b<0,1>.$$

Coordinate system dim 2 -- basis.png

Thus, representing a vector in terms of its components is just a way (a single way in fact) to represent it in terms of a pair of specified unit vectors aligned with the axis.

We use the following notation for these vectors in ${\bf R}^2$ and ${\bf R}^3$: $$i=<1,0,0>,\ j=<0,1,0>,\ k=<0,0,1>.$$ They are called the basis vectors. For every vector, we have the following: $$<a,b,c>=ai+bj+ck.$$

Coordinate system dim 3 -- basis.png

The basis vectors in ${\bf R}^n$ are denoted by $E_1,E_2,...,E_n$.

Of course, choosing a different Cartesian system will produce a new set of basis vectors! We can now understand the Cartesian system as the origin and the basis vectors instead of the origin and the axes. The choice of the basis vectors is dictated by the problem to be solves.

Example. Suppose we are to study the motion of an object sliding down a slope. Even though the gravitation is pulling it vertically down, the motion is restricted to the surface of the slope and, therefore, is linear. It is then beneficial to choose the first basis vector $i$ to be parallel to the surface and the second $j$ perpendicular.

Motion on an incline.png

The gravity force is then decomposed into the sum of two vectors. It is the first one that affects the object and is to be analyzed in order to find the acceleration (the Second Newton's Law). The second is perpendicular and is cancelled by the resistance of the surface (the Newton's Third Law). $\square$

Example. Even when we deal with the abstract spaces ${\bf R}^n$, such decompositions may be useful. For example, an investment advice might be to hold the proportion of stocks and bonds $1$-to-$2$. We plot each possible portfolio as a point on the $xy$-plane, where $x$ is the amount of stocks and $y$ is the amount of bonds in it. In order to evaluate how well portfolios follow this advice we can choose the first basis vector to be $i=<2,1>$ and the second perpendicular to it, $j=<-1,2>$.

Stocks and bonds.png

Then the first coordinate -- with respect to this new coordinate system -- of your portfolio reflects how well you have followed the advice and the second how much you've deviated from it. $\square$

9 Lengths of vectors

A vector is a directed segment and one of its attributes is its length. The meaning of this number is clear if we look at the vector $PQ$ as two points $P$ and $Q$.

Length of vector.png

Definition. The length of a vector is defined to be the distance between its initial and terminal points: $$||PQ|| = d(P,Q).$$ This number is also called the magnitude or the norm of the vector.

The notation resembles the absolute value and not by accident: it is the same thing in the $1$-dimensional case, ${\bf R}$.

The properties of distance will give us the properties of the magnitude.

First, the Positivity:

  • $d(P,Q)\ge 0$; and $d(P,Q)=0$ if and only of $P=Q$.

It follows that

  • $||PQ||\ge 0$; and $||PQ||=0$ if and only of $P=Q$,

or

  • $||A||\ge 0$; and $||A||=0$ if and only of $A=0$.

Second, the Symmetry:

  • $d(P,Q)=d(Q,P)$.

It follows that

  • $||PQ||=||QP||$,

or

  • $||A||=||-A||$.

Third, the Triangle Inequality:

  • $d(P,Q)+d(Q,S)\ge d(P,S)$.

It follows that

  • $||PQ||+||QS||\ge ||PS||$,

or

  • $||A||+||B||\ge ||A+B||$.
Triangle Inequality for vectors.png

That how the magnitude interacts with vector addition. What about scalar multiplication?

There is another convenient property, the Homogeneity:

  • $||c\cdot A||=|c|\cdot ||A||$.
Homogeneity for vectors.png

These properties are applicable to all dimensions and are used to manipulate vector expressions.

Example (Newton's Law of Gravity). Recall that the gravity between two objects of masses $M$ and $m$ located at points $O$ and $P$ is a function of three variables: $$F (P)= G \frac{mM}{d(O,P)^2}.$$ But gravity is a force and, therefore, a vector. If vector $X$ gives the location of the second object, we can re-write the law as follows: $$||F(X)|| = G \frac{mM}{||X||^2}.$$ Let's derive the vector form of the law. The relevant part of the law states that the force of gravity affecting either of the two objects is

  • directed towards the other object.

In other words, $F$ is points in the opposite direction to $X$, i.e., $-X$. That's all we need except for the coefficient: $$F(X)=C(-X).$$ We now use Homogeneity to find it: $$G \frac{mM}{||X||^2}=||F(X)||=|C|\cdot ||(-X)||=|C|\cdot||X||.$$ The final form is the following: $$F(X)=-G\frac{mM}{||X||^3}X.$$ Now that both the input and the output are $3$-dimensional! How do we illustrate this new kind of function? First, we think of the input as a point and the output as a vector and then we attach the latter to the former. Below, we plot vector $F(X)$ starting at location $X$ on the plane ($z=0$):

Gravitation vector field.png

The limit description of the asymptotic behavior takes the following form: $$||F(X)||\to +\infty \text{ as } ||X||\to 0\text{ and }||F(X)||\to 0 \text{ as } ||X||\to +\infty.$$ $\square$

Whenever only the directions of the vectors matter, the following is a convenient concept.

Definition. For any vector $X\ne 0$, its unit vector is given by: $$\frac{X}{||X||}.$$ Also any vector of length $1$ is called a unit vector.

When a Cartesian system is provided, we have the Euclidean metrics, i.e., the distance between points $P$ and $Q$ in ${\bf R}^3$ with coordinates $(x,y,z)$ and $(x',y',z')$ respectively is $$d(P,Q)=\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}.$$

Coordinate system dim 3 -- vector length.png

Theorem. The length of vector $A$ with components $(a,b,c)$ is $$||A||=\sqrt{a^2+b^2+c^2}.$$

It is easy to demonstrate independently that this formula satisfies the four properties of the length:

  • $||A||\ge 0$; and $||A||=0$ if and only if $A=0$.
  • $||-A||=||A||$.
  • $||A||+||B||\ge ||A+B||$.
  • $||c\cdot A||=|c|\cdot ||A||$.

10 Parametric curves

Functions process an input of any nature and produce an output of any nature.

In general, we represent a function diagrammatically as a black box that processes the input and produces the output: $$ \newcommand{\ra}[1]{\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{ccccccccccccccc} \text{input} & & \text{function} & & \text{output} \\ x & \mapsto & \begin{array}{|c|}\hline\quad f \quad \\ \hline\end{array} & \mapsto & y \end{array}$$

Convention. We will use upper case letters for the functions the outputs of which are (or may be) multidimensional, such as points and vectors: $$F,\ G,\ P,\ Q,\ ...,$$ and lower case letters for the functions with numerical outputs: $$f,\ g,\ h,\ ...$$

Functions in multidimensional spaces take points or vectors as the input and produce points or vectors of various dimensions as the output. We can say that the input $X$ is in ${\bf R}^n$ and the output $U=F(X)$ of $X$ is in ${\bf R}^m$: $$\begin{array}{lll} F:&P&\mapsto&U\\ &\text{in }{\bf R}^n&&\text{in }{\bf R}^m \end{array}$$ Then, the domain of such a function is in ${\bf R}^n$ and the range (image) is in ${\bf R}^m$. The domain can be less than the whole space. In fact, the function can be defined on the nodes of a partition of the subset of ${\bf R}^n$. Below we illustrate the four possibilities for $n=1,2$ and $m=1,2$:

Multidim functions.png

In addition to the usual functions, we see

  • a parametric curve $P:t\mapsto (x,y)$ or $P:t\mapsto <x,y>$,
  • a function of two variables $F:(x,y)\mapsto z$ or $F:<x,y>\mapsto z$, and
  • a vector field on the plane $V:(x,y)\mapsto <u,v>$.

We will review these three items starting with parametric curves. We need to learn how, instead of treating them one axis at a time, to study them as functions with multidimensional inputs and outputs. Vector algebra will be especially useful.

We will refer to as a parametric curve to

  • any function of the real variable, i.e., the domain lies inside ${\bf R}$, and
  • with its values in ${\bf R}^m$ for some $m=1,2,3...$.

In this section we will limit ourselves to the interpretation of these functions via motion. The independent variable is then time and the value is the location.

A point is the simplest curve. Such a curve with no motion is provided by a constant function.

A straight line is the second simplest curve.

We start with lines in ${\bf R}^2$. We already know how to represent straight lines on the plane; the first method is the slope-intercept form: $$y=mx+b.$$ This method does not include vertical lines: the slope is infinite! In our study of curves (specifically to represent motion) on the plane, there are no preferred directions and then it is unacceptable to exclude any straight lines. The second method is implicit: $$px+qy=r.$$ The case of $p\ne 0,\ q=0$ gives us a vertical line. The third method is parametric. It has a dynamic interpretation.

Example (straight motion). Suppose we would like to trace the line that starts at the point $(1,2)$ and proceeds in the direction of the vector $<2,3>$.

Tracing a line on the plane.png

We use motion as a starting point and as well as a metaphor for parametric curves, as follows. We start moving ($t=0$) from point $P_0=(1,2)$ under a constant velocity of $V=<2,3>$. Specifically, we move

  • $2$ feet per second horizontally, and
  • $3$ feet per second vertically.
Tracing a line on the plane 2.png

In terms of vectors, if we are at point $P$ now, we will be at point $P+V$ after one second. For example, we are at $P_1=P_0+V=(1,2)+<2,3>=(3,5)$ at time $t=1$. We then have already two points on our parametric curve $P$: $$P(0)=P_0=(1,2)\text{ and } P(1)=P_1=(3,5).$$ Of course, these are also the values of our function.

Let's find the formulas for this function. Early on, it's OK to do this component-wise. Then $$P(t)=(x(t),y(t)),$$ and $$x(0)=1,\ x(1)=3 \text{ and } y(0)=2,\ y(1)=5.$$ These functions must be linear; therefore, we have: $$x(t)=1+2t \text{ and } y(t)=2+3t.$$ This is a parametric curve... but not an acceptable answer if we are to learn how to use vectors!

The four coefficients, of course, come from the specific number that give us $P_0$ and $V$. Let's assemble the two coordinate function into one parametric curve: $$P(t)=(x(t),y(t))=(1+2t,\ 2+3t).$$ This is still not good enough; we still can't see the $P_0$ and $V$ directly! We continue by using vector algebra: $$\begin{array}{lll} P(t)&=(1+2t,\ 2+3t)&\text{...we use vector addition...}\\ &=(1,2)+<2t,\ 3t>&\text{...then scalar multiplication...}\\ &=(1,2)+t<2,\ 3>&\text{...and finally...}\\ &=P_0+tV. \end{array}$$ This is what we have discovered:

  • position at time $t$ = initial position $+$ $t\cdot$ velocity.

We stated this conclusion before! This time only the context has changed. $\square$

Warning: One can, of course, move along a straight line at a variable velocity.

The pattern is clear: the line starting at the point $(a,b)$ in the direction of the vector $<u,v>$ is represented parametrically as: $$P(t)=(a,b)+t<u,v>.$$ Similar for dimension $3$: the line starting at the point $(a,b,c)$ in the direction of the vector $<u,v,w>$ is represented as: $$P(t)=(a,b,c)+t<u,v,w>.$$ And so on.

At the next level, we'd rather have no references to neither the dimension of the space nor the specific coordinates.

Suppose $P_0$ is a point in ${\bf R}^m$ and $V$ is a vector. Then the parametric curve of the uniform motion through $P_0$ with the initial velocity of $V$ is the following: $$P(t)=P_0+tV.$$ Then, the line through $P_0$ in the direction of $V$ is the path (image) of this parametric curve.

Line as a parametric curve.png

Stated for $m=1$, the definition produces the familiar point-slope form! The rate of change is a single number (the slope) because the change is entirely within the $y$-axis. What has changed is the context: there are infinitely many directions in ${\bf R}^2$ for change. That is why the change and the rate of change is a vector. But the equation looks exactly the same... even though each letter may contain unlimited amount of information!

Example (prices). The definition applies to the abstract spaces. If ${\bf R}^m$ is the space of prices (of stocks or commodities), we might have $m=10,000$. The prices recorded continuously will produce a parametric curve and this curve might be a straight line. This happens when the prices are growing (or declining) proportionally but, possibly, at different rates. Also, in the short term this curve is likely to look like a straight line: the most recent change of each price is recorded is then the same change is predicted for the next time period. In each column we use the same recursive formula for the $k$th price: $$x_k(t+\Delta t)=x_k(t)+v_k\Delta t,$$ where $v_k$ is the $k$th rate of change.

Straight line of prices.png

The table is our $10,000$-dimensional curve! Can we visualize such a curve in any way? We pick two columns at a time and plot that curve on the plane. Since these columns correspond to the axes, we are plotting a “shadow” of our curve cast on the corresponding coordinate plane. They are all straight lines. $\square$

Exercise. Find a parametric representation of the line through two distinct points $P$ and $Q$.

In the physical space, a straight line is followed by an object when there are not forces at play. Even a constant force leads to acceleration which may change the direction of the motion.

Example (constant velocity). Recall these recursive formulas that gives the location as a function of time when the velocity is constant ($k=0,1,...$): $$\begin{array}{lll} x:&p_{k+1}&=p_k&+v\Delta t;\\ y:&q_{k+1}&=q_k&+u\Delta t. \end{array}$$

Line as a parametric curve recursive.png

These quantities are now combined into points on the plane: $$P_k=(p_k,q_k),$$ and the equations take a vector form too: $$P_{k+1}=P_k+V_k\Delta t.$$ $\square$

Example (thrown ball). Let's review the dynamics of a thrown ball. A constant force causes the velocity to change linearly, just as the location in the last example. How does the location change this time?

In the horizontal direction, as there is no force changing the velocity, the latter remains constant. Meanwhile, the vertical velocity is constantly changed by the gravity. The dependence of the height on the time is quadratic. The path of the ball will appear to an observer -- from the right angle -- as a curve:

Tx ty xy planes.png

A falling ball is subject to these accelerations, horizontal and vertical: $$x:\ a_{k+1}=0;\quad y: a_{k+1}=-g.$$ Now recall the setup considered previously: from a $200$ feet elevation, a cannon is fired horizontally at $200$ feet per second.

Cannon is fired horizontally.png

The initial conditions are:

  • the initial location, $x:\ p_0=0$ and $y:\ p_0=200$;
  • the initial velocity, $x:\ v_0=200$ and $y:\ v_0=0$.

Then we have two pairs of recursive equations -- for the location in terms of the velocity and the velocity in terms of acceleration -- independent of each other: $$\begin{array}{lll} x:& v_{k+1}&=v_0, & &p_{k+1}&=p_k&+v_k\Delta t;\\ y:& u_{k+1}&=v_k&-g\Delta t, &q_{k+1}&=q_k&+u_k\Delta t. \end{array}$$ These are the formulas in the vector notation: $$\begin{array}{llll} V_{k+1}&=V_{k}&+A&\cdot\Delta t,\\ P_{k+1}&=P_{k}&+V_{k+1}&\cdot\Delta t. \end{array}$$ $\square$

The advantage of the vector approach is that the choice of the coordinate system is no longer a concern!

Example (recursive formulas). In dimension $2$, for example, we don't have to align the $x$-axis with the direction of the throw and in dimension $3$ we don't have to align the $z$-axis with the vertical direction. Nonetheless, let's start with former case. A $6$-foot man throws -- straight forward -- a ball with the speed of $100$ feet per second. If the throw is along the $x$-axis and the $y$-axis is vertical, we have $$A=<0,-32>,\ V_0=<0,100>,\ P_0=(6,0).$$ This data goes into the first row of our table for the columns marked $x' ', y' '$, $x',y'$, and $x,y$ respectively.

Ball thrown excel 1.png

We apply the recursive formulas given above. In the spreadsheet,

  • the velocity is computed from the velocity
  • the location is computed from the acceleration,

What is the difference of our vectors approach from our previous treatment of the flight of a ball? Instead of three columns for $x' ', x', x$ and then three columns for $y' ', y', y$, one can see how the two components of acceleration, velocity, and location are combined into vectors contained in two columns each: $x' ',y' '$, then $x',y'$, then $x,y$. The formula is almost the same as before: $$\texttt{=R[-1]C+(RC[-2]-R[-1]C[-2])*R2C1}$$

Next an angled throw... The only change is the vector of initial velocity: $$A=<0,-32>,\ V_0=<100\cos \alpha,\ 100\sin \alpha>,\ P_0=(6,0),$$ where $\alpha$ is the angle of the throw.

Ball thrown excel.png

$\square$

Example (continuous motion). Now the continuous case... Starting with the physics, $$\begin{cases} x' '&=&0,\\ y' '&=&-g, \end{cases}$$ we integrate -- coordinate-wise -- once: $$\begin{cases} x'&=&&&v_x, & & x'(0)=v_x&\text{is the initial horizontal velocity},\\ y'&=&-gt&+&v_y,& & y'(0)=v_y&\text{is the initial vertical velocity}; \end{cases}$$ and twice: $$\begin{cases} x&=&&&v_xt&+&p_x, & & x(0)=p_x&\text{is the initial horizontal position},\\ y&=&-\tfrac{1}{2}gt^2&+&v_yt&+&p_y,& & y(0)=p_y&\text{is the initial vertical position}. \end{cases}$$ Thus, we have: $$\begin{cases} \text{depth }&=&\text{ initial depth }&+&\text{ initial horizontal velocity }&\cdot\text{ time },\\ \text{height}&=&\text{ initial height }&+&\text{ initial vertical velocity }&\cdot\text{ time }&-\tfrac{1}{2}g\cdot\text{ time }^2. \end{cases}$$ We take this solution to the next level by assembling these components into vectors just as in the last example. $$\begin{array}{ccccccc} \text{location}&=&\text{ initial location}&+&\text{ initial velocity }&\cdot\text{ time }&+&<0,-\tfrac{1}{2}g\cdot\text{ time }^2>. \end{array}$$ The last term needs work. The zero represents the zero horizontal acceleration while $-g$ is the vertical acceleration. Then the last term is the acceleration times $\frac{t^2}{2}$. Algebraically, we have: $$\begin{array}{ccccccc} \left[\begin{array}{l}x\\y\end{array}\right]&=&\left[\begin{array}{l}p_x\\p_y\end{array}\right]&+&\left[\begin{array}{l}v_x\\v_y\end{array}\right]&\cdot t&+&\left[\begin{array}{r}0\\-g\end{array}\right] &\cdot\frac{t^2}{2}. \end{array}$$ $\square$

The nature of the acceleration is irrelevant; we only need it to be constant.

Definition. Suppose $P_0$ is a point in ${\bf R}^m$ and $V_0,A$ are vectors. Then the parametric curve of uniformly accelerated motion through $P_0$ with the initial velocity of $V$ and acceleration $A$ is: $$P(t)=P_0+V_0\cdot t+A\cdot \frac{t^2}{2}.$$

We have an extra term, that disappears when $A=0$, in comparison to the uniform motion. Just as in the $1$-dimensional case, a constant acceleration produces a quadratic motion!

Exercise. Show that the path of this parametric curve is a parabola.

The values of a the function represented by a parametric curve lie in ${\bf R}^m$ as points but can also be seen as vectors. For example, we can re-write the familiar parametric curve of points: $$P(t)=P_0+V_0\cdot t+A\cdot \frac{t^2}{2},$$ as one of vectors: $$R(t)=R_0+V_0\cdot t+A\cdot \frac{t^2}{2}.$$ Instead of passing through point $P_0$ it passes through the end point of vector $R_0=OP_0$, which is the same thing. And, of course, the end of vector $R(t)$ is the point $P(t)$. The advantage of the latter approach is that it allows us to apply vector operations to the curves.

A more general approach to parametric curves, as well as their calculus, is presented in the next chapter.

Example (circle transformed). Recall how we parametrized the unit circle using the angle as the parameter. Here, the $x$- and $y$-coordinates of a point at angle $t$ is $\cos t$ and $\sin t$ respectively: $$x=\cos t,\ y=\sin t.$$ The values of $t$ may be the nodes of a partition of an interval such as $[0,2\pi]$ or run through the whole interval.

Circle parametrical.png

We can also look at this formula as a parametrization with respect to time. Then this is a record of motion with a constant speed or, in other words, a constant angular velocity. Now, this is the vector representation of this curve: $$R(t)=<\cos t,\sin t>.$$ So, applying vector operations to this curve will give as new curves, just as in the $1$-dimensional case (Part I). For example, using scalar multiplication by $2$ on all vectors means stretching radially the whole space. We then discover that the curve in the plane given by: $$Q(t)=2R(t)=2<\cos t,\sin t>,$$ is a parametric curve of the circle of radius $2$.

Circle stretched.png

Similarly, using vector addition with $W=<3,1>$ on all vectors means shifting the whole space by this vector. We then discover that the curve in the plane given by: $$S(t)=W+2R(t)=(1,2)+2<\cos t,\sin t>,$$ is a parametric curve of the circle of radius $2$ centered at $(1,2)$.

Circle shifted.png

And so on with other transformations of the plane. $\square$

11 Partitions of the Euclidean space

In Parts I and II, we used partitions of intervals, as well as the whole real line, in order to study incremental change. This time, we need partitions of the $n$-dimensional Euclidean space. The building blocks will come from partitions of the axes.

For dimension $2$, these are rectangles. An interval in the $x$-axis: $$[a,b]=\{x:\ a\le x\le b\},$$ and an interval in the $y$-axis: $$[c,d]=\{y:\ c\le y\le d\},$$ make a rectangle in the $xy$-plane: $$[a,b]\times [c,d]=\{(x,y):\ a\le x\le b,\ c\le y\le d\}.$$

Rectangle as product of intervals.png

A partition of the rectangle $[a,b]\times [c,d]$ is made of smaller rectangles constructed in the same way as above. Suppose we have partitions of the intervals $[a,b]$ in the $x$-axis and $[c,d]$ in the $y$-axis:

Partitions.png

We start with a partition of an interval $[a,b]$ in the $x$-axis into $n$ intervals: $$ [x_{0},x_{1}],\ [x_{1},x_{2}],\ ... ,\ [x_{n-1},x_{n}],$$ with $x_0=a,\ x_n=b$. Then we do the same for $y$. We partition an interval $[c,d]$ in the $y$-axis into $m$ intervals: $$ [y_{0},y_{1}],\ [y_{1},y_{2}],\ ... ,\ [y_{m-1},y_{m}],$$ with $y_0=c,\ y_n=d$.

Partition for Riemann sums dim 2.png

The lines $y=y_j$ and $x=x_i$ create a partition of the rectangle $[a,b]\times [c,d]$ into smaller rectangles $[x_{i},x_{i+1}]\times [y_{j},y_{j+1}]$. The points of intersection of these lines, $$X_{ij}=(x_i,y_{j}),\ i=1,2,...,n,\ j=1,2,...,m,$$ will be called the nodes of the partition. So, there are nodes and there are rectangles (tiles); is that it?

This is how an object can be represented with tiles, or pixels:

Tiles form an object 2.png

Now, are curves also made of tiles? Such a curve would look like this:

Line from tiles.png

If we look closer, however, this “curve” isn't a curve in the usual sense; it's thick! The correct answer is: curves are made of edges of the grid:

Digital Euclidean curve.png

We have discovered that we need to include, in addition to the squares, the “thinner” cells as additional building blocks. The complete decomposition of the pixel is shown below; the edges and vertices are shared with adjacent pixels:

Pixel decomposition.png

Example (dimension $1$). We start with dimension $n=1$:

Cubical grid 1d.png

In this simplest of partitions, the cells are:

  • a node, or a $0$-cell, is $\{k\}$ with $k=...-2,-1,0,1,2,3, ...$;
  • an edge, or a $1$-cell, is $[k, k + 1]$ with $k=...-2,-1,0,1,2,3, ...$. $\\$

And,

  • $1$-cells are attached to each other along $0$-cells.

$\square$

Example (dimension $2$). For the dimension $n=2$ grid, we define cells for all integers $k,m$ as products:

  • a node, or a $0$-cell, is $\{k\} \times \{m\}$;
  • an edge, or a $1$-cell, is $\{k \} \times [m, m + 1]$ or $[k, k + 1] \times \{m \}$;
  • a square, or a $2$-cell, is $[k, k + 1] \times [m, m + 1]$. $\\$

And,

  • $2$-cells are attached to each other along $1$-cells and, still,
  • $1$-cells are attached to each other along $0$-cells.
Cubical grid edges.png

Cells shown above are:

  • $0$-cell $\{3\}\times \{3\}$;
  • $1$-cells $[2,3]\times\{1\}$ and $\{2\}\times [2,3]$;
  • $2$-cell $[1,2]\times [1,2]$.

$\square$

Similarly for dimension $3$, we have boxes. Intervals in the $x$-, $y$-, and $z$-axes: $$[a,b]=\{x:\ a\le x\le b\},\ [c,d]=\{y:\ c\le y\le d\},\ [p,q]=\{z:\ p\le z\le q\},$$ make a box in the $xyz$-space: $$[a,b]\times [c,d]\times [p,q]=\{(x,y):\ a\le x\le b,\ c\le y\le d,\ p\le z\le q\}.$$

In dimension $3$, surfaces are made of faces of our boxes; i.e., these are tiles:

Digital surface.png

The cell decomposition of the box follows and here, once again, the faces, edges, and vertices are shared:

Voxel.png

Example (dimension $3$). For all integers $i,m,k$, we have:

  • a node, or a $0$-cell, is $\{i\} \times \{m\}\times \{k\}$;
  • an edge, or a $1$-cell, is $\{i \} \times [m, m + 1]\times \{k\}$ etc.;
  • a square, or a $2$-cell, is $[i, i + 1] \times [m, m + 1]\times \{k\}$ etc.;
  • a cube, or a $3$-cell, is $[i, i + 1] \times [m, m + 1]\times [k, k + 1]$.
Cubical complex in 3d.png

$\square$

Thus, our approach to decomposition of space, in any dimension, boils down to the following:

  • The $n$-dimensional space is composed of cells in such a way that $k$-cells are attached to each other along $(k-1)$-cells, $k=1,2, ...,n$.

The examples show how the $n$-dimensional Euclidean space is decomposed into $0$-, $1$-, ..., $n$-cells in such a way that

  • $n$-cells are attached to each other along $(n-1)$-cells,
  • $(n-1)$-cells are attached to each other along $(n-2)$-cells,
  • ...
  • $1$-cells are attached to each other along $0$-cells.

What are those cells exactly?

Definition. In the $n$-dimensional space, ${\bf R}^n$, a cell is the subset given by the product with $n$ components: $$P=I_1\times ... \times I_n,$$ with its $k$th component is either

  • a closed interval $I_k=[x_k,x_{k+1}],$ or
  • a point $I_k=\{x_k\}.$ $\\$

The cell's dimension is equal to $m$, and it is also called an $m$-cell, when there are $m$ edges and $m-n$ vertices on this list. Replacing one of the edges in the product with one of its end-points creates an $(n-1)$-cell called a boundary cellof $P$.

Definition. Replacing one of the edges in the product with one of its end-points creates an $(n-1)$-cell called a faceof $P$. Replacing several edges with one of their end-points creates an $k$-cell, $k<n$, called a boundary cell of $P$.

Skeleta of the cube.png

Thus, partitions of the axes -- into nodes and edges -- create a partition of the whole space -- into cells of all dimensions.

Example. Below, a $3$-cell is shown as a “room” along with all of the cells of dimensions $0,1,2$:

Cube and cells of dimension 0,1,2,3.png

They all come from the nodes and edges on the axes:

  • $0$: each of the joints of the “beams” is the product of three nodes;
  • $1$: each of the “beams” is the product of two nodes and an edge;
  • $2$: each of the “walls”, as well as the “floor” and the “ceiling”, is the product of two edges and a node;
  • $3$: the “room” is the product of three edges.

The $2$-cells here are the faces of the $3$-cell, the $1$-cells are the faces of the $1$-cells, etc. $\square$

Definition. Suppose each of the coordinate axes of the $n$-dimensional space, ${\bf R}^n$, has a partition. Then, the product of the partitions consists of the cells given by the product with $n$ components: $$P=I_1\times ... \times I_n,$$ with its $k$th component is either

  • an edge $I_k=[x_k,x_{k+1}],$ or
  • a node $I_k=\{x_k\},$ $\\$

in the partition of the $k$th axis. This combination of cells is called a partition in ${\bf R}^n$.

Definition. Suppose we have a partition in ${\bf R}^n$ and suppose a subset $D$ of ${\bf R}^n$ is the union of some of the cells in the partition. We say that this is a partition of $D$ provided: if a cell is present, then so do all of its boundary cells.

For example, any sequence of edges $Q_i,\ i=1,...,n$, of a partition can be seen as a curve. However, a partition of the curve also includes all of the end-points of the edges.

Digital Euclidean curve with nodes.png

Then a continuous curve consists of a sequence of consecutive edges or, which is the same, of a sequence of adjacent nodes: $$Q_i=P_{i-1}P_i.$$

Furthermore, for a partition of a surface made of faces, we must also have all the edges and the nodes of these faces, and so on.

Cubical complexes.png

We will carry out all calculus constructions within these partitions.

12 Discrete forms

In Parts I and II, we assigned numbers to points within cells in the $1$-dimensional case to represent such things as location -- nodes or $0$-cells -- and velocity -- secondary nodes or $1$-cells. We will continue to do so. In fact, we will study functions defined at points located at the cells of a particular dimension $m$ in a partition. Below we see $m=0,1,2$:

Partitions and functions.png

Firstly, these points -- secondary, tertiary, etc. nodes -- may be specified as a result of sampling a function defined on the whole region. Note that, in that case, one node may be shared by two adjacent cells.

Secondly, these points are used for mere bookkeeping. We then can choose them to be the end-points or corners or mid-points etc. In truth though, the quantities are assigned to the cells themselves. In other words, each cell is an input of these functions, as explained below.

Recall how we defined discrete forms for dimension $1$: within each of the pieces of a partition of the line this function is unchanged; i.e., it's a single number. This is how we plot the graphs of $0$- and $1$-forms over ${\bf R}^1$:

Forms as functions.png

There are more types of cells in the higher dimensional spaces, but the idea remains:

DiscreteKFormKCell.png

Definition. A discrete form of degree $k$ over ${\bf R}^n$, or simply a $k$-form, is a real-valued function defined on $k$-cells of ${\bf R}^n$.

And these are $0$-, $1$-, and $2$-forms over ${\bf R}^2$:

Forms as functions dim 2.png

To emphasize the nature of a form as a function, we can use arrows (${\bf R}^1$):

Forms as functions 2.png

Here we have two forms:

  • a $0$-form with $0\mapsto 2,\ 1\mapsto 4,\ 2\mapsto 3, ...$; and
  • a $1$-form with $[0,1]\mapsto 3,\ [1,2]\mapsto .5,\ [2,3]\mapsto 1, ...$.

A more compact way to visualize is this:

Forms as functions 3.png

Here we have two forms:

  • a $0$-form $q$ with $q(0)=2,\ q(1)=4,\ q(2)=3, ...$; and
  • a $1$-form $s$ with $s\Big([0,1] \Big)=3,\ s\Big([1,2] \Big)=.5,\ s\Big([2,3] \Big)=1, ...$.

We can also use letters to label the cells, just as before. Each cell is then assigned two symbols:

  • one is its name (a latter) and
  • the other is the value of the form at that location (a number).

This idea is illustrated for forms over ${\bf R}^1$ and ${\bf R}^2$ respectively:

Forms as functions 5.png

We have a $0$-form $q$ and a $1$-form $s$ in the former example:

  • $q(A)=2,\ q(B)=4,\ q(C)=3, ...$;
  • $s(AB)=3,\ s(BC)=.5,\ s(CD)=1, ...$.

We also have a $2$-form $\phi$ in the latter example:

  • $q(A)=2,\ q(B)=1,\ q(C)=0,\ q(D)=1$;
  • $s(a)=1,\ s(b)=-1,\ s(c)=2, s(d)=0$;
  • $\phi(\tau )=4$.

We can simply label the cells with numbers, as follows (in ${\bf R}^3$):

Forms as functions 4.png

These forms may represent the following characteristics of a flow of a liquid:

  • 1. a $0$-form: the pressure of the liquid at the joints of a system of pipes;
  • 2. a $1$-form: the flow rate of the liquid along the pipe;
  • 3. a $2$-form: the flow rate of the liquid across the membrane;
  • 4. a $3$-form: the density of the liquid inside the box.

These forms will be used to study functions of several variables in the upcoming chapters. However, the example of parametric curves -- and especially motion in space -- suggests that we may need the domain of these functions to be multi-dimensional. We saw a function defined, just like a $0$-form, at the nodes of the partition of the line, but with values in ${\bf R}^2$, unlike a $0$-form.

This is a generalization of the last definition.

Definition. Suppose $n$ and $m$ are given. Then a vector-valued discrete form $F$ of degree $k$, or simply a $k$-form, is a function defined on $k$-cells of ${\bf R}^n$ with values in ${\bf R}^m$.

As you can see, we will be using capital letters for vector-valued forms in accordance with our convention.

Note that discrete forms do not exactly match our list of functions: numerical functions, parametric curves, vector fields, and functions of several variables. From the same domain, we pick cells of different dimensions producing forms of different degrees.

This is an illustration of two vector-valued forms: a $0$-form and a $1$-form (for the latter, the vectors have to be moved to put the starting points at the origin); i.e., $n=1$ and $m=2$:

Parametric curve as a form.png

The former may represent the locations and the latter the velocities. Both can be seen as parametric curves.

Next is an illustration of a real-valued and a vector-valued $1$-forms; i.e., $n=2$, $m=1$ and $n=2$, $m=2$ respectively:

Real and vector-valued forms.png

The former may represent the a flow of water along a system of pipes and the latter the same flow with possible leaks. Both can be seen as vector fields.

The algebra of vectors allows us to reproduce the definitions from the last chapter in the new, multi-dimensional in both input and output, context. We just assume that a space of inputs ${\bf R}^n$ with a partition and a space of outputs ${\bf R}^m$ are given.

We need one more generalization; we introduce orientation of cells.

About the $0$-cells, we will simply allow them to appear with both positive and negative signs.

Definition. An oriented $0$-cell (or node) $A$ is a $0$-cell of the partition with its sign specified: $A$ or $-A$.

The choice will depend on the circumstances. The need for this will become clear shortly.

We have $1$-cells defined as products of a single edge and several nodes. The order of nodes doesn't matter: $$E=AB=BA,\ A=(a_1,...,a_n),\ B=(b_1,...,b_n)\ \Longrightarrow\ a_i= b_i,\ i=1,2,...,n, \text{ and } a_k \ne b_k \text{ for some } k.$$ We would like distinguish the two ways we can follow an edge.

Oriented grid.png

Definition. An oriented $1$-cell (or edge) $E$ is a $1$-cell of the partition with the order of its two nodes specified: $AB$ or $BA$. The cell is called positively oriented of the order of the nodes does with the directions of the axes: $$E=AB,\ A=(a_1,...,a_n),\ B=(b_1,...,b_n)\ \Longrightarrow\ a_k < b_k \text{ for some } k.$$ The cell is called negatively oriented of the order of the nodes goes against the directions of the axes: $$E=AB,\ A=(a_1,...,a_n),\ B=(b_1,...,b_n)\ \Longrightarrow\ a_k > b_k \text{ for some } k.$$

Notation: $$BA=-AB.$$

The orientation of higher-dimensional cells is addressed later.

The following is crucial.

Definition. The boundary of a $0$-cell is $0$, denoted by: $$\partial A=0;$$ The boundary of a $1$-cell is the difference of its end-points denoted by: $$\partial AB=B-A.$$

Even though any sequence of edges $E_i,\ i=1,...,n$, is seen as a curve, a continuous curve consists of a sequence of consecutive oriented edges or, which is the same, of a sequence of adjacent nodes: $$E_i=P_{i-1}P_i.$$

Cubical curve -- nodes and edges.png

What is the sum of the boundaries of the edges of $C$?

Definition. The boundary of a curve $C$ from $A$ to $B$ is $B-A$.

Recall that $0$-forms are defined on the positively oriented $0$-cells... and now they are instantly extended to the negatively oriented cells. Similarly, $1$-forms are defined on both positively and negatively oriented $1$-cells.

Forms as functions -- oriented cells.png

We amend our definition.

Definition. Suppose $n$ and $m$ are given. Then a real-valued or vector-valued discrete form $F$ of degree $k$, or simply a $k$-form, $k=0,1$, is a function defined on the positively and negatively oriented $k$-cells of ${\bf R}^n$ with values in ${\bf R}$ or ${\bf R}^m$ respectively so that: $$F(-a)=-F(a).$$

Now calculus.

Definition. The difference of a discrete $0$-form $F$ is a discrete $1$-form given by its values on each edge $E=AB$ of the partition: $$\Delta F \, (E)=F(B)-F(A).$$ A $1$-form is called exact when it is the difference of some $0$-form.

The picture above may serve as an illustration of this concept.

Definition. The sum of a discrete $0$-form $F$ along a collection $Q$ of oriented nodes $N_1,N_2,...,N_k$ is defined and denoted to be: $$\sum_{Q} F=F(N_1)+F(N_2)+...+F(N_k).$$

Definition. The sum of a discrete $1$-form $G$ along a collection $C$ of oriented edges $E_1,E_2,...,E_k$ is defined and denoted to be: $$\sum_{C} G=G(E_1)+G(E_2)+...+G(E_k).$$

The relation between the two operations remains the same as in the $1$-dimensional case: they cancel each other.

FTDC.png

Recall first that any sequence of edges $Q_i,\ i=1,...,n$, of a partition is seen as a curve, while a continuous curve consists of a sequence of consecutive edges or, which is the same, of a sequence of adjacent nodes: $$Q_i=P_{i-1}P_i.$$

Cubical curve.png

Theorem (Fundamental Theorem of Discrete Calculus of degree $1$). Suppose a partition of an $n$-cell in ${\bf R}^n$ is given. Suppose $F$ is a discrete $0$-form on this partition and suppose $A$ is a node of the partition. Then, for each node $X$ and any continuous curve from $A$ to $X$, we have $$\sum_{C} (\Delta F) =F(X)-F(A).$$

Proof. We just add all of these and cancel the repeated nodes: $$\begin{array}{lll} \sum_{C} G&=G(E_1)&+G(E_2)&+...&+G(E_k)\\ &=G(P_{0}P_{1})&+G(P_{1}P_{2})&+...&+G(P_{k-1}P_{k})\\ &=\big[F(P_{1})-F(P_{0})\big]&+\big[F(P_{2})-F(P_{1})\big]&+...&+\big[F(P_{k})-F(P_{k-1})\big]\\ &=-f(P_0)&&&+F(P_k)\\ &=F(X)-F(A). \end{array}$$ $\blacksquare$

In other words, if $F$ is the difference over every edge, it is the difference over any continuous curve.

FTDC dim 3.png

Corollary. Under the conditions of the theorem, we have $$\sum_{C} (\Delta F) =\sum_{\partial C}F.$$

But do they cancel each other in either order? Does this formula from Part II still make sense: $$\Delta\left( \sum_{C} G\right) =G(X)?$$ We will address this question in the following chapters.

13 Angles between vectors and the dot product

Once again, a Cartesian system pre-measures the space ${\bf R}^n$ so that we can do geometry through computations with the coordinates of points and the components of vectors. That's how we find the distances between points and the magnitudes of vectors. What about the angles?

Recall the coordinate system for dimension $1$. It is a correspondence:

  • location $P\ \longleftrightarrow\ $ number $x$.

What is the difference between the vectors $OP$ and $OQ$ ($P,Q$ are not equal to $O$) represented in terms of their components $x$ and $x'$? There can be only two possibilities:

  • if $P$ and $Q$ are on the same side of $O$ then the directions are the same,
  • if $P$ and $Q$ are on the opposite sides of $O$ then the directions are the opposite.
Coordinate system dim 1 -- directions.png

Then the theorem about the directions for dimension $1$ is stated as follows; the angle between the vectors $OP$ and $OQ$ with components $x\ne 0$ and $x'\ne 0$ is

  • $0$ when $x\cdot x'>0$; and
  • $\pi$ when $x\cdot x'<0$.

However, it is clear that only the directions of the vectors matter and not the sizes! We can then make the same conclusions using these numbers (vectors) instead: $$\frac{x}{|x|} \text{ and } \frac{x'}{|x'|} .$$ The advantage is that they can only take two possible values, $1$ and $-1$. And so does their product! We can then restate the theorem: the angle between the vectors $OP$ and $OQ$ with components $x\ne 0$ and $x'\ne 0$ is

  • $0$ when $\frac{x}{|x|}\cdot \frac{x'}{|x'|}=1$; and
  • $\pi$ when $\frac{x}{|x|}\cdot \frac{x'}{|x'|}=-1$.

Matching these four numbers, $$0\mapsto 1 \text{ and }\pi\mapsto -1,$$ we realize that this is the cosine! Indeed, $$\cos 0= 1 \text{ and }\cos\pi= -1.$$ We then have a new version of our theorem.

Theorem (Angles for dimension $1$). If $\widehat{QOP}$ is the angle between the vectors $OP$ and $OQ$ with components $x\ne 0$ and $x'\ne 0$ then $$\cos \widehat{QOP} = \frac{x}{|x|}\cdot \frac{x'}{|x'|}.$$

Now the coordinate system for dimension $2$. It is a correspondence:

  • location $P\ \longleftrightarrow\ $ a pair of numbers $(x,y)$.

What is the difference between the directions from the origin $O$ toward locations $P$ and $Q$ (other than $O$) represented in terms of their coordinates $(x,y)$ and $(x',y')$? We are talking about angle between the two directions.

Coordinate system dim 2 -- angles.png

The angle we are looking for is: $$\widehat{QOP}=\beta -\alpha.$$ The cosine of this angle can be found from the trigonometric functions of these two angles: $$\cos \widehat{QOP}=\cos\beta \cos\alpha+\sin\beta \sin\alpha.$$ We are already excluding the magnitudes of the vectors from consideration. Now, these angles are found in terms of the components of the vectors: $$\cos\alpha=\frac{x}{||OP||},\ \sin\alpha=\frac{y}{||OP||},\ \cos\beta=\frac{x'}{||OQ||},\ \sin\beta=\frac{y'}{||OQ||}.$$ Therefore, we express the angle in terms of the unit vectors of the two original vectors: $$\cos \widehat{QOP}=\frac{x}{||OP||}\cdot \frac{x'}{||OQ||}+\frac{y}{||OP||}\cdot \frac{y'}{||OQ||}=\frac{xx'+yy'}{||OP||\ ||OQ||}.$$ We will have a special name for the numerator of this fraction.

Definition. The dot product of vectors $<x,y>$ and $<x',y'>$ in ${\bf R}^2$ is defined by: $$<x,y>\cdot <x',y'>=xx'+yy'.$$

Thus, the dot product is computed, as other vector operations, component-wise.

We now re-state our theorem about the directions.

Theorem (Angles for dimension $2$). If $\gamma$ is the angle between vectors $A\ne 0$ and $B\ne 0$, then: $$\cos \gamma =\frac{A\cdot B}{||A||\ ||B||}.$$

Example. Let's verify the theorem. First the two basis vectors: $$i=<1,0>,\ j=<0,1>\ \Longrightarrow\ i\cdot j = 1\cdot 0 +0\cdot 1=0.$$ Indeed, they are perpendicular and $\cos \pi/2=0$. Similarly, $$<1,1>\cdot <-1,1> = 1\cdot 1 +(-1)\cdot 1=0.$$ However, $$<1,0>\cdot <1,1> = 1\cdot 1 +0\cdot 1=1.$$ To see the correct angle of $45$ degrees, we apply the formula from the theorem: $$\cos \gamma =\frac{<1,0>\cdot <1,1>}{||<1,0>||\ ||<1,1>||}=\frac{1}{1\ \sqrt{2}}=\frac{\sqrt{2}}{2}.$$ $\square$

Definition. The dot product of vectors $<x,y,z>$ and $<x',y',z'>$ in ${\bf R}^3$ is defined by: $$<x,y,z>\cdot <x',y',z'>=xx'+yy'+zz'.$$

The dot product is component-wise operation: $$\begin{array}{rcccccc} A&=<&x,&y,&z&>\\ \cdot\\ B&=<&u,&v,&w&>\\ \hline A\cdot B&=&x\cdot u+&y\cdot v+&z\cdot w& \end{array},\qquad A\cdot B= \left[\begin{array}{c}x\\y\\z\end{array}\right]\cdot \left[\begin{array}{c}u\\v\\w\end{array}\right]= \begin{array}{l}x\cdot u+\\y\cdot v+\\z\cdot w\end{array}\quad. $$

Example. It is once again easy to confirm that the basis vectors are perpendicular to each other: $$i\cdot j=<1,0,0>\cdot<0,1,0>=1\cdot 0+0\cdot 1+0\cdot 0=0,$$ and so on. $\square$

Example. What is the angle between the diagonal of a cube and any of its sides?

Cube with diagonal cut out.png

We see this angle as the angle between the vector $<1,1,1>$ and the basis vectors. Without any trigonometry, we have $$\cos\alpha=\frac{<1,1,1>\cdot <1,0,0>}{||<1,2,3>||\ ||<1,0,0>||}=\frac{1}{\sqrt{3}}.$$ $\square$

Definition. The dot product of vectors $A=<a_1,a_2,...,a_n>$ and $B=<b_1,b_2,...,b_n>$ in ${\bf R}^n$ is defined by: $$A\cdot B=a_1b_1+a_2b_2+...+a_nb_n=\sum_{k=1}^na_kb_k.$$

Below we see how this new operation compares with the other vector operations: $$\begin{array}{|r|ccccc|} \hline \text{vector addition}&A&+&B&=&C\\ &\text{vector}&&\text{vector}&&\text{vector}\\ \hline \text{scalar multiplication}&c&\cdot&A&=&C\\ &\text{number}&&\text{vector}&&\text{vector}\\ \hline \text{dot product}&A&\cdot &B&=&s\\ &\text{vector}&&\text{vector}&&\text{number}\\ \hline \end{array}$$ The last two might be confusing without context; for example, consider the three possible meanings of the following: $$0\cdot A=0.$$

Let's consider the properties of the dot product.

If we just set $Y=X$, we have the so-called Normalization property: $$||X||^2=X\cdot X.$$ Indeed, the angle of $X$ with itself if $0$: $$\cos 0 =1 =\frac{X\cdot X}{||X||\ ||X||}=\frac{X}{||X||}\cdot\frac{X}{||X||}.$$ Combined with the formula for the cosine of the angle, the result suggests that the dot product is independent from the Cartesian system. Certainly, this system is just a tool that we introduce into the space the geometry of which we study and we don't expect that changing the components of vectors will also change the distances and the angles!

Next Commutativity or Symmetry: $$A\cdot B=B\cdot A.$$ This means that the angle is between $A$ and $B$; i.e., the same from $A$ to $B$ as from $B$ to $A$.

Next Associativity: $$(cA)\cdot B=c(A\cdot B)=A\cdot (cB).$$ So, the effect of stretching on the dot product is a multiple and the angle doesn't change (for $c>0$).

Next Distributivity: $$A\cdot (B+C)=A\cdot B + A\cdot C.$$

Treated component-wise, the Commutativity, Associativity, Distributivity properties for the dot product of vectors follow from the Commutativity, Associativity, Distributivity for numbers.

Once again, these properties allow us to use the usual algebraic manipulation steps for numbers as long as the expressions make sense to begin with.

Back to geometry. What is the meaning of the angle between two vectors $A$ and $B$ in ${\bf R}^n$? After all in the abstract spaces, when $n>3$, there is no reality test for this concept and we can't verify the formulas we are to use! The answer is to reduce the multidimensional case to the case of $n=2$. Indeed, every two vectors define a plane and this plane has the same vector algebra operations -- including the dot product -- as the ambient space ${\bf R}^n$:

Every two vectors define a plane.png

In the meantime, the plane has the well-understood Euclidean geometry: the lengths of vectors and and the angles between vectors have a real-life meaning.

Theorem (Angles for dimension $n$). If $\gamma$ is the angle between vectors $A\ne 0$ and $B\ne 0$ in ${\bf R}^n$, then: $$\cos \gamma =\frac{A\cdot B}{||A||\ ||B||}.$$

Proof. Instead of trigonometric formulas we used for case $n=2$, we will rely on the algebraic properties of the dot product. We start with the Law of Cosines (cosine is what we are looking for anyway) which states: $$c^2=a^2+b^2-2ab\cos\gamma,$$ for any triangle with sides $a,b,c$ and angle $\gamma$ between $a$ and $b$.

Law of cosines.png

We interpret the lengths of the sides of the triangle in terms of the lengths of vectors: $$a=||A||,\ b=||B||,\ c=||A-B||.$$ Then we translate the law into the language of vectors: $$||A-B||^2=||A||^2+||B||^2-2||A||\ ||B||\cos\gamma.$$ Instead of solving for $\cos\gamma$, we expand the left-hand side: $$\begin{array}{llll} ||A-B||^2&=(A-B)\cdot (A-B)&\text{...by Normalization,}\\ &=A\cdot A+A\cdot (-B)+(-B)\cdot A+(-B)\cdot (-B)&\text{...by Distributivity,}\\ &=||A||^2-2A\cdot B+||B||^2&\text{...by Associativity and Normalization.} \end{array}$$ The Law of Cosines then takes the following form: $$||A||^2-2A\cdot B+||B||^2=||A||^2+||B||^2-2||A||\ ||B||\cos\gamma.$$ Now we cancel the repeated terms in the two sides of the equation and obtain the following: $$-2A\cdot B=-2||A||\ ||B||\cos\gamma.$$ $\blacksquare$

Warning: when one of the two vectors is zero, the angle between them does not make sense.

If we know the angle, can we rotate a given vector $V$? First, we limit ourselves to in dimension $2$; otherwise we'd have to specify the axis of rotation. Now, the problem has a trivial solution when this angle is $\pi$: it's just $-V$. There is also an easy solution when this angle is $\pi/2$ counterclockwise.

Rotation of vectors.png

Definition. The normal vector of a vector $V=<u,v>$ on the plane is given by $$V^\perp=<u,v>^\perp=<-v,u>.$$

This is easy to confirm: $$V\cdot V^\perp=<u,v>\cdot <-v,u>=u(-v)+vu=0.$$ We have then a special operation on vectors in dimension $2$.

From the inequality $$|\cos\gamma |\le 1,$$ we derive the following.

Corollary (Cauchy Inequality). For any pair of vectors $A\ne 0$ and $B\ne 0$ in ${\bf R}^n$, we have: $$|A\cdot B|\le ||A||\ ||B||.$$

In other words, if we rotate two vectors, the dot product reaches its maximum when they are perpendicular to each other.

14 Projections and decompositions of vectors

To find the its $x$-coordinate of a point on the $x$-plane we go vertically from that points until we reach the of $x$-axis:

Projection R2R.png

The result resembles shadows left by points, vectors, or objects on the $x$-axis with the light cast from above (or from the right for the $y$-axis):

Projections.png

It is called the projection of the point on the $x$-axis. Same for the $y$-axis.

Similarly, the projection of a vector on the $x$-axis gives its $x$-component. If several coordinate systems co-exit, a transition from one to another will require expressing the new coordinates of a point or the new components of a vector in terms of the old. We do that one axis or basis vector at a time.

For vectors, this is what happens with the original vector $i$. Any vector $A$ is expressed in terms of $i$ by finding $A$'s projection $P$ on the $x$-axis (component is $2$):

Orthogonal projection of vectors.png

Suppose next that we are introducing a new basis vector $i$. Then every vector $A$ needs to be expressed in terms of $i$ and some other vectors perpendicular to $i$. The question becomes: how much does $A$ “protrude” in the direction of $i$?

Orthogonal projection of vectors 2.png

Vector $A$ is expressed in terms of $i$ by finding $A$'s projection $P$ on the line created by $i$.

The idea applies to all vectors: every vector $A$ can be expressed in terms of any other non-zero vector $V$ and another vector perpendicular to $V$. If $P$ is the projection, what's the other vector? Here: $$A=P+(A-P).$$

Definition. Suppose $A$ and $V\ne 0$ are two vectors in ${\bf R}^n$. Then the orthogonal projection of $A$ onto $V$ is such a vector $P$ that

  • 1. $P$ is parallel to $V$ and
  • 2. $A-P$ is perpendicular to $V$.

Let's find an explicit formula.

First, “parallel” simply means a multiple! Therefore, the first property means that there is a number $c$ -- this is the one we are looking for -- such that: $$P=cV.$$ The second property is expressed in terms of the dot product: $$V\cdot (A-P)=0.$$ We substitute: $$V\cdot (A-cV)=0,$$ and use Distributivity and Associativity: $$V\cdot A-cV\cdot V=0.$$ Next we use Normalization: $$V\cdot A=c||V||^2.$$ Then, $$c=\frac{V\cdot A}{||V||^2}.$$ This is the multiple of $V$ that gives us $P$. Thus, we have the following result.

Theorem (Projection). The orthogonal projection $P$ of $A$ onto $V\ne 0$ is given by: $$P=\frac{V\cdot A}{||V||^2}V.$$

Notice that the formula -- as expected -- depends only on the direction of $V$: $$P=\left( \frac{V}{||V||}\cdot A\right) \frac{V}{||V||}.$$

15 Sequences of vectors and their limits

Vectors correspond to points: $$OP\ \longleftrightarrow\ P.$$ We are then able to discuss convergence of sequences of vectors. For convenience, we just restate the definition given earlier in this chapter replacing distances between points with magnitudes of differences of vectors: $$d(PQ)=||OQ-OP||.$$

Suppose $\{A_k:\ k=1,2,3...\}$ is a sequence of vectors in ${\bf R}^n$. First, our definition is equivalent to the following condition:

  • for each real number $\varepsilon > 0$, there exists a number $N$ such that, for every natural number $k > N$, we have

$$||A_k - A|| < \varepsilon .$$

Definition. We say that a sequence of $\{A_k:\ k=1,2,3...\}$ of vectors in ${\bf R}^n$ converges to another vector $A$ in ${\bf R}^n$, called the limit of the sequence, if: $$||A_k-A||\to 0\text{ as }k\to \infty,$$ denoted by: $$A_k\to A \text{ as }k\to \infty,$$ or $$A=\lim_{k\to \infty}A_k.$$

Sequence of vectors.png

We consider now the algebra of sequences and limits. Limits behave well with respect to the vector operations. Below we assume that the sequences are defined on the same set of integers.

We start with addition.

Sum Rule for sequences Rn.png

To graphically add two sequences, we plot parallelograms. Then, the diagonals of these parallelograms form the new sequence. Now, if either sequence converges to $0$, then so do these diagonals.

Theorem (Sum Rule). $$A_k\to 0 \text{ and } B_k\to 0 \ \Longrightarrow\ A_k+ B_k\to 0.$$

Proof. From the assumption and the definition it follows: $$||A_k||\to 0 \text{ and } ||B_k||\to 0.$$ Since these two numerical sequences converge to $0$, then so does their sum: $$||A_k||+||B_k||\to 0,$$ as we know from the Sum Rule for numerical sequences in Part I. From the Triangle Inequality we conclude: $$||A_k+B_k||\le||A_k||+||B_k||\to 0 .$$ Therefore, by definition $A_k+ B_k\to 0$. $\blacksquare$

Multiplying a sequence by a scalar simply stretches the whole picture uniformly.

Constant Multiple Rule for sequences Rn.png

Theorem (Constant Multiple Rule). $$A_k\to 0 \ \Longrightarrow\ cA_k\to 0 \text{ for any real }c.$$

Proof. From the assumption and the definition it follows: $$||A_k||\to 0.$$ Since this is a numerical sequence that converge to $0$, then so does its multiple: $$|c|\, ||A_k||\to 0,$$ as we know from the Constant Multiple Rule for numerical sequences in Part I. Then: $$||cA_k||=|c|\, ||A_k||\to 0 .$$ Therefore, by definition $cA_k\to 0$. $\blacksquare$

For more complex situations we need to use the fact that convergent sequences are bounded.

Theorem (Boundedness). $$A_k\to A \ \Longrightarrow\ ||A_k|| < Q \text{ for some real } Q.$$

Proof. The idea is that the tail of the sequence will fit into some ball around the limit; meanwhile, there are only finitely many terms left... Choose $\varepsilon =1$. Then by definition, there is such $N$ that for all $k>N$ we have: $$||A_k-A|| < 1.$$ Then, we have $$\begin{array}{lll} ||A_k||&=||(A_k-A)+A||&\text{ ...then by the Triangle Inequality...}\\ &\le ||A_k-A||+||A||&\text{ ...then by the inequality above...}\\ &<1+||A||. \end{array}$$ To finish the proof, we choose: $$Q=\max\{||A_1||,\ ...,\ ||A_k||,\ 1+||A||\}.$$ $\blacksquare$

The proof is illustrated below:

Boundedness for sequences Rn.png

The converse isn't true: not every bounded sequence is convergent. We will show later that, with an extra condition, bounded sequences do have to converge...

We are now ready for the general results on the algebra of limits.

Sum Rule for sequences Rn.png

Theorem (Sum Rule). If sequences $A_k ,B_k$ converge then so does $A_k + B_k$, and $$\lim_{k\to\infty} (A_k + B_k) = \lim_{k\to\infty} A_k + \lim_{k\to\infty} B_k.$$

Proof. Suppose $$A_k\to A \text{ and } B_k\to B.$$ Then, $$||A_k - A||\to 0 \text{ and } ||B_k-B||\to 0.$$ We compute: $$\begin{array}{lll} ||(A_k + B_k)-(A+B)||&= ||(A_k-A)+( B_k-B)||& \text{ ...then by the Triangle Inequality...}\\ &\le ||A_k-A||+|| B_k-B||&\\ &\to 0+0 & \text{ ...by SR for numerical sequences...}\\ &=0. \end{array}$$ Then, by the last theorem, we have $$||(A_k + B_k)-(A+B)||\to 0.$$ Then, by the first theorem, we have: $$A_k + B_k\to A+B.$$ $\blacksquare$

Dot Product Rule for sequences.png

Theorem (Dot Product Rule). If sequences $A_k ,B_k$ converge then so does $A_k \cdot B_k$, and $$\lim_{k\to\infty} (A_k \cdot B_k) = \big( \lim_{k\to\infty} A_k \big) \cdot \big( \lim_{k\to\infty} B_k \big).$$

Exercise. Prove the theorem.

The following is a corollary.

Theorem (Constant Multiple Rule). If sequence $A_k $ converges then so does $c A_k$ for any real $c$, and $$\lim_{k\to\infty} c\, A_k = c \cdot \lim_{k\to\infty} A_k.$$

Warning: it is considered a serious error if you use the conclusion (the formula) one of these rules without verifying the conditions (the convergence of the sequences involved).

We represent the Sum Rule as a diagram: $$\newcommand{\ra}[1]{\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\la}[1]{\!\!\!\!\!\xleftarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} \newcommand{\ua}[1]{\left\uparrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} \begin{array}{ccc} A_k,B_k&\ra{\lim}&A,B\\ \ \da{+}&SR &\ \da{+}\\ A_k+B_k & \ra{\lim}&\lim(A_k+B_k)=A+B \end{array}$$ In the diagram, we start with a pair of sequences at the top left and then we proceed in two ways:

  • right: take the limit of either, then down: add the results; or
  • down: add them, then right: take the limit of the result.

The result is the same!