Tools to use Calculus

    

The Cartesian Plane - why it works

   Warning: Concepts may appear Larger on this Side
Calculus is about stuff that moves. What happens to the quantity of gasoline in your car as you go to grandma's house? Well, there are as many factors as you want to consider, depending on what you want to know about the quantity of gasoline used.

Look at the gas gauge when you leave to visit grandma and subtract the reading you see when you arrive. This ignores the events of the drive over so how can you answer grandma when she asks, "How was your drive, my dears"?		   "I'm starting to get a handle on using Maple. Its like Oracle, which feels like a mixture of Java and Perl. And now that I'm looking at the math that is the start of Calculus its looking like Set Theory. They are all connected."

"Huh?"

Calculus - Finney & Thomas, Addison Wesley: 1990
Concepts of Modern Mathematics - Stewart, Dover: 1995
Maple 9.00 - Maplesoft, Waterloo Maple Inc.: 2003
Calculus 5e - Stewart, Thomson Learning Inc.: 2003
     
Before we try to answer grandma's question let's look at a brief overview of Calculus. Since Calculus moves I'm going to build a foundation to set it on. The overview will be at the end of this article.   "Take two steps back before you start."
"Then I can start."
"No, take two steps back before you start."
"But if I do that I will never start."
"Hmmm."
   
The Cartesian Coordinate Plane,   Tools to use Calculus - section topics
, is a tool for investigating the properties of a mathematical model that represents a person, place, object, thing, action, state of being, quantity or quality. It possess attributes which are named.

The Roman Numerals in the figure label the four quadrants of the Cartesian Plan. Traveling counterclockwise from the positive half of the x-axis, they are named First, Second, Third and Fourth. Any point is either in a quadrant or on the xy-axes. Now a point is a zero-dimensional entity (thing) which I am told I can't really talk about. But if I take one from axis x and one from axis y, I get an Ordered Pair (x,y) which I can talk about.

O.K. so a point is just there. Everyone accepts it but polite people don't talk about it. A line is a bunch of points that has specific properties. To demonstrate some properties of a line do the following:

Pick a number between - and +
     Forget it
Pick two numbers and add them together
Reverse the order of the numbers and add again
     Forget it
Pick two numbers and multiple them together
Reverse the order of the numbers and multiple
     Forget it
  
These properties are born out of Set Theory. A line without numbers has them. Labeling each point on a line with the name of a number gives you a Number Line.   n | - < n < + is read, "n is an element of the set of Real Numbers such that n is between negative and positive infinity". Stay tuned for more properties of n and .
A line is a one-dimensional object. It has length. By using many lines you can build models that have many dimensions. To construct a Cartesian Plane we only need two lines. Think about two Number Lines that do not share any of their points with each other. They do not know each other, so we are going to introduce them. To keep things friendly we will have them meet at zero and introduce them by name.
    
"X-axis this is Y-axis" and immediately they find they share a common point of interest. This point is the intersection of the two lines. Geometrical intersection corresponds to Set Theoretic intersection.   x-axis y-axis = {0} is read "The intersection of x-axis and y-axis is zero". Well, that's nice but I was looking for something more like (0,0) or the named point called Origin. Some way of getting xy-values at every point in the plane to do some Math.
Two lines cross at a single point. We say that in this situation these two one-dimensional objects define a two-dimensional object named a plane. Actually you can do this with one line and a point not on that line. Anyway, you need to account for the additional dimension. In this case starting at one and going to two. Mathematics does this and Calculus is built on a Geometry that is part of Set Theory.    
Since x-axis and y-axis are each sets of Real Numbers we can look at another operation defined by Set Theory. The Cartesian Product of x-axis and y-axis is the set of Ordered Pairs used to build a coordinate geometry in the Cartesian Coordinate Plane. We now have a two-dimensional object called a plane where each point is uniquely named by a coordinate in the form of (x,y).   x-axis x y-axis = { (x,y) | x and y } is read, "The Cartesian Product of x-axis and y-axis is the set of Ordered Pairs x,y such that x is an element of the set of Real Numbers and y is an element of the set of Real Numbers". Since , x-axis equals and the y-axis equals we really have x which is also written as . The Cartesian Product is also known as the Direct Product.
     
There be Distances here!    
The Cartesian Plane comes packed with the properties of the Real Numbers and those of Euclidean Geometry. Since each point is labeled with a coordinate we can explore and experiment with models of reality. A Mathematical Model is a lot cheaper to build than a bridge, canon or seat belt and you can see if its going to work before you build the real thing.

   The Cartesian Coordinate Plane is a two-dimensional Euclidean Space. A Space is a Set together with the distance function obtained by defining the distance between two elements. In our case the set is with a distance function based on the Pythagorean Theorem called the Euclidean Metric. It is also called the Euclidean Plane and can be written as E2.
Distances along the axes is simply a matter of subtracting one value from another. On the    
x-axis subtract one value of x from the other value of interest. The same is true for the values of y.    
Along the x-axis it looks like we are ignoring the y-values and along the y-axis it looks like we are ignoring the x-values. This is not quite true. Its just that along the x-axis all the values of y are constant as are all the values of x along the y-axis. The constant values in this case are zero. Since "Nothin' from nothin' leaves nothin'" is trivial that part of the calculation was not written down. What is important about this is that every horizontal and vertical line in the Cartesian Plane has a constant term that resolves to zero.

    
The distance between any two points on the Cartesian Plane can be measured from the graph or calculated from the coordinates of the points. This feature of the Cartesian Plane is due to the properties we built into the plane when we constructed it.   For any non-negative n, n-dimensional Euclidean space is the set n where the metric is the square root of the sum of the squares of the differences of the n-tuples of two coordinates.

It is sentences like this that gives mathematics a bad name. O.K., look at the equation and take it apart from the inside out.

The metric is ( (qi - pi)2 )1/2 where i = 1, ..., n.

Looking at the picture? To resolve the confusion:
Q = (x2, y2) = (q1, q2)
P = (x1, y1) = (p1, p2).
To Understand General Relativity    
    A Musical Interlude
    "I just found out that n-space is a real things. It has at least three other names: n-dimensional Euclidean space, Euclidean n-space and En."

"Cool. So we live in a 4-space sci-fi universe?"

"Well, yes and no."

"Please sir, may I have some vagueness with that uncertainty?"

"But of course. By definition, En is a metric space, and that makes it a topological space."
  "Topology. Right, like turning a donut into a coffee cup. Since a donut has one side which is its outside surface and one hole, you can morph the donut shape to look like a coffee cup leaving it topologically unchanged."
    "It gets better. En is the prototypical example of an n-manifold, unfortunately I don't know what that means."

"That's fine, neither do I."

"Anyway it's a differentiable n-manifold."

"So you want to apply Differential Calculus to a manifold and you don't know what a manifold is."

"Right."

"Good."

"Marvelous, now I've been told that for n ≠ 4, any differentiable n-manifold that is homeomorphic to En is also diffeomorphic to it".
  "Homeomorphic provides for a one-to-one mapping and back again. Kind of like this. Put a dot on the donut and you can find a corresponding dot on the coffee cup. If you reverse the process and start with that dot on the coffee cup you will find the dot you started with on the donut. Betcha' you can find a proof of this in Set Theory. You might what to look there for diffeomorphic as well."

"Fair enough. And for my next trick here is another fun fact that I think I need to understand General Relativity. Euclidean n-space can also be considered as an n-dimensional real vector space and in fact is a Hilbert space."
    "Well you've collected a lot of spaces that seem to hang together to build General Relativity except for the Hilbert Space that's essential for Quantum Mechanics."
    Back to the Business at Hand
Symmetry    
To get twice the information out of our efforts to learn things we will take a little time to understand some basics of symmetry. It is no accident that the figure here looks like a street map. Symmetry is a useful guide when noticed. Many times it can be explained but, often it is discovered like when you get that "Wow" feeling. Here are three types of symmetries and a tool to go along with each of them.    
     
Symmetry about the y-axis shows that the function of x, f(x), produces the same values for y when the domain of x is divided at x = 0.

If f(x) = f(-x) then,
f(x) is symmetric about the y-axis.		   "Hey, what about my needs", the function mumbled and grumbled. He paced in Nowhere hands clasped behind his back, shoulders slumped forward and his gaze fixed downward. "With just a bit of understanding so many could avoid confusion". He sat on Nothing and listed his needs:

(1) a domain D, I can use that for input
(2) a target T, that's where I'll put my output
(3) a rule, every x D specifies a unique y T

"It's because of me", proclaimed f(x)!
     
Symmetry about the x-axis shows that the relation of x, r(x), produces both positive and negative values of equal magnitude of y over the entire domain of x.

If r(x) = ±y then r(x) is symmetric about the x-axis.		   "You can pick your friends but you can't pick your relations", the relation laughed at everyone, or was he laughing at himself?

"ill formed ideas" time, x D | r(x) = ±y T

Well, this ensures that the result of a relation is ambivalent. The relation is still laughing. But at whom, he or me?
     
Symmetry about the Origin shows that for every point ( x, y ) on the graph of the function there is also a point ( -x, -y ).

If f(x) = -f(-x) then,
f(x) is symmetric about the Origin.

A double negative is a little hard to look if you like, use this test written as follows,
-f(x) = f(-x).		   The xy-axes of the Cartesian Coordinate Plane provide the Ordered Pairs designating names for locations. The coordinate ( 0, 0 ) is called the Origin.
     
x2 + y2 = 1 can be rewritten to solve for y in terms of x,
r(x) = ( 1 - x2 )1/2.

Notice that x2 is always positive and must be less than or equal to one in order to solve for r(x).
D = { -1 ≤ x ≤ 1 }.

The initial step in r(x) is to square x so we know that r(x) = r(-x) and r(x) is symmetric about the y-axis. We can cut the size of D to solve for in half.

The final step in r(x) results in a square root so we know that r(x) = ±y and is symmetric about the x-axis.

I have just wasted an enormous amount of time trying to think of an interesting way to explain symmetry about the Origin using this example. So, (-x)2 + (-y)2 = 1 = x2 + y2 does the job.		    
     
Intercepts    
Some functions produce a graph that touch or cross an axis. If such an event occurs when x = 0, it is called the y intercept. Likewise, when y = 0, it is called the x intercept. These points are easy to find if they exist. Combined with symmetry we can learn many things about the nature of the function. Mathematical Models may appear complex but through some effort to learn their properties we can gain a greater understanding of what they represent.    
    "Wait, hurry up. Listen, do you smell that? Could be what? I think I see."
   
     

Lines - section two

   Tools to use Calculus - section topics