Help with Integrals? - Shin's Story

[shintashi]

Hi.

I've been having a lot of difficulty wrapping my head around

(1,0) is the same as the set {1,{0}}

1. I do not know what a Set is, and the definitions I've found do not make any sense.
2. I still do not understand integrals
3. I do not know what this {1,{0}} means.
4. all my attempts to get some one familiar with integrals to help have been answered with "it would take too much time" or "why do you want to learn integrals?".

Apologies if this is misplaced.

-Shin

 

[neutrino]

What exactly does not make sense?

A set, loosely defined, is a finite or infinite collection of objects, which may have some common attribute(s). As I said, it is not a very rigorous definition, because I have not defined the term 'collection', but you get idea.

As for integrals, it will be clear if you have good idea of what functions are. Again, you should tell us more on what/which part of the topic you don't understand.

 

[shintashi]

What exactly does not make sense?

A set, loosely defined, is a finite or infinite collection of objects, which may have some common attribute(s). As I said, it is not a very rigorous definition, because I have not defined the term 'collection', but you get idea.

As for integrals, it will be clear if you have good idea of what functions are. Again, you should tell us more on what/which part of the topic you don't understand.

by common attribute do you mean denominate numbers?, and by set do you mean something like (1,2,3..n) or something else? I'm not certain that I know what a function is. F of X is a function of variable X, and the f(x) is like a brief computer program for input (x) and output.. I am not sure what the output is written as, but I understand the stuff inbetween is the equation.

can you break down a basic example of an integral into stupidspeak?

 

[neutrino]

by common attribute do you mean denominate numbers?

I don't know what you are referring to by "denominate numbers", but some examples of a set with common properties are: All numbers between 1 and 10,The set of all integers, the set of all numbers x which satisfies a certain equation, like x²-5x+6=0, etc.

and by set do you mean something like (1,2,3..n) or something else?

When writing down a set is roster notation, one usually use the curly braces {}. So the set of all integers in roster notation would be {...-3,-2,-1, 0, 1, 2, 3,...}.

the f(x) is like a brief computer program for input (x) and output

You roughly have the right idea, but I think you may need to a bit more reading and do some problems.

can you break down a basic example of an integral into stupidspeak?

Well, you need to get a grip on functions before I can tell you what integrals are, even in stupid speak. ;)

 

[shintashi]

I don't know what you are referring to by "denominate numbers", but some examples of a set with common properties are: All numbers between 1 and 10,The set of all integers, the set of all numbers x which satisfies a certain equation, like x²-5x+6=0, etc.


When writing down a set is roster notation, one usually use the curly braces {}. So the set of all integers in roster notation would be {...-3,-2,-1, 0, 1, 2, 3,...}.



You roughly have the right idea, but I think you may need to a bit more reading and do some problems.


Well, you may need to get a grip of functions before I can tell you what integrals are, even in stupid speak. ;)

denominate numbers are numbers attached to objects or units of measure, such as 10 meters, 10 dollars, etc.

I'll work on functions, its just very odd. I tested out of all math requirements for my degree (which arrived yesterday) but for transfer I wanted to be prepared to do higher level math, so I've been reading up on the history of Math, Archimedes, Zeno, Euler, Leibniz, Berkeley, hilbert, Newton, and next is Fermat. Euler introduced a lot of terms that our public library has essentially nothing useful on, so we have to order books through library loan, by name specifically, or as they recommend "use the internet".

 

[cybercrypt13]

First of all I would like to tell you that you're not ready for Integrals if you're not completely understanding sets and functions but I'll attempt to explain what I know.

1) A set is any group of numbers (to keep it simple) just like the other guy said. It can be made up of anything but let's say you have a set of 1 to 5 which would be 1,2,3,4,5...

2) A function: Well, to make it real simple thing of your x,y coordinate system. X is left to right and Y is up and down. If I were to give you an X and give you a function such as f(x)=x+4*3, then I'm telling you that in order for you to find the Y value of any X value that I give you, you have to plug it into the function f(x) "stated f of x" and it will tell you were your Y point is.

So instead of me giving you a bunch of points on a graph that you have to plot out one at the time, I can give you a function or formula that if you take any X value it will give you the Y value.

3) Now, before you get anywhere close to Integrals you have to first move through what Derivatives are. This is a rather complex conversation on its own, but basically if you were to think of drawing a smooth curved line through all your points and I were to ask you to draw me a tangent line at any particular point on the graph, you wouldn't know how. Derivatives help you do this. They allow you to take any point on the graph and the original equation you were given and determine its tangent line.

4) It turns out tangent lines are much more important than you would think. Given a function, a first derivative of that function will give you the velocity curve of that function. In other words, the derivative will tell me a particles velocity at any point on the curve. Then taking its second derivative you'd be able to determine its acceleration.

So, in case you are still with me, let's say that I give you a question such as a car is going 50mph and it stops at a rate of 22ft/s^2. Can you tell me the distance it will take it to stop? Well, armed with the knowledge of what a derivate can do, you can then use Integrals to determine your original derivative to work the problem.

An Integral is quite simply the ability to take a function and determine what its original derivative was. Remembering that the first derivative is velocity and the second is acceleration, the above question gives you acceleration. You take its integral and you'd have the original velocity equation. Plug in what you know is the velocity of 50mph in this case, and then take its integral and you'd be at a distance formula to sove for S.

You're not going to understand this stuff unless you start working problems and a quick read over some material is not going to get you to that point. You need lots of practice... :-)

Have fun and I hope that helps.

glenn

 

[Data]

Here are some rough ways to think about it (there will be some technical loopholes in what I'm going to tell you, and these aren't at all rigorous. But those are things that you won't have to worry about for a while.):

A set is a collection of objects. Typically sets are denoted by something like {x | P}, by which we mean that $x$ is in the set if and only if $x$ satisfies property $P$. Here are some examples:

$$\mathbb{Z} = \{ x | x \mbox{ is an integer}\}$$

$$\mathbb{N} = \{ x | x \mbox{ is a positive integer}\}$$

$$\mathbb{R} = \{ x | x \mbox{ is a real number}\}$$

$$GL_2(\mathbb{R}) = \{ x | x \mbox{ is a } 2\times 2 \mbox{ matrix with real entries and a nonzero determinant}\}$$

$$C(\mathbb{R}) = \{ f | f \mbox{ is a continuous function with domain and codomain } \mathbb{R} \}$$

That last example probably will not help you if you don't know what a function is, so let's take a look at those. Let $X$ and $Y$ be sets.

By a function $f$ with domain $X$ and codomain $Y$ (denoted $f: X \rightarrow Y$), we mean a rule for assigning an object in $Y$ to every object in $X$.

So, given any $x$ in $X$ (denoted $x \in X$), the function $f$ associates some (single) $y \in Y$ to $x$. We denote this by $f(x) = y$, and write f maps x to y (functions are also frequently referred to as maps), or y is the image of x under f.

Notice that there aren't many restrictions on what $f$ can do. The only real restriction is that for every $x \in X$, we assign exactly one $y \in Y$. This invites looking at types of functions that do satisfy some other properties:

For example, according to our definition, we could have two elements (read: objects) of $X$, call them $x_1$ and $x_2$, to which the function $f$ assigns the same element $y$ of $Y$, ie. with $f(x_1) = f(x_2)$. Suppose we prohibit such an occurrence: ie. we ensure that our function satisfies $f(x_1) \neq f(x_2)$ whenever $x_1 \neq x_2$.

It turns out that this special class of functions is very interesting a lot of the time. We call such functions injective, or, more colloquially, "one-to-one."

Another interesting property comes along in a different way: Note that while the definition assures us that each $x \in X$ has an associated $y \in Y$, that it says nothing about the reverse statement - given any $y \in Y$, there is not necessarily any $x \in X$ with $f(x) = y$. Now suppose we add the restriction to $f$ that given any $y \in Y$, we can always find some $x \in X$ with $f(x) = y$.

We call functions satisfying this restriction surjective, or "onto." Surjective functions also form an interesting subclass. Even further, imagine that we restrict $f$ such that it is both injective and surjective. We call such functions bijective, or "invertible" (you should try to figure that one out for yourself!).

Here are some examples of functions:

$$f: \mathbb{R} \rightarrow \mathbb{R} \mbox{, defined by } f(x) = x \ \mbox{ for every } x \in \mathbb{R}$$

$$f: GL_2(\mathbb{R}) \rightarrow \mathbb{R} \mbox{, where the image of any matrix in the domain is its top left entry}$$

$$f: \mathbb{N} \rightarrow \mathbb{N} \mbox{, defined by } f(x) = x^2 \mbox{ for every } x \in \mathbb{N}$$

$$f: \mathbb{N} \rightarrow \mathbb{N} \mbox{, defined by } f(1) = f(2) = 1, \ f(n) = f(n-1) + f(n-2) \mbox{ for every } n \in \mathbb{N} \mbox{ with } n \geq 3$$

Of these examples, one is injective but not surjective, one is surjective but not injective, one is bijective, and one is neither injective nor surjective. You should try to figure out which is which (the last example is defined recursively. You should look up what that means if you aren't sure).

Here are some examples of things that are not well-defined functions, even though they may appear to be so. You should try to figure out what it is about their proposed definitions that does not match the definition I gave above for a function.

$$f: \mathbb{R} \rightarrow \mathbb{R} \mbox{, defined by } f(x) = \log x \mbox{ for every } x \in \mathbb{R}$$

$$f: \{ x | x>0 \mbox{ and } x \in \mathbb{R}\} \rightarrow \mathbb{R} \mbox{, defined by } f(x) = y \in \mathbb{R} \mbox{ whenever } y^4 = x \mbox{ and } x\in \mathbb{R}$$

$$f: C(\mathbb{R}) \rightarrow C(\mathbb{R}) \mbox{, defined by } f(g) = 1/g \mbox{ for every } g \in C(\mathbb{R})$$

(when $g: \mathbb{R} \rightarrow \mathbb{R}$ is a function, denote by (1/g) the function mapping $x$ to $1/g(x)$ whenever $g(x) \neq 0$)

 

[shintashi]

thanks everyone for the help.

notation 1: concerning derivatives - i had a vague idea of derivatives from a book called "calculus made easy" but got deadlocked at delta/change which I later learned in Macroeconomics. I didn't learn what a "Sigma/sum" was until physics, and only an hour ago while reading another calculus book and after weeks of extensive wiki articles (summation, infintessimals, and multisets to name a few) and other books and articles, got this far.

notation 2: data - your x (E) X symbol is scary and I only know of it being some kind of math version of axiomatic logic for element or equivelence. While doing a cursory examination of greek math symbols I discovered that the same symbols used in one field have completely different meanings in another field. (my GF is studying for a Biology/Genetics degree and most of the symbols in Calculus are in Biology, and have nothing in common).

I will make it a point to read your article several times till I get it. Thank you for the effort.

-shin

 

[neutrino]

That 'x E X' just means that x 'is a member/element of' the set X.

 

[Data]

I (very) roughly defined a set up there. If $X$ is a set, the notation

$$x \in X$$

just means that $x$ is one of the objects in $X$ (for which we say x is an element of X).

And yes, you are certainly correct that almost every notation you'll see in mathematics, or physics, or any other discipline, is reused elsewhere, and usually many times, with completely different meanings. There is an overabundance of fields of research and a shortage of symbols! You might ask "well, why not come up with some more symbols?". The fact is that with a little bit of experience, and some context, it is not very difficult to guess what most notations mean.

I couldn't count the number of times that I've been assigned problems with undefined notations in them (physicists are quite lax about such things!). Even in obscure cases it's possible to guess what they mean - I have very rarely found the need to ask for clarification from a professor (in a particle physics course earlier this term, an expression of the form $\langle \psi_1|j_0(rk/2)|\psi_2 \rangle$ appeared, with no explanation anywhere for what the $j_0$ could mean. A classmate asked me what I thought. My reply was "it must be a Bessel function." They were incredulous; we hadn't seen any Bessel functions before in the course, so why should they pop up out of nowhere now? I was quite happy to be vindicated a few days later!).