As an introductory text, and I have a bunch, look at (an earlier edition of) Joseph Gallian's Contemporary Abstract Algebra. It will always be on my shelf. A second ood alternative is Dummit and Foote's. As far as Analysis, the standard I believe is Rudin's Principle's of Mathematical Analysis. For a slightly softer introduction (but still rigorous), Bartle's Real Analysis is fine book that is easy to read.
The two subjects are staples for higher mathematics, and that is why most uni's require their graduate students to be familiar with them. In a nutshell, algebra is the study of mathematical structures and their influences, implications, and generalizations. A more advanced treatment of sets and their operations, but far more general then what an average US highschool student would consider "algebra." As an example: consider the set of Integers {...2,1,0,1,2,...}. This set is naturally endowed with addition (primary) and multiplication (secondary).. The secondary operation "distributes" over the primary. One wishes to generalize the properties of this kind of set by saying it is a Ring, and more specifically what they call an Integral Domain. Another example: the rational numbers... this is the set of Integers together with their multiplicative inverses. One generalizes this to a (countable) Field which is a set that contains objects which have additive and multiplicative inverses, indentities and inherits all features from the Ring associated to it.
Introdictory Analysis is essentially a rehash of The Calculus, going far deeper, seeing and proving its foundations. In fact, many call Introductory Analysis "Advanced Calculus." One inportant thing to note is that Real Analysis is fundamentally different from "Real Algebra" for the simple fact that one cannot create the real numbers as an "algebraic evolution" from that of the rationals. The notion of limit is required. But even so, these two subjects interact heavily. The Reals form a Field, and hence can be studied algebraically as well. The set of all real polynomials forms a Ring. The set of all 11 continuous functions forms a Group (a set with only one operation instead of two, but still maintains inverses). The set of all continuous functions forms a Monoid (a set with an operation, which need not have inverses). The "operation" or "multiplication" in the latter two is actually function composition. To say that a set has a binary operation means that when you perform the operation on two things of this type, you get another one of the same type.


08232010, 07:40 AM #61
Last edited by MiscMathematician; 08232010 at 07:57 AM.
** Official Math Thread http://goo.gl/VpMlfQ **

09052010, 04:54 PM #62
Halp!
Supposed to find the limit as x approaches 3+, 3, and 3. I understand how limits work but I'm confused by the system of equations. If I plug 3.001 into the first equation, then x is greater than 3. WHAT MEAN?"Watch your thoughts, for they become words. Choose your words, for they become actions. Understand your actions, for they become habits. Study your habits, for they will become your character. Develop your character, for it becomes your destiny."

09052010, 05:02 PM #63
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3+ is what u approach from the right, so just plug in 3 for the equation where x>3 to get the limit.
3 use the equation for x<3
and for 3 just use the equation where x=3
for you brahs that know about the fourier series, i have a question
if f(x)=sin^2(x), how do you show that the fourier series is 1/2cos(2x)/2. i know that the 2 are equivalent, but going through and solving for the coefficients i keep getting 1/2, but not with the other part of the answer.**** Football Crew
TB Lightning

09052010, 05:28 PM #64

09052010, 05:50 PM #65

09052010, 05:59 PM #66
Ok, I put 3 in as the limit for all answers and it says they're all correct now, which makes sense. The limits are both 3 as x approaches 3 and 3+, so that makes sense that the limit as x approaches 3, is 3 since it's the same on both sides. I still don't understand how that works for the function 0, x=3 though.
"Watch your thoughts, for they become words. Choose your words, for they become actions. Understand your actions, for they become habits. Study your habits, for they will become your character. Develop your character, for it becomes your destiny."

09052010, 06:06 PM #67

09052010, 06:34 PM #68

09052010, 08:04 PM #69

09062010, 03:16 AM #70

09062010, 04:40 AM #71

09062010, 10:23 AM #72
Reps for not just answers, but explanation of this stuff so I can actually understand wtf is going on. It's labor day weekend and I'm pretty sure I'm not gonna be able to get a hold of my instructor, but this online hw is due 1am tomorrow. I have waded through hours of videos and none of them have shown problems like these, nor do I know how to enter these into wolframalpha.
1.)
If possible, choose k so that the following function is continuous on any interval:
2.)
Let
Find each point of discontinuity of f, and for each give the value of the point of discontinuity and evaluate the indicated onesided limits. If you have more than one point, give them in numerical order, from smallest to largest.
3.)
Let
Find:
4.)
Let f be defined by
Find (in terms of m) 1+ and 1
Find the value of m so that
"Watch your thoughts, for they become words. Choose your words, for they become actions. Understand your actions, for they become habits. Study your habits, for they will become your character. Develop your character, for it becomes your destiny."

09062010, 11:37 AM #73
Let k=20. The function is continuous at 2 then, and everywhere else.
2.)
Let
Find each point of discontinuity of f, and for each give the value of the point of discontinuity and evaluate the indicated onesided limits. If you have more than one point, give them in numerical order, from smallest to largest.
Since both functions are continuous, the only (potential) point of discontinuity will be on the bounds of the piecewise definition. Plug in x=5. If the values agree, it is everywhere continuous, if they do not, there is a single discontinuity.
3.)
Let
Find:
Sorry, should be 2(9+6h+h^2) (too lazy to retype the tex)
4.)
Let f be defined by
Find (in terms of m) 1+ and 1
Find the value of m so that
Plug in 1 into the first and the second. You will get the two values it is asking for (in general the + sign means use the function for which x > that value,  for <). Set these two values equal to each other and solve for m.** Official Math Thread http://goo.gl/VpMlfQ **

09062010, 01:05 PM #74

09062010, 01:56 PM #75
That's what I thought before. I factored this in my head the first time and forgot to cancel out the (x2) so I got lost.
I'm still not understanding this. If they're both continuous, then how can there be discontinuity? If I plug in 5, then I get 35 on one function and 25 on the other. Other than that, they're both infinite functions.
So, I worked this out before and got (6h+h^2)/h but it doesn't accept that as an answer, nor does it accept 6h+h, 6, or 6. Also when h>0 then the limit is 6, but I'm not seeing how it comes out as a negative.
I'm still doing something wrong. I got 1.6 for x>1+ and 4 for x>1 with m=0. The problem is that when m=0, the limit of 1+ is 8 and the limit of 1 is 8.
Sorry for asking so many dumb questions. It's been a few semesters since I took algebra."Watch your thoughts, for they become words. Choose your words, for they become actions. Understand your actions, for they become habits. Study your habits, for they will become your character. Develop your character, for it becomes your destiny."

09062010, 02:01 PM #76

09062010, 02:05 PM #77

09062010, 02:14 PM #78
[QUOTE=Errorist;543664583]That's what I thought before. I factored this in my head the first time and forgot to cancel out the (x2) so I got lost.
I'm still not understanding this. If they're both continuous, then how can there be discontinuity? If I plug in 5, then I get 35 on one function and 25 on the other. Other than that, they're both infinite functions.
Pretend the function is a road and when you plug in your x, you get your current altitude. The road is all nice until x=5. But then you fall 10 meters and die. This discontinues your life.
So, I worked this out before and got (6h+h^2)/h but it doesn't accept that as an answer, nor does it accept 6h+h, 6, or 6. Also when h>0 then the limit is 6, but I'm not seeing how it comes out as a negative.
I'm still doing something wrong. I got 1.6 for x>1+ and 4 for x>1 with m=0. The problem is that when m=0, the limit of 1+ is 8 and the limit of 1 is 8.
Sorry for asking so many dumb questions. It's been a few semesters since I took algebra.** Official Math Thread http://goo.gl/VpMlfQ **

09062010, 02:20 PM #79

09062010, 03:08 PM #80
Ok, that makes a little bit more sense, but there's an infinite amount of Xs where this occurs. With that, it also says "If you have more than one point, give them in numerical order, from smallest to largest. If you have extra boxes, fill each in with an x."
So lets say the first point is 5. According to the left hand side of the limit will be 35 and the right hand side will be 25. How many and which of these discontinuities am I supposed to list if there's an infinite amount of them? It's asking for 3 different points. Should I just give it the 1 point with it's left and right hand side limits, then fill the rest of the boxes up with Xs?
Oh ok, I forgot to distribute the  to 6h+h^2 when I opened up the parentheses. Also, I was confused with which h to cancel out. Is the h being canceled or factored out? Because I didn't know I could cancel out 1 of each. I thought it was just one of them.
Ah ****. I originally got 0=7m because I added 8 to both sides instead of subtracting. I need to stop trying to do **** in my head."Watch your thoughts, for they become words. Choose your words, for they become actions. Understand your actions, for they become habits. Study your habits, for they will become your character. Develop your character, for it becomes your destiny."

09062010, 03:47 PM #81
Sweet I'm gonna be doing a lot of Dynamical systems in my masters. Any particular interests? I quite like dealing with bifurcation theory and stable/unstable manifolds although I would like to understand chaos better.
Hey man could you maybe elaborate on where you said "but there's an infinite amount of Xs where this occurs". There is only one discontinuity in that function which is at x=5. Which others do you think there are?
If you have a fraction of the form...
then this can also be written as so everything gets divided by the denominator.Last edited by Muckle_Ewe; 09062010 at 03:53 PM.

09062010, 04:06 PM #82
 Join Date: Aug 2007
 Location: Miami, Florida, United States
 Age: 32
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Just gotten into it, but it may play a big role in my thesis. Im using the approach of a reconstructed phase space from a 1dimensional time series with the time embedding approach. Then in phase space I can detect desired features of my time series (i.e. in phase space oscillations have a distinct). Im working on realtime algorithms though so its not so mathematical as stability theory and bifurcations. Chaos is interesting, when Ive read about it, it is a lot more enjoyable than most topics

09062010, 04:07 PM #83
It was asking for at least 3 points of discontinuity, except it was just to trick me. I put in just the 1 point with both of its one sided limits and kept the rest of the entry boxes blank and it said it was correct.
Oh thanks, duh. I have this habit of either forgetting or over complicating the simple stuff after I've been introduced to harder stuff. I do this on quizzes and tests also. I'll be able to calculate the hard problems and will miss or flat out forget how to do the easiest problems. It's embarrassing."Watch your thoughts, for they become words. Choose your words, for they become actions. Understand your actions, for they become habits. Study your habits, for they will become your character. Develop your character, for it becomes your destiny."

09152010, 09:50 PM #84
K wtf.
I got this for the derivative and I know it's right
but I can't figure out f'(1)."Watch your thoughts, for they become words. Choose your words, for they become actions. Understand your actions, for they become habits. Study your habits, for they will become your character. Develop your character, for it becomes your destiny."

09152010, 10:20 PM #85

09152010, 10:27 PM #86

09162010, 01:50 PM #87
I understand HOW to use the rules in calculus, but I'm having difficulty knowing WHEN to use them. Is there an easy way of determining which rules need to be applied, like certain things I should look for that are a dead give away? For example, I thought I was supposed to use the quotient rule on a problem the other day because it was in fractional form, but I was instead supposed to use the power rule.
"Watch your thoughts, for they become words. Choose your words, for they become actions. Understand your actions, for they become habits. Study your habits, for they will become your character. Develop your character, for it becomes your destiny."

09162010, 02:14 PM #88

09162010, 03:59 PM #89

09162010, 04:42 PM #90
 Join Date: Aug 2009
 Location: United Kingdom (Great Britain)
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I don't really understand your question completely and this advice i'm about to give may come as no surprise.
But you need to do problems  lots of problems!
It's great that you're using this thread for help. Use the advice and techniques people have given here and do example problems. The best way to learn anything is through practice and there are a lot of calculus examples on the internet.
I can't quite remember if I've posted this before, but Paul's Online Notes is absolutely one of the best calculus resources on the internet.
http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspxIf you do what you've always done, you'll get what you always got
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