1. Here's a link to the proof:
http://math.stanford.edu/~lekheng/flt/wiles.pdf

I'm lost at the first sentence.

He claimed that he had a 'truly marvelous' proof, however the margin in his copy of Arithmetica was 'too narrow' to contain it, so it was never found. His proof could have easily been flawed.
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4. Here's a link to the proof:
http://math.stanford.edu/~lekheng/flt/wiles.pdf

I'm lost at the first sentence.
Christ it's like he refers to at least twenty other incomprehensible papers every paragraph.

5. Going from the third to last to the second to last line will only give non-zero solutions. It should be obvious from the third to last line that x = 0 is another solution.
he said positive solutions
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he was either completely wrong or trolling (probably the latter)

7. he said positive solutions
so he did
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8. Can someone explain the Tau function to me?
http://en.wikipedia.org/wiki/Tau-function
If there are no characters on top of the sigma and pi, does that mean n goes to infinity? Are those sequences at all?
What does Tau: N arrow Z mean? Best I can figure it means the natural numbers map onto the integers and this is somehow related to the function's values.
Why is this notation necessary if there already existed a function that was the twenty-fourth root of this one?

9. If there are no characters on top of the sigma and pi, does that mean n goes to infinity? Are those sequences at all?
Yes, it just means for all N greater or equal to zero, and I dunno what you mean by "Are those sequences at all?" but there are plenty of convergent infinite series.

What does Tau: N arrow Z mean?
It means the function Tau takes the set N and maps it to Z

So N is the domain of the function and Z is the co-domain. It's pretty common (and useful) notation.
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10. Yes, it just means for all N greater or equal to zero, and I dunno what you mean by "Are those sequences at all?" but there are plenty of convergent infinite series.
He couldn't tell if they were actually sequences or something else that uses as similar notation because just putting n greater than or equal to 1 like that isn't the way it's usually taught.
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11. Explaining proofs using the formal limit definition to a first year calc student is difficult.
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12. Deltas and epsilons everywhere
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13. Deltas and epsilons everywhere
I think calc classes should teach very basic formal logic. Just quantifiers, connectives, and implication would do. I think that would help make things like the formal limit definition much clearer. I can't tell you how many people misunderstand that the epsilon represents ALL possible positive reals, but delta can be anything per epsilon that makes the statement true. Picking up basic formal logic on my own for my real analysis class made formal definitions like that seem so obvious whereas I didn't have a fucking clue what they meant when I was taking calc 1.

Edited:

That or delay the formal definition until analysis because if kids are just being taught to manipulate symbols and not understanding how something actually works, they're not really gaining anything. Particularly when it's something with few actual applications like the limit definition. In calc 1, we were shown how to prove something using it and told explicitly we didn't really need to understand it.
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14. Hey guys, I'm starting Numerical Analysis and Probability Theory and Statistics next semester. Anything I should know before hand? Which one requires the most work? I was thinking of getting the books already now and bringing one during my vacation.

15. I think calc classes should teach very basic formal logic. Just quantifiers, connectives, and implication would do. I think that would help make things like the formal limit definition much clearer. I can't tell you how many people misunderstand that the epsilon represents ALL possible positive reals, but delta can be anything per epsilon that makes the statement true. Picking up basic formal logic on my own for my real analysis class made formal definitions like that seem so obvious whereas I didn't have a fucking clue what they meant when I was taking calc 1.

Edited:

That or delay the formal definition until analysis because if kids are just being taught to manipulate symbols and not understanding how something actually works, they're not really gaining anything. Particularly when it's something with few actual applications like the limit definition. In calc 1, we were shown how to prove something using it and told explicitly we didn't really need to understand it.
I never had that, and what you describe is EXACTLY what troubled me when I tried to understand the epsilon-delta proofs.

16. I know this thread is quite old but I would like to know if this is good. I never did matrix nor Jacobian before but I'm learning by myself and I decided to try on spherical coordinates.

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17. i had a maths competition today with a few tough questions that I couldn't answer, and I thought you guys might be interested to see 'em

3. Find a function f(x) whose derivative is equal to its inverse.
5. The equilateral triangle ABC has a circumscribed circle. A line originating at A passes through the line BC at the point D and through the circle at the point Q. Show that 1/DQ = 1/CQ + 1/BQ.

there was another one i didn't get but it was lame

18. snippppp

19. Ok well let y=f(x), dy/dx=f'(x), and we also have x= f^-1(y). Now we need f such that f^-1(y)=f'(x), so we need x = dy/dx xdx = dy so y=(x^2)/2 + c. (defined for x>0 or x<0)
does that work? unless I'm missing something the derivative of that is x and the inverse is sqrt(2x-c)
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20. He just didn't do it right.

y=f(x)
1/y=f^-1(x)
dy/dx=f'(x)

1/y=dy/dx => ydy=dx => y^2/2=x => y=sqrt(2x)

dy/dx=sqrt(2x)/2x=y^-1
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21. He just didn't do it right.

y=f(x)
1/y=f^-1(x)
dy/dx=f'(x)

1/y=dy/dx => ydy=dx => y^2/2=x => y=sqrt(2x)

dy/dx=sqrt(2x)/2x=y^-1
1/y = f^-1(X).... huh?

22. He just didn't do it right.

y=f(x)
1/y=f^-1(x)
dy/dx=f'(x)

1/y=dy/dx => ydy=dx => y^2/2=x => y=sqrt(2x)

dy/dx=sqrt(2x)/2x=y^-1
Uh, x = f^-1( y )
f^-1 is used to denote the inverse of function f.
Where y = f( x ), 1/y = f( x )^-1 not f^-1( x ).

23. My FP2 exam is tomorrow. I assume you'd all find it trivial if you did it now, correct?

24. My FP2 exam is tomorrow. I assume you'd all find it trivial if you did it now, correct?
Our school didn't do FP2 :saddowns:
I've got M2 tomorrow, should be a cakewalk.

25. 3. Find a function f(x) whose derivative is equal to its inverse.
That's an interesting question. Tricky because the inverse of a function is not easy to work with
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26. Our school didn't do FP2 :saddowns:
I've got M2 tomorrow, should be a cakewalk.
Don't you need to do FP1 and FP2 for the Further Maths A Level?

27. Don't you need to do FP1 and FP2 for the Further Maths A Level?
No, there are required modules sure but you can pick and choose.
We did FP1, FP3, FP4, S2 and M2

28. My FP2 exam is tomorrow. I assume you'd all find it trivial if you did it now, correct?
I have FP2 tomorrow as well, Edexcel?

29. does that work? unless I'm missing something the derivative of that is x and the inverse is sqrt(2x-c)
Oops yeah I fucked up, used f^-1(y) instead of f^-1(x)

I'm thinking since we have: f'(x) = f^-1(x)

Could try go somewhere with: f(f'(x)) = 1

Or differentiate both sides w.r.t x:

f''(x) = 1/(f'(x))

Nevermind, I think its got something to do with the golden ratio, lets see if this works...

Okay think about this, differentiating something in form ax^n gives you anx^(n-1), inverting something in form ax^n gives you something in form (x/a)^(1/n) so for powers we have n-1=1/n, solving gives the golden ratio, then we just need to find a.

After calculating a I get:

f(x) = (2/1+rt5)^(2/(rt5 - 1))*x^((1+rt5)/2) I can't be fucked to check if it works.

30. How hard is Further Maths AS/A2 in general? I'm on track to self-study it because of a clusterfuck in my scheduling, and I was wondering if it was feasible for me to complete both in the span of a year.

31. My FP2 exam is tomorrow. I assume you'd all find it trivial if you did it now, correct?
Nope. I've just finished my first year of physics at university and I found FP2 harder than anything I've had to do this year in maths. Hope it went well

Edited:

How hard is Further Maths AS/A2 in general? I'm on track to self-study it because of a clusterfuck in my scheduling, and I was wondering if it was feasible for me to complete both in the span of a year.
I did both in one year, but I had quite a bit of help. It is difficult, but extremely useful for uni physics.

Edited:

To clarify a little further: further maths AS is piss easy, A2 is rock hard.
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32. I did both in one year, but I had quite a bit of help. It is difficult, but extremely useful for uni physics.

Edited:

To clarify a little further: further maths AS is piss easy, A2 is rock hard.
Thanks

33. You're welcome - also if you get stuck, post problems in this thread so we can have a go at them
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34. How hard is Further Maths AS/A2 in general? I'm on track to self-study it because of a clusterfuck in my scheduling, and I was wondering if it was feasible for me to complete both in the span of a year.
it's very hard, but good module choices and practise mean that it is doable. I personally prefer the mechanics modules, so am looking at c3, c4, d1, m2, fp1 and fp2 and have done c1, 2 s1 and m1.
I'm probably only going to do the further maths as level.

35. got 81% in first year of physics. YAY!
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36. Is the kth degree taylor series of a function guaranteed to be the best fit polynomial of degree k for that function?
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37. Is the kth degree taylor series of a function guaranteed to be the best fit polynomial of degree k for that function?
I don't think so

38. Is the kth degree taylor series of a function guaranteed to be the best fit polynomial of degree k for that function?
If we're just talking about best fit then I would say: if all the derivatives up to, and including the kth derivative exist and are continuous on an open interval which totally includes the region we want to approximate, I believe it will approximate it better than any other degree k polynomial through that point, in general, not for a specific point.

Of course for any specific point on the curve (as opposed to best fit in a general region) you wanted to approximate, you could do better by simply using a line that passes through the curve at the point that you want to approximate, and the point on the curve where you're taking the Taylor series around.

I'm not sure exactly how to quantify how good a fit it is, possibly by finding the area between the approx and actual curve.

39. i have decided that statistics are the würst
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40. If we're just talking about best fit then I would say: if all the derivatives up to, and including the kth derivative exist and are continuous on an open interval which totally includes the region we want to approximate, I believe it will approximate it better than any other degree k polynomial through that point, in general, not for a specific point.

Of course for any specific point on the curve (as opposed to best fit in a general region) you wanted to approximate, you could do better by simply using a line that passes through the curve at the point that you want to approximate, and the point on the curve where you're taking the Taylor series around.

I'm not sure exactly how to quantify how good a fit it is, possibly by finding the area between the approx and actual curve.
You have a good point. I think the problem is with my question, not with the answer. Every Taylor approximation is only going to be a "good fit" for a certain distance from the point that is being approximated around. It's also going to depend on how well behaved the function is.

I need to find a better way to phrase my question, I think.