Random math questions

[#1procrastinator]

Ah thanks laser. How're you finding the calculus part by the way? the notes look insane lol

[Lasercookie]

Ah thanks laser. How're you finding the calculus part by the way? the notes look insane lol

Yeah, I'm finding it pretty interesting. Its mostly the suggested questions from that Trench Real Analysis book where I've been spending the most time on. I've also been reading through Spivak Calculus (the last part of it is basically what we're covering now) and a couple of other books too.

I wish the Linear Algebra part of the course would speed up the pace though, I really hope we're done with chapter 4 of the book now.

[#1procrastinator]

Yeah, I'm finding it pretty interesting. Its mostly the suggested questions from that Trench Real Analysis book where I've been spending the most time on. I've also been reading through Spivak Calculus (the last part of it is basically what we're covering now) and a couple of other books too.

I wish the Linear Algebra part of the course would speed up the pace though, I really hope we're done with chapter 4 of the book now.

Haha, most of the time I've spent so far on the calculus part is trying to decipher the notes!

One plus is that the pace of LA gives me time to wrap my head round the calc stuff 

[#1procrastinator]

Having a go at Trench's questions...all feedback is appreciated 

Question 7 from 4.1

7) Suppose that (finite) and, for each for large n. Show that

Der Versuch
By hypothesis,   such that and

by the triangle inequality (should I reorder that around? I added first rather than add and subtract to use the triangle inequaliy).

and hence by definition of the limit.

[#1procrastinator]

Assignment question:



If I'm not wrong, we're supposed to prove this , right? If so, anyone feel like pointing me in the right direction? I'm thinking perhaps mean value theorem only because I've seen them used to prove inequalities but still not quite sure how to apply that here.

[Lasercookie]

Assignment question:

(Image removed from quote.)

If I'm not wrong, we're supposed to prove this , right? If so, anyone feel like pointing me in the right direction? I'm thinking perhaps mean value theorem only because I've seen them used to prove inequalities but still not quite sure how to apply that here.

I was stuck on this one too. The way I've gone about it is to have a close look at the hint (you can see what it says in the hint is true by substituting in the equation we have for the unit tangent vector and using the various properties of dot products), the equation we're told to consider and also look at the results you proved in the previous parts of the question (you can see they carry over the multivariable case if you start off with x(t,s) etc. instead of x(s) with your proof of those).

I had this theorem pointed out to me, but you can interchange the order that you take the partial derivatives http://en.wikipedia.org/wiki/Symmetry_of_second_derivatives#Formal_expressions_of_symmetry I haven't been able to prove this yet, I'm assuming it's in the textbook somewhere.

[#1procrastinator]

Thanks - did you end up using the mean value theorem at all? I probably should finish off the other parts- did not immediately see how they were relevant to the last one (aside from the fact these are all parts of one question :p) until you pointed it out.

Wasn't that proven in the lectures? I'd thought of that too but it didn't really help, plus I wasn't 100% sure that the functions satisfied the necessary conditions. But x is infinitely differentiable, so that implies continuity therefore the symmetry between the second partials exist...yeah? But perhaps I should do all the previous questions before attempting this one hahaha

---

By the way, in the section on arc length parametrisation in the notes


Is the variable 's' meant to be time or length? Because there's  which seems to say s is a dummy variable for time, but in line 4 seems to imply that s is a length.

[nubs]

Is it possible to express the solutions to 1+e^x*cos(x) = 0 as a general solution?
Or 1+e^x*sin(x)=0?

Something similar to sin(x)=0, so the general solution would be x=n*pi where n is an element of Z

Thanks!

[#1procrastinator]

I was stuck on this one too. The way I've gone about it is to have a close look at the hint (you can see what it says in the hint is true by substituting in the equation we have for the unit tangent vector and using the various properties of dot products), the equation we're told to consider and also look at the results you proved in the previous parts of the question (you can see they carry over the multivariable case if you start off with x(t,s) etc. instead of x(s) with your proof of those).

I had this theorem pointed out to me, but you can interchange the order that you take the partial derivatives http://en.wikipedia.org/wiki/Symmetry_of_second_derivatives#Formal_expressions_of_symmetry I haven't been able to prove this yet, I'm assuming it's in the textbook somewhere.

I got up to a point where I had to show that the but I couldn't finish it in time. Have huge doubts on the validity of my arguments though.

[#1procrastinator]

Find the Jordan canonical form of an nxn matrix A whose entries are all equal to 1.

I suppose first thing you to do would be to calculate det(A-eI)=0 to find the eigenvalues e?
I don't know my determinant stuffs well but from playing around with an eigenvalue calculator, would the eigenvalues be 0,...,n?

[kamil9876]

It's good to firstly look at the rank. We have as all the rows are the same. This means that the kernel of is dimensional. Thus we have linearly independent eigenvectors with eigenvalue . This tells us that there is at most one more eigenvalue and the corresponding eigenspace is one dimensional. We can find it by inspection, but another way would be to use the fact that the sum of the eigenvalues (counted with multiplicity) is the trace. Thus so we have that the eigenvalues are lots of and one . Since we have linearly independent eigenvectors, this tells us that the JNF is diagonal, so the JNF is the diagonal matrix with diagonal 0,0,0,0..0,n

[#1procrastinator]

Thanks kamil - but how did you get the eigenvalue 0?

[kamil9876]

Because, as I said, has kernel of dimension . Therefore we can find linearly independent vectors such that , i.e . So these are eigenvectors with eigenvalue .

[#1procrastinator]

This one feels like it should be really easy - express in terms of hyperbolic functions

[lzxnl]

This one feels like it should be really easy - express in terms of hyperbolic functions

Isn't artanh x [tex] ln\frac{1-x }{1+x} [/tex} or something?