Is there a non-ugly proof of the following identity:
\langle Ax,y \rangle = \langle x,A^*y \rangle
where A is an nxn matrix over, say, \mathbb{C}, A* is its conjugate transpose, and \langle \cdot , \cdot \rangle is the standard inner product on \mathbb{C} ^n.
Does anyone know how many chapters Courant & John's "Introduction to Calculus and Analysis, Volume II" has? Because I've found a 4 chapter reprint, and was wondering if I should get it.
Find all p \geq 0 such that
\sum_{k=1}^{\infty} \frac{1}{k \, (\log (k+1))^p}
converges.
It looks like the integral test is the most likely candidate, but I haven't been able to make any progress using it. I'd appreciate a push in the right direction.
Edit:
I've managed to prove...
I have the definition that if F is a finite field then a \in F is a primitive root if ord(a) = |F|-1.
Now what I don't understand is how exactly are there \phi(|F|-1) primitive roots?
(Note: This material is supposed not to use any group theory.)
I've been away for quite a while, and now I can't find the book forum. :confused:
Anyway, I'm looking for a book that covers introductory abstract algebra. The course outline is:
- (commutative) rings, fields, Z, Q, R, C, Z[i], polynomial rings, properties of Z.
- division algorithm...
I'm having trouble with the following limit:
\lim_{n \rightarrow \infty} 2^n \arcsin \frac{k}{2^n u_{n}} \text{, where \emph{k} is constant.}
I'm given that \lim u_{n} = u, where u is constant.
Apparently the book says the answer is \frac{k}{u}, but I can't figure out why.
Points A and B have position vectors (3i + j)m and (5i + 4j + 2k)m respectively. A particle moves from rest at the point A to the point B under the action of a constant force F newtons only. Given that the work done by the force in moving the bead from A to B is 34 Nm, find F.
I've found that...
A particle P describes the curve with polar equation r = a e^{\theta \sqrt{3}} \cosh 2\theta in such a manner that the radius vector from the origin rotates with uniform angular speed \omega. Show that the resultant acceleration of the particle at any instant makes an angle of 30 degrees in the...
A uniform solid cylindrical drum of mass 1.5kg and radius 0.5m is free to rotate about a fixed, smooth, horizontal axis which coincides with the axis of the cylinder. The axis is at a height of 2m above a horizontal table, and a light string of AB of length 4m has one end attached to the...
Can anyone help me with this question?
A uniform circular disc has mass 4m and radius r. A particle of mass m is attached to the end of the disc at point A of its circumference. The loaded disc is free to rotate about a horizontal axis which is tangential to the disc at the point B, where AB...
How do I find the moment of inertia of a solid formed by rotating the curve of y=sinx about the x-axis in the interval [0, pi]?
I've tried to set up integrals by summing up cylinders parallel to the y-axis but to no avail.
A small ring of mass M can move freely on a smooth, circular hoop, of radius R. A couple of light inextensible strings that pass over smooth pegs situated below the center of the hoop at the same horinzontal level are attached to the ring. Their other ends are attached to particles of mass m...
I'm having trouble with the following question. Can anyone please give me a push in the right direction?
A body consists of equal masses M of inflammable and non-inflammable material. The body descends freely under gravity from rest. The combustible part burns at a constant rate of kM per...
https://www.amazon.com/exec/obidos/ASIN/052138835X/ref=sib_rdr_dp/102-8371051-2688163"&tag=pfamazon01-20
Is this book worth it? It seems like an interesting read.
I'm having trouble with the following limit:
\lim_{x \rightarrow 0} \frac{1 - \sqrt{1 - 4x^2}}{x^2}
At the back of the book (Apostol Volume I), it says the answer is 1/2, but I get 2. Can anyone clarify?
Thanks.
I was asked to prove, using induction, that 34n+2 + 26n+3 is divisible by 17. I tried to do it, but I couldn't get anywhere. Can someone give me a push in the right direction?
Here's my attempt:
f(k) = 34k+2 + 26k+3
f(k+1) = 34k+6 + 26k+9
And now, all I have to do is prove that f(k+1) -...
If you have a particle that is attached to two elastic strings moving with SHM. How can I determine whether or not the particle performs complete oscillations?
For example I have a particle that's at a point equidistant from A and B (which are the ends of the two strings), and say that the...
What classes did you take in college, and when did you take them (as a freshman, sophomore, etc.)?
I'm finishing up my senior high school year and I'm wondering what awaits me. I'd appreciate any replies, especially from math majors.
If a spaceship is travelling at c and a beam of light is emitted through it. The beam of light would be travelling at c relative to the object. Correct?
How about if there's an observer, what would the speed of the beam of light be relative to him. c?
And if the spaceship was instead...
I need help understanding electric fields. I basically suck at answering questions related to them (I get 95% of them wrong :cry:).
What I do know about them is:
The electric field vector moves from a positive charge (+q) to a negative charge (-q). (Is this the same case in a capacitor? I...
How come whenever I load the index page certain forums are expanded while others are collapsed, and whenever I collapse and expand the ones I want it doesn't seem to remember and returns to the way it was? It wasn't like this 2 days ago. :cry:
I've been considering applying to a few Canadian universities (in Ontario), but I'm somewhat oblivious to the application process there.
I went the online application route and made myself a user name and everything on the ouac website. Now, from what I understood, I can choose a bunch of...
Is there a method one can use to obtain the prime factorization of a certain number?
For example:
Find the prime factorization of 49 + 39. [MathFest 2004]
I realize that I can re-write the expression as 29.29+39, but that's about as far as I can go. :cry:
I didn't know where else to post this, so sorry if this is the wrong subforum.
Anyway, I bought myself a physics textbook for self-teaching purposes. I decided to take a look at its Fields chapters, and now I know how to derive Coulomb's Law using F=eE and Φ=∫ E dS. Right, so how advanced is...