Homework Statement
Let \lambda_0 \in \mathbb{C} be an eingenvalue of the n \times n matrix A with algebraic multiplicity m , that is, is an m-nth zero of \det{A-\lambda I} . Consider the perturbed matrix A+ \epsilon B , where |\epsilon | \ll 1 and B is any n \times n matrix...
Homework Statement
Prove \sin{10} , in degrees, is irrational.
Homework Equations
None, got the problem as is.
The Attempt at a Solution
Im kinda lost.
Homework Statement
I've been reading some introductory PDE books and they always seem to motivate the search for solutions of the first order quasilinear PDE by the method of characteristics, by introducing a flow model; i was thinking, could someone give an example of a phenomenon modeled by...
Sorry for the inactivity, my computer decided to self-destruct under the heat.
Well, that (a_1,a_2,...,a_n) does not lie in our rectangle centered about the point x is not much of a problem, since said rectangle was constructed inside our original and arbitrary rectangle, not necessarily...
Let R be an open rectangle such that x \in R , R=(a_1,b_1)\times ... \times (a_n,b_n) . If x=(x_1,...,x_n) , we construct an open rectangle R' with sides smaller than 2\min{(b_i - x_i, x_i-a_i)} for 1\leq i \leq n , and centered about the point x . By construction R' \subset R and...
Homework Statement
I have been self studying Spivak's Calculus on Manifolds, and in chapter 1, section 2 (Subsets of Euclidean Space) there's a problem in which you have to find the interior, exterior and boundary points of the set
U=\{x\in R^n : |x|\leq 1\}.
While it is evident that...
I get it now, thanks.
Now, at the risk of seeming kind of stubborn, imagine you've just been given the definition of compact sets and you were immediately asked to prove this (which is the case with Spivak's book), how would you do it without constructing the functions \phi and \psi , which...
While I don't doubt there's nothing wrong with your argument, I am not familiar with homeomorphisms and your proof seems a little out of my grasp right now. I am trying to prove it by means of covers, I suppose A is a cover of \{x\}\times B , and I want to prove there is a finite subcollection...
I was reading Spivak's Calculus on Manifolds and in chapter 1, section 2, dealing with compactness of sets he mentions that it is "easy to see" that if B \subset R^m and x \in R^n then \{x\}\times B \subset R^{n+M} is compact. While it is certainly plausible, I can't quite get how to handle...
We do not know if f and g are continuous, we only assume them to be integrable, so it is not necessarily true that
\int (f-\lambda g)^{2}=0
implies (f-\lambda g)^{2}=0, since f-\lambda g could be zero except at an isolated number of points (it's integral would still be zero but the...
Homework Statement
In an effort to keep me from spending all summer lying on the couch, I recently started reading Michael Spivak's Calculus on Manifolds; while working on problem 1-6 I got stuck on a technical detail and I was wondering if anyone could provide a little insight.
Problem 1-6...
In an effort to keep me from spending all summer lying on the couch, I recently started reading Michael Spivak's Calculus on Manifolds; while working on problem 1-6 I got stuck on a technical detail and I was wondering if anyone could provide a little insight.
Problem 1-6 says:
Let f and g...
How do you represent the Fourier transform of a periodic signal as the input in a LTI system?
More precisely, a sequence of triangle pulses symmetric to the origin.
I know what the Fourier transform is 2\pi (sum (Xn delta(omega - (n)omega0)))
I was told to reduce the Xn to...
Homework Statement
You are given a standard Wheatstone bridge, prove that the bridge is balanced if and only if R_x = R_3 \frac{R_2}{R_1} . Subindexes depend on the names assigned to each resistance. Proving that if the bridge is balanced THEN the resistors satisfy said relationship is easy...
As other people pointed out, do the dot product for all three vectors and you'll get a system of 4 unknowns and 3 equations whose answer is most likely a 4-3=1-dimensional subspace of R^4. The answer should be a line.
I have been doing some self study on differential equations using Tom Apostol's Calculus Vol. 1. and I got stuck on a problem (problem 12, section 8.5, vol. 1).
Homework Statement
Let K be a non zero constant. Suppose P and Q are continuous in an open interval I.
Let a\in I and b a real...
Homework Statement
Let A be a square matrix with right inverse B. To prove A has a left inverse C and that B = C.
Homework Equations
Matrix multiplication is asociative (AB)C=A(BC).
A has a right inverse B such that AB = I
The Attempt at a Solution
I dont really know where to...
Changes in potential gravitational energy over a change in height are determined only by the initial and final heights, and you've got both. Just compute U (or V, however you want to call it) at height 1 and then compute U at height 2, and substract. Hint: you gotta consider gravity is acting...
You've got two different accelerations in this problem, what you gotta do is compute the value and direction of the total acceleration experienced by the passenger.
Hint: where's the 3.25^2/42 acel. pointing to? and the 0.555 one?
I hope this helps.
I was wondering if someone could tell me (in pedestrian terms) what are the crystal structures of aluminum, copper and iron as a function of the temperature.
I just read the chapter on crystalline arrays of Callister's Materials Ecience & Eng. but the book just says that Cu and Al have a FCC...
Homework Statement
What's the formula that better describes the temperature as a function of time for an enclosed body of water with certain initial temperature T_a immersed in another body of water of initial temperature T_b?
More clearly, I performed an experiment in which I put a...
The criteria's not too strict, I just don't see any teacher taking seriously info quoted from wikipedia or yahoo answers. The sources are alright as long as they at least state the authors name, and would be even better if they in turn stated their sources.
Homework Statement
Let f:A\subset{\mathbb{R}}^{n}\mapsto \mathbb{R} be a linear function continuous a \vec{0} . To prove that f is continuous everywhere.
Homework Equations
If f is continuous at zero, then \forall \epsilon>0 \exists\delta>0 such that if \|\vec{x}\|<\delta then...
Homework Statement
I was asked by my lab teacher to investigate the effects on the human body of being exposed to very high and very low pressures, and while I have found some good sources of info it is not that easy to find good and quotable sources, I was wondering if anyone could offer some...