Sorry i wasnt able to get help in the hw department. figured id try here.
Homework Statement
For a coordinate patch x: U--->\Re^{3}show thatu^{1}is arc length on the u^{1} curves iff g_{11} \equiv 1
The Attempt at a Solution
So i know arc legth of a curve \alpha (t) = \frac{ds}{dt} =...
Homework Statement
Let \gamma be a stright line in a surface M. Prove \gamma is a geodeisc
The Attempt at a Solution
In a plane we know a straight line is the shortest distance between two point. I am not sure if this applies to straight lines on a surface.
Further more, there...
Homework Statement
For a coordinate patch x: U--->\Re^{3}show thatu^{1}is arc length on the u^{1} curves iff g_{11} \equiv 1
The Attempt at a Solution
So i know arc legth of a curve \alpha (t) = \frac{ds}{dt} = \sum g_{ij} \frac {d\alpha^{i}}{dt} \frac {d\alpha^{j}}{dt} (well thats actually...
Homework Statement
Given g\equiv g_{ij} =
[-1 0;
0 1]
Show that A= A^{i}_{j} =
[1 2
-2 1]
is symmetric wrt innter product g, has complex eigenvalues, but eigenvectros have zero length wrt the complex inner product.
The Attempt at a Solution
Im sure this is just a simple...
Homework Statement
Few questions here, nothing super tough, just cant get it/ want verification.
1. The following experiment is repeated twice: a fair coin is flipped repeatedly until it lands heads. Let X be the number of flips required in the first trial and Y the number required in...
Homework Statement
A biased coin lands heads with probabilty 2/3. The coin is tossed 3 times
a) Given that there was at least one head in the three tosses, what is the probability that there were at least two heads?
b) use your answer in a) to find the probability that there was...
Homework Statement
A vector d is a direction of negative curvature for the function f at the point x if dT \nabla ^2f(x)d <0. Prove that such a direction exists if at least one of the eigenvalues of \nabla ^2 f(x) is negative
The Attempt at a Solution
Im having trouble with this...
Homework Statement
Find the decomposition of the standard two-dimensional rotation representation of the cyclic group Cn by rotations into irreducible representations
The Attempt at a Solution
Ok i did this directly, finding complementary 1-dimensional G-invariant subspaces. but...