I am reading Thurston's book on the Geometry and Topology of 3-manifolds, and he describes the metric in the Poincare disk model of hyperbolic space as follows:
... the following formula for the hyperbolic metric ds^2 as a function of the Euclidean metric x^2:
ds^2 = \frac{4}{(1-r^2)^2} dx^2...
On a finite-dimensional vector space over R or C, is every norm induced by an inner product?
I know that this can fail for infinite-dimensional vector spaces. It just struck me that we never made a distinction between normed vector spaces and inner product spaces in my linear algebra course...
Homework Statement
Find the expansion of 1 - x^2 on the interval 0 < x < 1 in terms of the Eigenfunctions
J_0 ( \sqrt{ \lambda_k ^{(0)}} x)
(where \lambda_k ^{(0)} denotes the kth root > 0 of J_0) of
(x u')' + \lambda x u = 0
u(1) = 0
u and u' bounded.
Homework Equations
Hint from...
circle x sphere = ???
Is the product space S^1 \times S^2 related (e.g. homeomorphic or homotopy equivalent) to a more familiar topological space? I am currently looking at maps from S^1 \times S^2 into other spaces, and I am having a really hard time visualizing what I am doing. Any thoughts...
Realistically speaking, what league of schools does a hardworking, smart but not brilliant math major have a shot at for a PhD in pure math? Let's say I might apply with 8 graduate courses at a "lower Ivy", coauthored 1-2 papers, solid recommendations.
I am not getting my hopes up for Stanford...
After participating in a more algebra-oriented REU last summer, I would love to try working in some area of geometry of topology this year. Unfortunately this seems to be a pretty rare field for REU topics. Here are the programs I have found so far that have geometry or topology projects...
Homework Statement
Assume that f(z) is analytic and that f'(z) is continuous in a region that contains a closed curve \gamma. Show that
\int_\gamma \overline{f(z)} f'(z) dz
is purely imaginary.
Homework Equations
If f(z) is holomorphic on the region containing a closed curve \gamma or if...
I am looking for two rotation matrices M1 and M2, which describe a rotation by an arbitrary angle around the axes passing through (0,0,0) and (1,1,1), and (1,0,0) and (2,1,1). All relative to the standard basis. How would I approach this problem?
I just need a short reminder from Calculus. Suppose you have a linear functional \alpha from C1[-1,1] to \Re, given by
\alpha(f) = \int_{-1}^{1}f(t)g(t)dt
for some fixed continuous function g. What is \frac{d \alpha}{d f}?