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comment Relation between Harmonic vector field and Harmonic 1-form
Perhaps you don't mean to have the word "unit" in your definition of a harmonic vector field?
Apr
30
comment Nice way to express $H^{-1}(\mathbb{S}^1)$
Paul, I agree that your explanation is more explicit and clearer.
Apr
30
comment Nice way to express $H^{-1}(\mathbb{S}^1)$
Isn't this what Christian Remling suggested?
Apr
30
comment Nice way to express $H^{-1}(\mathbb{S}^1)$
And why does Johannes Hahn's situation imply the use of charts? You can define $H^{-1}$ as the space of bounded linear functionals on $H^1$.
Apr
30
comment Nice way to express $H^{-1}(\mathbb{S}^1)$
I don't understand your question. Indeed, the $L^2$ inner product is undefined on $H^{-1}$, no matter what definition you use of the latter.
Apr
30
comment Nice way to express $H^{-1}(\mathbb{S}^1)$
There's no reason why Fourier series have to be restricted to ones in $L^2$.
Apr
21
comment Compact manifolds locally bi-Lipschitz to Euclidean space
Benoit, thanks. I had tried for a few minutes the approach you suggest but wasn't successful. Of course that says more about me than the idea.
Apr
21
comment Compact manifolds locally bi-Lipschitz to Euclidean space
Nice answer. Is there a way to do the last step without the exponential map and valid for a continuous metric? Would a smoothing argument work?
Apr
20
comment Compact manifolds locally bi-Lipschitz to Euclidean space
You don't need the exponential map. It's true even if the metric is only continuous and is analogous to the fact that any continuous function on a compact manifold is uniformly continuous. Just take any (finite) cover of $M$ of balls. Since $g$ is positive definite on the closure of each ball, there exists a positive constant $a > 0$ such that $a^{-1} \le g \le a$ and therefore the ball with the Riemannian metric structure is bilipschitz to the Euclidean ball.
Apr
20
comment Tameness in $\mathbb{R}^{n^2}$ of the subset consisting of matrices of positive determinant
Very naive question, since I don't understand the last desingularization step: Could this be done on induction by dimension? The answer given seems to give a correct answer if the limiting marix $M$ has corank 1. Is there an argument now for corank 2?
Apr
19
comment Boundary Conditions for Sine Gordon on some K< 0 surfaces
One possible place to specify boundary conditions is where the surface becomes singular (say, a cusp). This is where one of the principal curvatures blows up. If you have explicit formulas for an embedding, you should be able to identify where this is. This is also where each asymptotic direction converges to one of the principal directions.
Apr
13
comment Flat coordinates of a Riemannian metric
You have an explicit formula for $g$ and you want an explicit formula for the change in coordinates?
Apr
6
comment On the definition of fundamental vector field
This looks like a nice set of lecture notes: empg.maths.ed.ac.uk/Activities/GT/Lect1.pdf. The bottom of page 7 has a 1-line concrete explanation of the fundamental vector field. It's better than my answer below.
Apr
6
answered On the definition of fundamental vector field
Apr
4
comment Is there a complex structure on the 6-sphere?
Dennis, many thanks for this. I did not know the last statement regarding even complex dimensions.
Mar
30
comment Nonlinear elliptic problem involving the p-laplacian, Hölder inequality
A simple way to check for obvious errors is to check that the inequality is invariant under rescaling of each function appearing in it, as well as rescaling space. And why is there a $dx$ at the end of the right side?
Mar
24
comment Why do we teach calculus students the derivative as a limit?
By the way, viewing a derivative as a "sensitivity" is nothing but a way to describe the linear approximation of a function. But I believe it's a way that makes it easier to understand and use the approximation than the usual formula we teach.
Mar
24
comment Why do we teach calculus students the derivative as a limit?
No. The traders just know that there is a black box called the Black-Scholes formula (or more sophisticated variants) that, given a volatility and stock price, spits out the option price. They want to hedge their position by shorting the right amount of stock that will offset any changes to the option price. The derivative tells them the right "hedge ratio". It's all very simple and uses nothing but the basic concept of a derivative as a sensitivity. This simple view is useful in many other contexts. So I believe it's shame to flood students with so many other things but not this.
Mar
23
comment Why do we teach calculus students the derivative as a limit?
I would hope that anyone who has learned calculus properly would know immediately (without having to think about the graph) that, under reasonable circumstances, the option price would change by approximately by $0.40. In fact, option traders, including those who have no formal training in calculus, do use the derivative in exactly this way. They even know that the second derivative tells them whether the estimate is an over- or underestimate. And they understand that there are extreme circumstances where the estimate is not useful at all. This is what I mean by working knowledge of calculus.
Mar
22
comment Why do we teach calculus students the derivative as a limit?
That's not my experience at all. Here is a simple example from finance: The price of an option on a stock is a function of the stock price. Suppose that when the stock price is 100, the derivative of the option price with respect to the stock price is 2. Suppose today's stock price moved by \$0.20. Estimate the change of the option price. Will a student say immediately that the option price changes by \$0.40? Or will they try to figure out what rule or formula they should use to figure this out?