Wikipedia:Reference desk/Archives/Mathematics/2010 June 26

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June 26

Vector calculus

I found a question in my textbook that really confuses me: Find a C¹ function f: R³ --> R³ such that takes the vector i + j + k emanating from the origin to i - j emanating from (1,1,0) and takes k emanating from (1,1,0) to k - i emanating from the origin.

I am confused because I thought vectors are displacable, so it doesn't matter where it originates from. 173.179.59.66 (talk) 08:35, 26 June 2010 (UTC)

I guess that "i − j emanating from (1,1,0)" means (i − j) − (i + j). Bo Jacoby (talk) 10:06, 26 June 2010 (UTC).

Why is that? 173.179.59.66 (talk) 11:24, 26 June 2010 (UTC)

This is an interesting question. It sounds like we should think of the tangent bundle Tℝ³. We have two vectors based at two points, and a function ƒ : ℝ³ → ℝ³. The first vector, written above as i + j + k is based at the origin, and we want it to be mapped to the vector i - j based at (1,1,0). This means that ƒ(0,0,0) = (1,1,0) and the differential at the origin takes i + j + k to i - j, i.e.

${\mbox{d}}f_{\mathbf {0} }:T_{\mathbf {0} }\mathbb {R} ^{3}\to T_{f(\mathbf {0} )}\mathbb {R} ^{3}\ \ {\mbox{is such that}}\ \ ({\mbox{d}}f_{\mathbf {0} })(1,1,1)=(1,-1,0).$

Practically, you need to evaluate the Jacobian matrix of ƒ, and then evaluate it at x = y = z = 0. You'll also need to do the same for the second vector. You want ƒ(1,1,0) = (0,0,0) and

$({\mbox{d}}f_{(1,1,0)})(0,0,1)=(-1,0,1).\,$

Then you just need to check the differentiability of ƒ, and prove it's C¹. •• Fly by Night (talk) 11:11, 26 June 2010 (UTC)

Oh, I don't really need a solution to the problem...but if you care, there's a slightly easier but much less elegant approach. 173.179.59.66 (talk) 11:20, 26 June 2010 (UTC)

I see... so you mean that you thought that all vectors were unmovable? Well, given a manifold (just think of the plane if you like), there is something called a tangent bundle. This is a fibre bundle whose fibres are tangent spaces. The tangent space at a given point is the space of all vector based at that point that are also tangent to the manifold. So in the case of the plane, the tangent space at (1,0) is the space of all vectors based at (1,0). Let's say we have a map from the plane to the plane. If it carries (1,0) to (0,1) then the differential takes all of the vectors based at (1,0) to the space of vectors based at (0,1). The differential gives a linear map between the tangent space to the plane at (1,0) and the tangent space to the plane at (0,1). So vectors are very much movable. More generally, if f : M → N is a differentiable map between two manifolds, M and N, then the differential

${\mbox{d}}f_{x}:T_{x}M\to T_{f(x)}N$

is a linear map between tangent spaces. So in some sense it actually moves vectors from one manifold onto another manifold. •• Fly by Night (talk) 11:30, 26 June 2010 (UTC)

Contingent events

What techniques do actuaries or others use to determine the probability of contingent events?--220.253.96.217 (talk) 13:53, 26 June 2010 (UTC)

Have you had a search of Wikipedia using the search box at the top of the page? Try sticking in 'contingent events'. Bayes Theorem may also be of use. Dmcq (talk) 14:08, 26 June 2010 (UTC)

I think the concept you're asking for is conditional probability. 75.57.243.88 (talk) 17:19, 26 June 2010 (UTC)

Limits

I want to show that a function tends to zero as x tends to zero. Let

$f(x):={\frac {e^{-1/x^{2}}}{x^{3}}}$

How do I show that ƒ(x) → 0 as x → 0? I've tried L'Hôpital's rule and power series expansion, but I keep falling flat. For example:

$f(x)\sim {\frac {1}{x^{3}}}\left(1-{\frac {1}{x^{2}}}+{\frac {1}{2!\,x^{4}}}-{\frac {1}{3!\,x^{6}}}+\cdots \right)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k!\,x^{2k+3}}}$

But that doesn't exactly tell me much. L'Hôpital's rule just gives higher order denominators with each and every iteration. Any ideas? •• Fly by Night (talk) 14:55, 26 June 2010 (UTC)

I would solve it using the Landau notation. I'm strapped for time at the moment so 'mafraid I can't show you how. 76.229.193.242 (talk) 15:02, 26 June 2010 (UTC)

The substitution y = 1/x² simplifies the limit to $\lim _{x\to 0}{\frac {e^{-1/x^{2}}}{x^{3}}}=\lim _{y\to +\infty }{\frac {y^{3/2}}{e^{y}}}$. You should be able to solve the latter easily with whatever method you are used to.—Emil J. 15:16, 26 June 2010 (UTC)

That is, the manipulation is only valid for $\lim _{x\to 0+}f(x)$, but if you show that to be zero, then $\lim _{x\to 0-}f(x)$ is also zero because f is an odd function.—Emil J. 15:21, 26 June 2010 (UTC)

Easy as that! How foolish I feel now. Thanks EmilJ. •• Fly by Night (talk) 15:33, 26 June 2010 (UTC)

power series

why is it that your article says $f^{\prime }(x)=\sum _{n=1}^{\infty }a_{n}n\left(x-c\right)^{n-1}=\sum _{n=0}^{\infty }a_{n+1}\left(n+1\right)\left(x-c\right)^{n}$ and not $f^{\prime }(x)=\sum _{n=1}^{\infty }a_{n}n\left(x-c\right)^{n-1}=\sum _{n=0}^{\infty }a_{n+1}\left(n+1\right)\left(x-c\right)^{n-2}$

In the second summation you're replacing n everywhere by n+1. It couldn't be more obvious than that. Zunaid 22:22, 26 June 2010 (UTC)