Wikipedia:Reference desk/Archives/Mathematics/2010 November 23

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November 23

Area of a surface of revolution

I have looked in several calculus books and they all give formulas for the area of a surface of revolution (if x and y are defined parametrically on the interval [a, b]) as:

$A=2\pi \int _{a}^{b}y(t){\sqrt {(x'(t))^{2}+(y'(t))^{2}}}\,dt$ if it is rotated around the $x$-axis

and

$A=2\pi \int _{a}^{b}x(t){\sqrt {(x'(t))^{2}+(y'(t))^{2}}}\,dt$ if it is rotated around the $y$-axis

What about when the curve is rotated about some other line (including things such as y = x)? I realize the formula only holds if the plane curve does not cross the axis of rotation. Is it just

$A=2\pi \int _{a}^{b}r(t){\sqrt {(x'(t))^{2}+(y'(t))^{2}}}\,dt$

where $r(t)$ measures the distance from the axis of rotation to the plane curve (along lines perpendicular to the axis of rotation)? Thanks. StatisticsMan (talk) 01:57, 23 November 2010 (UTC)

Yes, the formula is $A=\int 2\pi R(s)ds$ where the curve is parameterized by the arc length s You'd get your formulas after change of variable $ds=|v(t)|dt$ (Igny (talk) 03:40, 23 November 2010 (UTC))

dot product

your article says a dot product is dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number obtained by multiplying corresponding entries and adding up those products. But how are vectors sequences of numbers? I thought vectors were just defined as a segment with a magnitude and direction, or x and y hat components ? (I'm mostly familiar with them from a vector standpoint. —Preceding unsigned comment added by 24.92.78.167 (talk) 04:22, 23 November 2010 (UTC)

The article Cartesian coordinate system explains how a vector is described by coordinates. Bo Jacoby (talk) 07:39, 23 November 2010 (UTC).

And for a concise answer, a vector with x and y components can be described by the sequence of two numbers $(x,y)$. A 3D vector can be described as $(x,y,z)$. -- Meni Rosenfeld (talk) 09:06, 23 November 2010 (UTC)

Isn't it more usual to use (i,j) and (i,j,k) to describe 2D and 3D floating vector components, so that the (x,y) and (x,y,z) coords can be added to describe the endpoint of a 2D or 3D fixed vector ? Perhaps this is only the case in my engineering background, and not in pure mathematics. StuRat (talk) 19:35, 29 November 2010 (UTC)

There are two concepts that need to be distinguished here. There is the algebraic operation

$\mathbf {a} \cdot \mathbf {b} =\sum _{i=1}^{n}a_{i}b_{i}$

and then there is the geometric operation

$\mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\,\left\|\mathbf {b} \right\|\cos \theta$

If the components of a and b are co-ordinates relative to an orthonormal basis then these two operations give the same result. Otherwise (if, for example, a and b are represented in polar co-ordinates) then the operations are not the same. Gandalf61 (talk) 16:42, 23 November 2010 (UTC)

If you know how to compute the square and the sum and the difference and the half, then you can always compute the product by the formula

$a\cdot b=\left({\frac {a+b}{2}}\right)^{2}-\left({\frac {a-b}{2}}\right)^{2}$

If $a=(a_{1},a_{2})$ and $b=(b_{1},b_{2})$ are vectors, then you know how to compute the sum and the difference and the half, and the square is

$a^{2}=|a|^{2}=a_{1}^{2}+a_{2}^{2}$,

so the product is

$a\cdot b={\frac {(a_{1}+b_{1})^{2}+(a_{2}+b_{2})^{2}}{4}}-{\frac {(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}}{4}}=a_{1}b_{1}+a_{2}b_{2}$

Thus the dot product definition is motivated. If two vectors $a$ and $b$ are at right angle to one another, then the two diagonals $|a+b|$ and $|a-b|$ are equal, and so the dot product is zero.

$a\cdot b=0$.

Bo Jacoby (talk) 01:53, 24 November 2010 (UTC).

Groups presentations

Say I have a group with presentation $\langle x,y\mid x^{11}=1,xy=yx^{-1}\rangle$. I think it's pretty obvious that in this group, x has order 11, but I'm having a hard time showing this with any sort of rigour. It seems so clear that the second relation can't possibly affect the order of x (and that, in fact, if we had a presentation with just the second relation, x and y would have infinite order), but I just can't see how to prove it. I want to use von Dyck's theorem to say that there exists a group such that x is order 11 and the second relation is satisfied, and inject my group in there, forcing the order of x to be 11, but again, I've not really been rigorous about showing that such a group does definitely exist. I suppose I'm also just wondering in general how to show that relations in a presentation don't interact with each other in an 'interesting' way (or if they do...). Any assistance would be appreciated, Icthyos (talk) 17:01, 23 November 2010 (UTC)

Rewrite the second relation as x^y = x⁻¹. Then it is clear that it is a presentation of the semidirect product $C_{11}\rtimes _{\varphi }\mathbb {Z}$, where

$\varphi _{n}(u)={\begin{cases}u^{-1}&n{\text{ odd,}}\\u&n{\text{ even,}}\end{cases}}$

with x being the generator of the C₁₁, and y the generator of the Z. In particular, the order of x is indeed 11. (Of course, you don't need to know that the semidirect product has exactly this presentation, you only need to know that it satisfies the given relations. For this purpose, the simpler product $C_{11}\rtimes C_{2}$ would also work, giving the dihedral group D₁₁.)—Emil J. 17:14, 23 November 2010 (UTC)

I don't think there is any method which would work in general. It is even algorithmically undecidable whether a given finite presentation defines a nontrivial group. However, looking for semidirect products often helps.—Emil J. 17:23, 23 November 2010 (UTC)

I see, thank you. I tend to forget about semidirect products, as it seems like quite a mysterious construction to me (I've not really met them that much). I'm always quite surprised when I learn that a group can be expressed as a semidirect product, it's a bit like a black box to me, for now at least. Is it fair to say that as soon as we add another relation on top of x⁸ = 1 into our presentation (apart from something like y⁴=1), we can no longer immediately read off what the order of x is...even if it seems intuitively obvious? We need to search for something like a semidirect product, as above, to determine what the group actually is? Thanks again, Icthyos (talk) 20:24, 23 November 2010 (UTC)

I'm a bit lost. Where does x⁸ = 1 come from, and how does it relate to the presentation above? If you add x⁸ = 1 on top of x¹¹ = 1, it already forces x = 1.—Emil J. 14:32, 24 November 2010 (UTC)

I'm confused too, I have no idea where that came from either! I meant x¹¹ = 1, rather than x⁸. Apologies! Icthyos (talk) 14:58, 24 November 2010 (UTC)

Ah, OK. Yes, putting in more relations can make the order of x smaller than it looks. This holds even for relations that seemingly do not involve x. Your y⁴ = 1 happens not to break anything (it gives $C_{11}\rtimes C_{4}$), but if you add y³ = 1 instead, it makes x = 1 (because x = x^y3 = ((x^y)^y)^y = x⁻¹, thus x² = 1).—Emil J. 16:25, 24 November 2010 (UTC)

Am I missing something? To me this looks trivial. Since x¹¹=1, it follows that the order of x must divide 11. That means it's either 11, or 1. If it were 1, that would mean x=1; surely it's routine to refute that possibility. --Trovatore (talk) 22:08, 25 November 2010 (UTC)

Oops, the above presentation is actually an altered form of the one I'm working with (I wanted to see if I could extract the method, and use it on my own problem), but raising x to a prime power was a bit foolish of me. In my original presentation, x is raised to a non-prime, leading to the difficulty. Icthyos (talk) 09:21, 26 November 2010 (UTC)

Even if it is prime, in general it's not routine to refute the possibility that x = 1, that's kind of the point.—Emil J. 13:24, 26 November 2010 (UTC)

But in the specific case it's pretty trivial. Any expression equal to 1 would have to be a product of terms x¹¹, x⁻¹¹, xyxy⁻¹, yx⁻¹y⁻¹x⁻¹. The only way to get an x without 10 other xs around requires it to be bracketed by y and y⁻¹ terms. There's a little messing around to make it rigorous but it's basically routine I think. --Trovatore (talk) 20:26, 26 November 2010 (UTC)

Your product may also involve arbitrary conjugates of the terms you listed. I'm pretty sure that if at all possible, any rigorous proof based on this would be far more complicated than constructing the semidirect product.—Emil J. 15:37, 29 November 2010 (UTC)

A somewhat ugly equation

$(1+nx)^{p}(1-x)^{1-p}=1\,\!$

Can someone solve this assuming n and p are known and I'm looking for x (I'd appreciate even a solution for n=1 in case the above equation too impractical to solve). It's effectively an extension of Kelly criterion and expected utility, I'd just like to know how far can I go before the bets become -EU. Thanks 78.0.251.166 (talk) 17:51, 23 November 2010 (UTC)

There's no closed-form elementary solution to this. You need to solve it numerically. -- Meni Rosenfeld (talk) 19:04, 23 November 2010 (UTC)

x=0 seems to give 1.166.137.10.83 (talk) 20:28, 23 November 2010 (UTC)

Of course, I was talking about the root the OP was interested in. -- Meni Rosenfeld (talk) 07:23, 24 November 2010 (UTC)

Yes, x=0 is one solution. If p is a positive integer then the other p−1 solutions satisfy the polynomial equation

$\sum _{k=0}^{p-1}\left(n^{k+1}{\binom {p}{k+1}}+(-1)^{k}{\binom {p-1}{k+1}}\right)x^{k}=0$

Bo Jacoby (talk) 02:35, 24 November 2010 (UTC).

For $p>1$ there are no real solutions (other than 0), so I think the OP wants $0<p<1$ which is not an integer. -- Meni Rosenfeld (talk) 07:23, 24 November 2010 (UTC)

I see. Asking www.wolframalpha.com , say, (1+nx)^p (1-x)^(1-p)=1,p=1/5,n=8 , the answer is numerical: x=0.229303. Asking more generally (1+nx)^p (1-x)^(1-p)=1,p=1/5, the answer is a uselessly complex analytic expressions for x as a function of n. Bo Jacoby (talk) 13:38, 24 November 2010 (UTC).

Thanks, I think I'm just gonna make a table for useful n's and p's I guess. I was hoping for an equation where you can plug in n and p and get x but I can do without it. FWIW p is the probability of the event, n is the odds we're getting in the form of n-to-1 (if n=1 it's an even money bet; e.g. if n=4 if we risk $1, we stand to win $4). x is the part of our bankroll we're betting. For example if we have $10 and x=0.3, we should bet $3. The best tradeoff between risk of ruin and expected value is the Kelly criterion which is fairly easy to calculate and prove but the border where EU=1 is more useful, as you cannot always bet the Kelly amount but it's important to know if you're making a +EU or -EU bet. 93.140.46.149 (talk) 17:12, 24 November 2010 (UTC)

Don't give up hope. As p is not an integer the above summation should run to k=infinity instead of to k=p-1. Discarding the terms k>1 gives the approximate equation

$n^{1}{\binom {p}{1}}+{\binom {p-1}{1}}+\left(n^{2}{\binom {p}{2}}-{\binom {p-1}{2}}\right)x\approx 0$

which is linear and solvable

$x\approx {\frac {n/(1-p)-1/p}{(n^{2}-1)/2+1/p}}$

Plugging in p=1/5 and n=8 gives x=0.14 which is a very crude approximation to 0.23, while p=1/2 and n=2 gives x=0.57 which is a more reasonable approximation to 0.50. Including the term k=2 gives a quadratic equation which might provide sufficient approximation. Bo Jacoby (talk) 11:15, 25 November 2010 (UTC).

The series expansion of (1+nx)^p doesn't converge when nx>1, so this won't work. Sorry. Bo Jacoby (talk) 23:56, 25 November 2010 (UTC).

Line integral

I'm no mathematician - and I'm useless when it comes to calculus. I have a 3D volume (let's say it's a unit sphere, centered on the origin) containing some fluid of variable opacity. Let's say that the density varies at each point according to some simple function like maybe:

  density = 1 - sqrt(x2+y2+z2)

...a spherical blob that's utterly opaque in the middle and utterly transparent at the edges...kinda like an idealized raincloud. I want to know - for an arbitary line passing through the unit sphere (like a laser beam), how much of the light would be occluded by the fluid...(forgetting anything messy and real-worldish like scattering and re-radiation). Let's suppose the line is described by a point somewhere on the surface of the unit sphere - and a vector describing its direction.

I think I need a path integral - integrating the density function along the line...but I have absolutely no idea how to do that.

Part two of the question (assuming there is a relatively easy answer to part 1) is: If I wanted to approximate the density of a real fluid - what kinds of density functions over the unit sphere produce relatively, simple functions for the line integral?

If I happen to have picked a particularly difficult function for my example - ignore it and pick something easier.

216.136.51.242 (talk) 18:02, 23 November 2010 (UTC)

To the best of my knowledge, opacity doesn't work that way. There's no "utterly opaque" - no matter how dense the substance is, light will still go through if the layer it has to pass is thin enough. The opaqueness at a point can take any nonnegative value, and the portion of the light that passes is $e^{-A}$ where A is the line integral of the density along the path taken by the light.

So to answer your question about computing line integrals along straight lines, you first need to find the points where the light enters and leaves, $(x_{1},y_{1},z_{1}),\ (x_{2},y_{2},z_{2})$, and the length of the segment between them, $L={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}}$. Then the line integral will be $L\int _{0}^{1}f(x_{1}+(x_{2}-x_{1})t,y_{1}+(y_{2}-y_{1})t,z_{1}+(z_{2}-z_{1})t)\ dt$. Putting in your particular f will give you a normal one-dimensional integral which may or may not be easy to solve. If it can't be solved analytically you can always find the integral numerically. With your example density function, taking endpoints $(1,0,0),\ (0,1,0)$, you get ${\sqrt {2}}\int _{0}^{1}\left(1-{\sqrt {t^{2}+(1-t)^{2}}}\right)\ dt={\frac {{\sqrt {2}}-\mathrm {arcsinh} (1)}{2}}\approx 0.26642$. -- Meni Rosenfeld (talk) 19:22, 23 November 2010 (UTC)

Isn't a black hole utterly opaque ? StuRat (talk) 19:29, 29 November 2010 (UTC)

Research Papers by James Stewart of Textbook Fame

I saw the unsourced statement at James Stewart: "Stewart's research focuses on harmonic analysis and functional analysis." which I don't doubt at all, but I can't find links to actual research or articles in refereed journals actually done by him. The bio at stewartcalculus.com likewise states his prior activity in research without any links (which is understandable, if not helpful- its purpose being PR and not reference). He seems to be a guest professor at best because U. of Torronto doesn't have a faculty page (that I can find) for him. I just want to see a detailed CV and publication history of this man who has made millions from his textbooks. 20.137.18.50 (talk) 18:52, 23 November 2010 (UTC)

I did some sleuthing; his advisor, Lionel Cooper (mathematician), worked on operator-theory for quantum mechanics. I found a few papers in computational chemistry by a James J. P. Stewart, published in the late 1980s, such as Calculation of the nonlinear optical properties of molecules, Optimization of parameters for semiempirical methods. These sound like applied mathematics - numerical optimization and function analysis applied to computational chemistry. It's possible they're the same Stewart (though one article shows an affiliation with USAF Academy, seemingly incongruous with the textbook Stewart). Clearly, the man's most famous work is Calculus - which is an excellent calculus textbook; but I'm surprised that he'd have published his work in applied computational-chemistry. It doesn't seem like the standard career-track for a renowned mathematics education expert. Nimur (talk) 22:42, 23 November 2010 (UTC)

In search of a function

For some time, I've been trying to come up with a function from $\mathbb {R} ^{+}$ to $[0,1[$ that satisfies these conditions:

• $f(0)=0$
• The function is strictly increasing, in other words, $x_{2}>x_{1}\Rightarrow f(x_{2})>f(x_{1})$
• The function must asymptotically approach 1, in other words, $\forall x\in \mathbb {R} ^{+}:0\leq f(x)<1$

(Note: $\mathbb {R} ^{+}$ includes 0.) The only function I've come up with is ${\frac {x}{x+1}}$ or trivial variations thereof. What are other such functions? JIP | Talk 20:48, 23 November 2010 (UTC)

How about ln([1 + 1/x]^x)? —Anonymous Dissident^Talk 20:55, 23 November 2010 (UTC)

You may want to consider 2*arctan(x)/pi.--Eliokim (talk) 21:24, 23 November 2010 (UTC)

You'll could do well with variations on sigmoid functions. Simply shift and scale to make them meet your requirements. e.g. the logistic function $P(t)={\frac {1}{1+e^{-t}}}$ can be modified to, e.g. $P(t)={\frac {1+e^{a}}{e^{a}}}({\frac {1}{1+e^{-(t-a)}}}-{\frac {1}{1+e^{a}}})$, which, for most a, meets your requirements. -- 140.142.20.229 (talk) 21:36, 23 November 2010 (UTC)

In fact, I should mention the shift-and-scale approach can be applied to almost any function that monotonically approaches a finite limit from below as x->inf: Pick a point (a,b) on the function f(x) for which the curve is monotonic for all x>a. Find h, the limit as x->inf for g(x) = f(x-a) - b. Your origin-including, inf+ asymptote=1 function is k(x) = 1/h * ( f(x-a) - b ). You can even use functions which approach the limit from above - just use -f(x) in the procedure above. Functions with finite limits as x->-inf? Use f(-x). If you're clever, even finite limits as x->0 can be used by employing 1/f(x) as above. (All this providing you can calculate the limits of the shifted function, of course. For this computer algebra systems like WolframAlpha are quite handy.) -- 140.142.20.229 (talk) 21:54, 23 November 2010 (UTC)

Another fundamentally different function is $1-e^{-x}$. --Tardis (talk) 22:14, 23 November 2010 (UTC)

This can be generalized, of course: $1-f(0)/f(x)$ for any strictly increasing unbounded f with $f(0)>0$. I used $f=\exp$ and you used the successor function S, but you can use $\cosh ,S\circ \sinh ,1+\cdot ^{p}\;\forall p>0,\exp \circ \exp ,\Gamma \circ S\circ S,{\log }\circ S\circ S,S\circ {\log }\circ S$, etc., etc. Then you can compose with any strictly increasing unbounded g with $g(0)\geq 0$: $1-f(g(0))/f(g(x))$, etc., etc. --Tardis (talk) 22:36, 23 November 2010 (UTC)

"$\forall x\in \mathbb {R} ^{+}:0\leq f(x)<1$" are not other words for "asymptotically approaches 1". For example, ${\frac {x}{2x+1}}$. -- Meni Rosenfeld (talk) 07:19, 24 November 2010 (UTC)

I know, I realised that after I wrote my original question. I knew what I meant, I just didn't write it out properly. The correct criterion is $\forall e\in ]0,1[:\exists x_{0}\in \mathbb {R} ^{+}:x>x_{0}\Rightarrow |1-f(x)|<e$. In other words, if you choose a limit, no matter how small, sooner or later the function is going to come closer to 1 than that limit. JIP | Talk 20:34, 24 November 2010 (UTC)

What do you consider "trivial variations"? If f is a function like you describe, then take $g(x)={\frac {f(x)}{1-f(x)}}$ and you get $f(x)={\frac {g(x)}{g(x)+1}}$. So if a monotonous transformation of the variable is considered trivial, then all such functions are trivial variations of ${\frac {x}{x+1}}$. Also, all such functions are $f(x)=h\left({\frac {x}{x+1}}\right)$ for some monotonous h. -- Meni Rosenfeld (talk) 07:39, 24 November 2010 (UTC)

With "trivial variations" I was meaning ${\frac {x}{x+n}}$ where $n\in \mathbb {R} ^{+},n>0$. I suppose I could also have extended it to ${\frac {x}{c(x+n)}}$, where $c\in \mathbb {R} ^{+},c\geq 1$, but then the function would not have asymptotically approached 1, but a smaller number. JIP | Talk 20:24, 24 November 2010 (UTC)

Here's another one: $f(x)=1-2^{-\lfloor x\rfloor }+{\frac {x-\lfloor x\rfloor }{2^{\lceil x\rceil }}}$. This formula looks pretty ugly, but it's just a piecewise linear function, with $f(0)=0$, $f(1)=1/2$, $f(2)=3/4$, $f(3)=7/8$, and so on; see the graph from Wolfram Alpha. —Bkell (talk) 16:15, 29 November 2010 (UTC)