Wikipedia:Reference desk/Archives/Mathematics/2010 September 12

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September 12

Laplace's Equation

So, I have a function $\phi$ satisfying $\nabla ^{2}\phi$ in V and $\phi =0$ on S where V is a region in $R^{3}$ and S is the bounding surface and I have to prove that $\phi =0$ everywhere in V. Not sure what I've done in airtight so just want to check it here. Basically, I've used a combination of Green's First and Second Identities to deduce that $\iint _{S}\phi \nabla \psi .d\mathbf {S} =0$ and since there are no restrictions on $\psi$, it is a completely arbitrary function and so the only way this is always 0 is if $\phi =0$. Is this a solid argument or does it need tightening up, or is it just plain wrong? Thanks asyndeton talk 13:34, 12 September 2010 (UTC)

I'm not sure I follow your argument. The integral is over S, but we want to know what φ does on V. We already know φ is zero on S. Rckrone (talk) 15:04, 12 September 2010 (UTC)

You're right, that was a stupid thing to say. The only other point of significance I can see in my, almost indecipherable, working is that $\iiint _{V}\nabla \phi .\nabla \psi dV=0$. This seems like an even weaker argument but can you again apply the arbitrariness of $\psi$ and deduce that $\phi$ must be 0? asyndeton talk 15:11, 12 September 2010 (UTC)

Your argument based on the `arbitrary' nature of $\psi$ seems shaky to me. The function $\phi$ is a solution of Laplace's equation, so it is by definition a Harmonic function, which is a fairly restricted class of functions that have some nice properties. You can probably use Harmonic_function#Maximum_principle to get your result, but our article doesn't seem to mention how this fact is derived.SemanticMantis (talk) 16:09, 12 September 2010 (UTC)

It seems clear why it works: ∇φ is divergence free in V so if it's non-zero somewhere we can consider the curve σ that follows the direction of ∇φ, along which φ is strictly increasing. σ can't be a closed curve so it has to meet S at each end and then φ along σ must have a strict maximum at one of those end points. If φ=0 on S, then this would be a contradiction and so ∇φ is uniformly zero on V and φ is constant and so zero. That doesn't really seem rigorous though. I'm sure there must be some way to use the vector calculus theorems to make this argument succinctly but I'm very bad at vector calculus. Rckrone (talk) 16:46, 12 September 2010 (UTC)

Thanks for your answer. I'll have to think on this for a while because it's slightly more sophisticated than I can cope with at first sight! asyndeton talk 09:00, 13 September 2010 (UTC)

Polynomial division

Hi. I've got a puzzle that asks me to divide $x^{101}-x+1$ by $x^{2}-3$and find the remainder. What is a concise way of doing that? Obviously polynomial long division and even synthetic division won't help much here. PS: I might have made a minor error in the coefficients or terms, since I'm recalling the puzzle by memory. If I seem to be you can make slight alterations, but I know for sure that the dividend is a trinomial with a first term with a huge exponent, a linear second term, and a constant third term, and that the divisor is the difference between a quadratic first term and a constant. Thanks. 24.92.78.167 (talk) 17:17, 12 September 2010 (UTC)

We have $x^{101}-x+1=q(x)(x^{2}-3)+(ax+b)$, and we want to find the rational numbers a and b, preferably without finding q. Try substituting appropriate values for x. Algebraist 17:26, 12 September 2010 (UTC)

You can do it in you head by working mod $x^{2}-3$. Basically just pretend x² is equal to 3, so x¹⁰⁰ is equal to 3⁵⁰. That reduces the expression to linear and you're done.--RDBury (talk) 18:42, 12 September 2010 (UTC)

If any further hint is needed, notice that what RDBury is saying is really the same thing as what Algebraist is saying. But if you have further questions, keep talking. Michael Hardy (talk) 22:54, 12 September 2010 (UTC)

I had some time and started the long division looking for a pattern, and I found that the exponent decreases by two every time and the coefficient increases one exponent. However I'm intrigued by RDBury's comment that 'you can do it in your head by working mod x^2-3'. What is meant by this? Why is it possible to pretend x^2 is equal to 3? Thanks. 24.92.78.167 (talk) 01:05, 13 September 2010 (UTC)

So the basic idea is that to find the remainder of $x^{101}-x+1$ when divided by $x^{2}-3$, we want to throw out anything that's a multiple of $x^{2}-3$ and see what's left. One way to do this is to say that now we are operating in an environment where $x^{2}-3$ is congruent to zero, and any multiple of it is zero as well. We can write this $x^{2}-3\cong 0.$ Now any two polynomials are congruent if their difference is a multiple of $x^{2}-3.$ So for example $x^{2}-2\cong 1$ since their difference is $x^{2}-3$ and $x^{3}\cong 3x$ since their difference is $x(x^{2}-3)$, which is a multiple of $x^{2}-3$. In particular we have $x^{2}\cong 3$ which is to say that in this environment $x^{2}$ and 3 are interchangeable. So we can freely turn all the $x^{2}$s into 3s to get our polynomial $x^{101}-x+1$ into a congruent one of the form we want, namely one with only a linear and constant term. This congruent polynomial differs from $x^{101}-x+1$ by a multiple of $x^{2}-3$ so it's the remainder we want. This idea is pretty central to dealing with quotient rings of polynomial rings but that might not mean much if you haven't seen any abstract algebra. Modular arithmetic is also relevant, although note that most of that article deals with setting some number congruent to zero while here we are setting a polynomial congruent to zero. Still the idea is the same. Rckrone (talk) 01:46, 13 September 2010 (UTC)

Summation Convention

Following on from my question on LaPlace's equation, I have to use Cartesian coordinates to show that for a function $\psi =\psi (r)$, which depends only on the radial coordinate $r=|\mathbf {x} |$

$\nabla \psi ={\frac {1}{r}}{\frac {d\psi }{dr}}\mathbf {x}$ and $\nabla ^{2}\psi ={\frac {1}{r}}{\frac {d^{2}(r\psi )}{dr^{2}}}={\frac {2}{r}}{\frac {d\psi }{dr}}+{\frac {d^{2}\psi }{dr^{2}}}$

Normally, I can work quite well with the summation convention and the first falls quite nicely into place but I seem to have a fundamental problem when it comes to finding the second one. If I show my working can someone point out the mistake? Caveat lector, my chain rule and use of total/partial derivatives may be a bit sloppy.

$\nabla ^{2}\psi$

$=\nabla .\nabla \psi$

$={\frac {\partial }{\partial x_{i}}}{\frac {\partial \psi }{\partial x_{i}}}$

$={\frac {\partial }{\partial x_{i}}}{\frac {x_{i}}{r}}{\frac {d\psi }{dr}}$

$={\frac {1}{r}}{\frac {d\psi }{dr}}{\frac {\partial x_{i}}{\partial x_{i}}}+x_{i}{\frac {d\psi }{dr}}{\frac {\partial }{\partial x_{i}}}{\frac {1}{r}}+{\frac {x_{i}}{r}}{\frac {\partial }{\partial x_{i}}}{\frac {d\psi }{dr}}$

$={\frac {1}{r}}{\frac {d\psi }{dr}}+x_{i}{\frac {d\psi }{dr}}{\frac {\partial r}{\partial x_{i}}}{\frac {d}{dr}}{\frac {1}{r}}+{\frac {x_{i}}{r}}{\frac {\partial r}{\partial x_{i}}}{\frac {d}{dr}}{\frac {d\psi }{dr}}$

$={\frac {1}{r}}{\frac {d\psi }{dr}}-x_{i}x_{i}{\frac {d\psi }{dr}}{\frac {1}{r^{3}}}+{\frac {x_{i}}{r}}{\frac {x_{i}}{r}}{\frac {d^{2}\psi }{dr^{2}}}$

$={\frac {d^{2}\psi }{dr^{2}}}$

Where's it going wrong? Thanks asyndeton talk 20:02, 12 September 2010 (UTC)

Here's how it should go:

$\sum _{i}{\left({\frac {1}{r}}{\frac {d\psi }{dr}}{\frac {\partial x_{i}}{\partial x_{i}}}+x_{i}{\frac {d\psi }{dr}}{\frac {\partial }{\partial x_{i}}}{\frac {1}{r}}+{\frac {x_{i}}{r}}{\frac {\partial }{\partial x_{i}}}{\frac {d\psi }{dr}}\right)}$

$={\frac {3}{r}}{\frac {d\psi }{dr}}+\sum _{i}{\left(-{\frac {d\psi }{dr}}{\frac {x_{i}^{2}}{r^{3}}}+{\frac {x_{i}^{2}}{r^{2}}}{\frac {d^{2}\psi }{dr^{2}}}\right)}$

$={\frac {3}{r}}{\frac {d\psi }{dr}}-{\frac {1}{r}}{\frac {d\psi }{dr}}+{\frac {d^{2}\psi }{dr^{2}}}$

I couldn't figure out what the mistake was for the longest time until I realized the formula in question had to be dimension dependent. Specifically $\nabla ^{2}\psi ={\frac {n-1}{r}}{\frac {d\psi }{dr}}+{\frac {d^{2}\psi }{dr^{2}}}$ in Rⁿ and so the derivation needed to reflect that somehow. Rckrone (talk) 04:34, 13 September 2010 (UTC)

Urgh, now you say it, it's so obvious that I should have had a $\delta _{ii}$ in there. My punishment for trying to be clever and skipping steps out. Many thanks. asyndeton talk 08:58, 13 September 2010 (UTC)

Probability Question

I am usually good in solving probability problems, but I stumbled upon the following question online and I have no idea how to approach it.

"You pick two cards (one after the other with no replacement) from a deck of 52 playing cards. What is the probability that the second card has a higher rank than the first one".

Please note that I am curious how to approach such a problem - I am not interested in a numerical answer.

Thanks Help Desk - where we get a chance to brush up on our math, after graduating from college :)

78.101.151.192 (talk) 20:44, 12 September 2010 (UTC)

Looks like 24/51. For the lowest ranked card, the chance of something higher is 48/51. For the highest ranked card, chance of higher is 0/51. For a mid-ranked card it would be 24/51, and as it's linear, that would be the overall probability. David Biddulph (talk) 21:12, 12 September 2010 (UTC)

First, compute the probability of drawing two cards of equal rank. Then compute the conditional probability of the second card having a higher rank under the contition that the ranks are not equal. Then combine these using Bayes' theorem. – b_jonas 21:50, 12 September 2010 (UTC)

I do not understand how Bayes' theorem comes in. There is a 48/51 chance of drawing two cards with unequal rank, and if you draw two cards with unequal rank then there's a 1/2 chance the first one will be lower, since it's completely symmetric among the two. Then you multiply. Rckrone (talk) 22:54, 12 September 2010 (UTC)

Rckrone: You're right, it's not Bayes' theorem, but the Law of total probability. Sorry. – b_jonas 11:26, 13 September 2010 (UTC)

Generally, to approach such a problem, begin with a similar problem which can be solved by counting rather than by computing. Solve this one: "You pick two cards (one after the other with no replacement) from a deck of 8 playing cards. What is the probability that the second card has a higher rank than the first one". Draw an 8 by 8 table like a chessboard where the rows represent the first card (i) and the columns represent the second card (j). Each of the 64 fields represent the drawing of two cards. The 8 fields i=j are discarded because the same card cannot be picked twice. The 28 fields in the upper right triangle represent that the second card has a higher rank than the first one, i<j. The 28 fields in the lower left triangle represent that the second card has a lower rank than the first one, i>j. (Assuming that different cards have different rank. I am not sure what you mean by the rank of a card. If different cards may have equal ranks, then mark the corresponding fields on the chessboard). The probability you are seeking is the number of favorable cases divided by the number of possible cases. Now you can approach your original problem. Bo Jacoby (talk) 07:44, 13 September 2010 (UTC).