Wikipedia:Reference desk/Archives/Mathematics/2009 May 2

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May 2

Derivatives of functions of more than one variable

If we have a function $f(x,y)$, then we can say, by definition, that ${\frac {\partial f}{\partial x}}=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x,y)-f(x,y)}{\Delta x}}$ and, similarly, that ${\frac {\partial f}{\partial y}}=\lim _{\Delta y\to 0}{\frac {f(x,y+\Delta y)-f(x,y)}{\Delta y}}$. My question: is there a name for the function as both $\Delta x$ and $\Delta y$ simultaneously approach 0, i.e. (using the Euclidean distance as a measure of separation from the origin) what is: $\lim _{(\Delta x,\Delta y)\to (0,0)}{\frac {f(x+\Delta x,y+\Delta y)-f(x,y)}{\sqrt {(\Delta x)^{2}+(\Delta y)^{2}}}}$? Irish Souffle (talk) 14:00, 2 May 2009 (UTC)

I don't think so, because that limit isn't well defined. It may well vary depending on the path you approach (0,0) by - for example, if you let Δx tend to zero first and then Δy, you'll get the partial derivative wrt y. The other way round, you get it wrt x. If you let them tend to zero together, you'll get various answers depending on exactly how you do it (you could have Δx=Δy or Δx=2Δy or any number of other options). You may find directional derivative interesting, though. It's the same basic idea, changing both x and y, but you specify precisely how you do it so you get a well defined answer. --Tango (talk) 14:08, 2 May 2009 (UTC)

One can say more than "may vary depending on the path". In every case in which ƒ is differentiable except those in which all directional derivatives are zero, it does vary depending on the path. Michael Hardy (talk) 14:32, 2 May 2009 (UTC)

Are you sure you're not thinking of the total derivative? If you know the total derivative you can find the partial derivative in any direction. -- BenRG (talk) 22:28, 2 May 2009 (UTC)

What makes you think he is thinking of anything other than what he said? What he said was perfectly clear and easy to understand. --Tango (talk) 23:00, 2 May 2009 (UTC)

It's a perfectly clear and unambiguous limit, but I don't think it's clear at all that it's the limit the OP meant to write down. I mean, if $f(x,y)={\sqrt {x^{2}+y^{2}}}$ then that limit exists at the origin and equals 1. No one has mentioned that yet, presumably because we all agreed that it wasn't what the OP meant to ask about, but it's clearly true given the question as written. Maybe the OP really wanted to know what you can say about all the limits as (Δx,Δy) approaches 0 from different directions, in which case it might be worth mentioning the interesting fact that the total derivative doesn't only give you the partial derivatives along the axes, as it might appear at first glance, but actually gives you the partial derivative in any direction. Or maybe not, I don't know. It did no harm to mention it. It was just a reply I dashed off. I'm not perfect. Give me a break. -- BenRG (talk) 12:56, 3 May 2009 (UTC)

Well indeed with a $\limsup$ in place of the $\lim$ that quantity has a meaning and sometimes a name; it's the (maximal) "slope" of the functional f, a kind of substitute of $\|\nabla f(x)\|$ for non-smooth f. --pma (talk) 16:57, 3 May 2009 (UTC)

Binomial expansion

I have to expand (1 + 4x)^(1/2) up to x^2 and i keep getting 1 + 2x - 1/8 x^2 but the book says the last term should be -2x^2. Am I wrong or is the book wrong? --212.120.248.41 (talk) 17:00, 2 May 2009 (UTC)

You wrong, book correct ;) See Binomial series. When expanding, remember to raise to the various powers not "x" only but the 4 too.....--84.220.230.175 (talk) 17:12, 2 May 2009 (UTC)

He is not doing a binomial series. I believe the person is asking about the power series expansion up to the quadratic term. Is this correct? If we call f(x) = (1 + 4x)^(1/2). The first term of the power series has coefficient f(0) = 1. The second term has coefficient f'(0). Here $f'(x)=1/2(1+4x)^{-1/2}(4)=2(1+4x)^{-1/2}$ so that f'(0) = 2. Thus, your first two terms are correct. Then, the third term has coefficient $f''(0)/2!$. We have $f''(x)=-(1+4x)^{-3/2}(4)$, so $f''(0)=-4$ which means $f''(0)=-4/2=-2$. So, the book is correct. StatisticsMan (talk) 19:33, 2 May 2009 (UTC)

Sure he (or she) is doing a binomial series. That's precisely what that is. It's equivalent to doing a Taylor expansion (which is what you did), but generally you wouldn't do it from first principles for a binomial, the specific formula is easier (see the article). --Tango (talk) 19:50, 2 May 2009 (UTC)

Good point, thanks. StatisticsMan (talk) 20:02, 2 May 2009 (UTC)

Essential singularity

Hello, I had a problem on a test recently and I got it right but I just want to be sure that my method was correct. The question was about the function

$\sin({\frac {1}{z}})$

and asked what type of singularity it had at 0. I answered essential, which is right, but my reasoning was that $\sin(z)$ has a power series expansion with an infinite number of terms of positive degree. Therefore, $\sin({\frac {1}{z}})$ is just the same series with ${\frac {1}{z}}$ plugged in. So, it has an infinite number of terms of negative degree and is therefore an essential singularity. Does this make sense? And, would the same method prove that $e^{1/z}$, or any such function, has an essential singularity at 0. Again, I know it is an essential singularity in this case, but is this a proof of it? I'm just not sure if such an "operation" is valid with power series. Thanks StatisticsMan (talk) 19:23, 2 May 2009 (UTC)

It certainly makes sense, although I'm not sure it's 100% rigorous (analysis isn't my thing!). I think the standard way to prove that function has an essential singularity is to show that on an arbitrarily small interval around 0, it takes all values between -1 and 1, meaning there is no way to continuously extend the function to 0 (so it isn't a removable singularity, it's clearly not a pole because sine is bounded on the reals). Someone better at analysis will hopefully come along and tell us if your approach can be made rigorous. --Tango (talk) 19:58, 2 May 2009 (UTC)

Yea, a simple way to do it is just pick two simple paths, like along the positive x-axis and along the positive y-axis, and do the limit. I don't know if that works specifically here, but hopefully you can see the two are different so it's not removable and one is finite so it's not a pole. StatisticsMan (talk) 20:01, 2 May 2009 (UTC)

Your method is correct, and rigorous. The series you have calculated is the Laurent expansion of sin(1/z) about the origin, and there is a theorem that says that the principal part of the Laurent expansion (the part with the negative powers) about an isolated singularity is an infinite sum if and only if the singularity is essential. The article on essential singularity also talks about this. 76.126.116.54 (talk) 20:37, 2 May 2009 (UTC)

I'm still unclear on this. I know about Laurent series. And, I know a singularity is essential if and only if the Laurent series has an infinite number of negative power terms. I guess I'm just asking, how do I know that just plugging in 1/z to both the function and the series will give me a new function and a new series, and that the series will match up with the function? I want to understand why it is okay. Thanks! StatisticsMan (talk) 20:43, 2 May 2009 (UTC)

Let $w=1/z$. We all agree that:

$\sin(w)=w-{\frac {w^{3}}{3!}}+{\frac {w^{5}}{5!}}-{\frac {w^{7}}{7!}}+\cdots$

There is no reason we can't substitute 1/z in the place of w. --COVIZAPIBETEFOKY (talk) 20:59, 2 May 2009 (UTC)

(e.c.) What's the matter? since $\scriptstyle \sin(z)=\sum _{k=0}^{\infty }c_{k}z^{k}$ for all $\scriptstyle z\in \mathbb {C}$, in particular $\scriptstyle \sin(z^{-1})=\sum _{k=0}^{\infty }c_{k}z^{-k}$ for all $\scriptstyle z\in \mathbb {C} \setminus \{0\}$.

Thus this is the Laurent series for $\scriptstyle \sin(1/z)$. You may check in your book (yes! we deliver!): thm 10.21 for the classification of isolated singularities and exercise 25 of chapter TEN for the existence and unicity of the Laurent expansion. These two imply the theorem quoted by the Anon. --pma (talk) 21:06, 2 May 2009 (UTC)

I'm not sure what book you're talking about :) But, I guess this makes sense. I just want to make sure I'm not doing something that seems right but is not. Thanks for the help. StatisticsMan (talk) 00:24, 3 May 2009 (UTC)

It was Real and Complex Analysis, that you quoted above. In any case the point is the classification of singularities plus the unicity of the Laurent exp.; by now sure all that is clear to you. --pma (talk) 06:25, 3 May 2009 (UTC)

Good point. I have the first edition of that book and I did quote it. We are using Greene and Krantz in class and I just bought Conway as well to study for the qualifying exam. I will probably read through Rudin some as well to get more perspective, and it's over both real and complex so it would help in both. I already knew about uniqueness and classification of singularities. It's just like taking a derivative of a power series. Although, for a holomorphic function, it's perfectly allowable to differentiate the power series and you end up with the derivative of the function, it's something you have to prove. So, I just wanted to be sure this was not the same sort of thing. StatisticsMan (talk) 14:21, 3 May 2009 (UTC)

calculating rate constant k

Hi I am sure many of you will be familiar with the equation for working out the rate constant, k of a reaction: k = Ae^-Ea/RT. Where A is a constant for a reaction e is the number e, Ea is activation energy R is gas constant we'll call it 8.31 (accuracy not terribly important here) and T is temperature in kelvin. I thought I will try and prove to myself that increasing the temperature increased k but found that when I did some calculations that the opposite was true, that is, when I increased the temp k decreased! I'll write what I did I must have done something wrong. I made up some theoretical values e.g. A = 1 Ea = 1 J mol^-1 T = 250 kelvin and R = 8.31. so I worked it out: k = 1e^-1/8.31x250 = 8.60x 10^-14. I thought perhaps increasing the temp e.g. to 300 kelvin would give a larger number but I got 2.09 x 10^-16 which is a smaller number. Whats going on here? Apologies if this should be on the science desk but I thought it best to post it here. Thanks in advance to anyone who can help! —Preceding unsigned comment added by 92.17.247.142 (talk) 19:51, 2 May 2009 (UTC)

You have made an error in your calculations. When you needed to compute $k=e^{-{\frac {1}{8.31\cdot 250}}}=\exp \left[-{\frac {1}{8.31\cdot 250}}\right]$, you actually computed $k=\exp \left[-{\frac {1}{8.31}}\cdot 250\right]$. Eric. 131.215.159.99 (talk) 20:31, 2 May 2009 (UTC)

Many thanks for your reply but I am still a little confused. I tried doing the calculations again by combining R and Temperature before I did the equation. So at 250 kelvin R x T = 8.31 x 250 = 2077.5 and at 300 kelvin R x T = 2493. So I tried putting these into my equation: 1e^-1/2077.5 = 0.9996. Putting the values at 300 kelvin into the equation gives me the same answer! Please help! —Preceding unsigned comment added by 92.16.104.118 (talk) 08:35, 3 May 2009 (UTC)

The two values are 0.999518768 and .999598957; the second number is slightly larger than the first number. The reason that these two reaction rates are very close to each other is that the number you have chosen for the activation energy, 1 J / mole = 0.001 kJ / mole, is unfortunately an unrealistically small number for most interesting chemical reactions. (As a reference point, a biological reaction might often have an activation energy (when catalyzed) in the range 20 - 200 kJ / mole; formation of a peptide bond is about 150 kJ / mol.) At very low activation energies, changing temperature has a rather small effect on the rate of reaction; this can be seen more easily from the perspective of statistical mechanics (compare with the Boltzmann distribution). To be more specific, consider the population of each state of the reactants at a particular temperature; those states whose energy exceeds a threshold (the activation energy) may possibly go on to form product. If the threshold is very low, then most states are already above this threshold, and raising the temperature has little effect on the population above vs. below the threshold; raising the temperature mostly shifts the population that is already above the threshold to being even farther above the threshold. Eric. 131.215.159.91 (talk) 11:15, 3 May 2009 (UTC)

Thank you very much Eric - your help is much appreciated :) —Preceding unsigned comment added by 92.16.30.113 (talk) 21:30, 3 May 2009 (UTC)