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October 2[edit]

Time derivative of the anomalies[edit]

Are there three types of angular velocity in motion on a ellipse, each corresponding to the time derivative of mean anomaly, true anomaly and eccentric anomaly?--188.26.22.131 (talk) 12:41, 2 October 2012 (UTC)[reply]

Yes. See Kepler's laws of planetary motion. Bo Jacoby (talk) 13:13, 2 October 2012 (UTC).[reply]

That shows how you use them to calculate the motion, but I'm not sure that's what the OP is looking for. Or maybe I just don't see it... Ssscienccce (talk) 17:35, 2 October 2012 (UTC)[reply]

The mean anomaly is

M=nt={\frac {2\pi }{P}}t.

So the mean motion is

{\frac {dM}{dt}}=n={\frac {2\pi }{P}}

where P is the period.

The eccentric anomaly E satisfies

M=E-\varepsilon \cdot \sin E.

So

{\frac {dE}{dt}}={\frac {n}{1-\varepsilon \cdot \cos E}}.

The true anomaly θ satisfies the equation

\tan({\frac {\theta }{2}})={\sqrt {\frac {1+\varepsilon }{1-\varepsilon }}}\cdot \tan({\frac {E}{2}})

Differentiate to obtain an equation for ${\frac {d\theta }{dt}}.$ Bo Jacoby (talk) 06:35, 4 October 2012 (UTC).[reply]

How much mathematics is essential?[edit]

My question comes down to what math should cover- is math the fundamental operations of addition, subtraction, division, and multiplication, coupled in applications such as equations etc.

And is calculus, differential equations etc. mostly experimentally derived workings that are useful in other subjects such as physics and finances?

So if i were to pick up a calculus book, most of the operations will be in the form of subtraction, division, miltiplication and division and just mostly operations of what we know? I know in linear algebra, there are directional shifts in matrices in regards to division, subtraction, addition, multiplication; but what is the real thing I should I know? — Preceding unsigned comment added by 108.50.249.125 (talk) 16:58, 2 October 2012 (UTC)[reply]

Know for what purposes ? If you get a job which doesn't require additional math skills, then the core set that everyone needs is addition, subtraction, multiplication, division, fractions, conversions, percentages, and interest rate calculations (using a formula). Some basic geometry, like calculating areas and volumes, would also be useful. I've never found the need to use algebra, trigonometry or calculus in "real life" (outside of work). StuRat (talk) 18:23, 2 October 2012 (UTC)[reply]

And I use all three routinely in my hobbies. —Tamfang (talk) 02:27, 3 October 2012 (UTC)[reply]

http://xkcd.com/1050/ . Just sayin'. --Trovatore (talk) 02:36, 3 October 2012 (UTC)[reply]

Calculus generally involves integration and differentiation which are more advanced mathematical operations than the arithmetic ones you have listed. Understanding those more advanced operations is essential to much of physics and most forms of engineering, though many people lead perfectly happy and successful lives while having no idea what calculus is about. Any decent introductory calculus text can teach you integration and differentiation, but these are not concepts that any student would be expected to already know before reaching calculus. So, in that sense, calculus is certainly not something that people already know simply because they have learned arithmetic. As others have said, what you "should" know can't really be defined without some idea of what you want to be able to do. Dragons flight (talk) 11:07, 3 October 2012 (UTC)[reply]

I agree with StuRat that his list is a good list of what everyone needs to know about math. And I agree with Tamfang that for many people, life can be immeasurably enriched by hobbies that use algebra, trig, and calculus. It's not clear to me what the context of your (the OP's) question is -- if you are a high school or college student wondering what areas of math to expose yourself to, one thing you should know is that the more math you learn, the higher-paying job you'll probably be able to get, the more your employment is likely to be shielded from economic recessions, and the more comfortable your life is likely to be. Duoduoduo (talk) 14:58, 3 October 2012 (UTC)[reply]

Another possibility for the list: I think that knowing some basic statistics is also extremely important for "real life", at least up to the point where you can comprehend that there are better methods of judging hypotheses than recounting a few anecdotes ("I took <insert quack cure> and my <insert malady> got better!"), but that these methods are not infallible ("Well, those 'scientists' were wrong about X, so why should I listen to what they say about Y?"). Though, of course, many would argue that statistics isn't a subset of maths. I think you could make the case that a bit of exposure to how "real" maths works (looking for patterns, proof, abstraction, etc.) is also important.

As to the question about calculus being "experimentally derived workings that are useful in other subjects", it is true that much of calculus was developed for the purpose of solving problems in other fields, and initially in a very informal way, but the advent of mathematical analysis put calculus on a firm theoretical footing and led to lots of beautiful mathematics. 130.88.99.231 (talk) 13:08, 4 October 2012 (UTC)[reply]

I agree that some basic knowledge of statistics is valuable. However, it's not the math that everyone needs to know (calculating standard deviations and such). What people need to know about statistics, they don't seem to teach in math class. There's how to ask an unbiased question (not "In the US Presidential election, do you intend to vote for the Muslim candidate or the Christian ?"), the importance of having a high level of participation among the people you poll (so no mailing out surveys, getting 1% back, and drawing conclusions from that 1%), the importance of polling a sample representative of the population (no surveys at country clubs), etc. StuRat (talk) 04:31, 6 October 2012 (UTC)[reply]

Calculus and differential equations[edit]

How do both relate? Are differential equations a part of Calculus? Are differential equations based on Calculus? Are differential equations related to, but independent of Calculus? — Preceding unsigned comment added by 80.58.205.34 (talk) 16:59, 2 October 2012 (UTC)[reply]

Differential equations are part of calculus, but they are ordinarily taught in a separate course, which students take after Calculus I and Calculus II. In other words, they are an advanced topic in calculus. Looie496 (talk) 19:53, 2 October 2012 (UTC)[reply]

Integral problems are a subset of differential equations, in particular those of the type

y'(x)=g(x)

, so that the solution is

y(x)=\int g(x)dx,

, and the main challenge is in solving the integral. In a generic differential equation, the left hand side is more complex. Sjakkalle (Check!) 20:13, 2 October 2012 (UTC)[reply]

"There is no such subject as calculus." — Paul Halmos Sławomir Biały (talk) 20:28, 2 October 2012 (UTC)[reply]

It's all very well to deny the existence of calculus, but there are limits! Marnanel (talk) 20:38, 2 October 2012 (UTC)[reply]

I find your humor to be quite derivative. Sławomir Biały (talk) 21:36, 2 October 2012 (UTC)[reply]

I found it partially derivative, but I am not so anti-derivative as you seem to be. 86.176.211.202 (talk) 02:41, 3 October 2012 (UTC)[reply]

Fast test for non-factorable polynomials of a special form[edit]

I have an arbitrary degree polynomial where every coefficient is one. Like x^117+x^87+x^30+x^23+x^13+x^4+x^2+x+1. Is there a fast, straightforward method for determining the factorability of such polynomials? Sebastian Garth (talk) 19:44, 2 October 2012 (UTC)[reply]

Just as an example, x^7+x^5+x^4+x^3+x^2+1 = (x^2-x+1) (x^5+x^4+x^3+x^2+x+1) and in general x^2-x+1 times (x^k+x^(k-1)+...+x+1) is (x^(k+2)+x^k+x^(k-1)...x^3+x^2+1) so there is at least one factorable polynomial with only unit coefficients for any positive integer degree greater than two.Naraht (talk) 19:57, 2 October 2012 (UTC)[reply]

Similarly (x^4-x^3+x^2-x+1) times (x^k+x^(k-1)+...+x+1) gives a similar result.Naraht (talk) 20:22, 2 October 2012 (UTC)[reply]

You may want to start at http://www.math.sc.edu/~filaseta/irreduc.html and some of the papers referenced.Naraht (talk) 20:40, 2 October 2012 (UTC)[reply]

Thanks, yes that looks somewhat promising. Or how about this: is there an efficient method for detecting whether or not there is a real root? Or is that just as difficult as factoring? Sebastian Garth (talk) 01:01, 3 October 2012 (UTC)[reply]

I can see some cases where the existence of a real root is clear:

If the constant term is 0 there is obviously a real root at x = 0. So from now on let's assume the constant term is 1, so f(0) = 1.
If the leading term (highest degree) has odd degree then there is a real root < 0.
If there are more terms with odd degree than with even degree then f(-1) < 0 so there is a real root between -1 and 0.
If there are the same number of terms with odd degree as with even degree then there is a real root at x = -1.
At the other extreme, if all terms have even degree, or if there is only one term with odd degree and it is not the leading term, then there are no real roots.

So the ambiguous case is when the leading term has even degree and there are more terms with even degree than odd degeer, but at least two terms with odd degree. I can't think of a simple method of handling that case (at least, not without using a root finding algorithm). Gandalf61 (talk) 09:36, 3 October 2012 (UTC)[reply]

If you sort the terms and find that every odd power is bracketed both above and below by an even power (i.e. the 5th power in x¹² + x⁵ + x⁴ is bracketed), then one can group terms and conclude there are no real roots. That reduces Gandalf's unresolved case to polynomials having two or more consecutive odd powers. Actually, one can use the same argument to say that if every set of N consecutive odd powers is bracketed both above and below by at least N consecutive even powers, then there are no real roots. Dragons flight (talk) 10:47, 3 October 2012 (UTC)[reply]

The number of real roots of the polynomial can be computed easily using Sturm's theorem.—Emil J. 12:44, 3 October 2012 (UTC)[reply]

That article looks like it could *really* use an example to be more clear. Say counting the number of real roots on a quintic equation..Naraht (talk) 13:09, 3 October 2012 (UTC)[reply]

I've added this to the references of Sturm's Theorem: Baumol, William. Economic Dynamics, section "Sturm's Theorem". Not a web resource, but I remember it has a very good example--maybe you can find it in your library. Duoduoduo (talk) 15:08, 3 October 2012 (UTC)[reply]

I've added an example of using Sturm's Theorem here. Thanks for the suggestion, Naraht. Duoduoduo (talk) 20:01, 4 October 2012 (UTC)[reply]

See also Descartes' rule of signs, which can sometimes prove that there exists at least one real root. Duoduoduo (talk) 15:12, 3 October 2012 (UTC)[reply]

Back to the original question, on factorization: see Factorization of polynomials and the various wikilinks therein. Duoduoduo (talk) 15:36, 4 October 2012 (UTC)[reply]