Wikipedia:Reference desk/Archives/Mathematics/2014 June 2

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June 2

SOIL

WHAT IS EROSION — Preceding unsigned comment added by 41.189.160.39 (talk) 02:08, 2 June 2014 (UTC)

See our erosion article. If you have further questions of this sort, you might do best to ask them at the science reference desk, since this desk is more for mathematics, and people who come here are somewhat less likely than people at the science desk to know the answers to your questions. Nyttend (talk) 04:04, 2 June 2014 (UTC)

3.14

Is there any evidence to suggest that pi (3.14 etc) might not be infinite? — Preceding unsigned comment added by Wholegraingood4u (talk • contribs) 11:04, 2 June 2014 (UTC)

I'm not sure what you mean by infinite but it has been proved long ago and many times that pi is irrational so it doesn't have a finite decimal representation. 11:18, 2 June 2014 (UTC)

It is definitely smaller than 4, so it is not infinite. Maproom (talk) 23:05, 2 June 2014 (UTC)

Pi is defined as a ratio, and the properties of the number have been worked out from that definition. I'm just clarifying that pi = 3.14... is a derived fact, not the definition. The fact that pi has no decimal expansion with finitely many digits is indisputable. However, if you want to be a bit silly, we can play around with the radix of our notation system. For example in base 10, pi=3.14... , but, in base pi, pi= 1, pi=10, exactly. One confusing thing then is that integers have infinite expansions in a base pi system! See Non-integer_representation#Base_.CF.80 for a little more info. SemanticMantis (talk) 15:33, 3 June 2014 (UTC)

Surely pi is 10 in base pi? AndrewWTaylor (talk) 17:02, 3 June 2014 (UTC)

Indeed, I've corrected my post. SemanticMantis (talk) 17:49, 3 June 2014 (UTC)

Area of a Figure Given by Implicit Polynomial Equation

The figure in question is this. (I hope the link will work). Its implicit symmetrical equation is $A(x^{8}+y^{8})+B(x^{6}y^{2}+x^{2}y^{6})+C~x^{4}y^{4}=r^{8},$ with A = 0.3, B = -3.3, C = 10, and r = 2. My question would be, what kind of integration to use ? Rotating it doesn't help, since the perpendiculars on the axes will always intersect its graphic in more than 1 point, even if I only take a quarter of the entire figure, so the usual Cartesian integration seems to be a no-go. I was thinking polar integration, but how on earth to express it in polar coordinates is beyond me. Or perhaps some other useful parameterization ? But I can't think of any... Can anyone give me a nudge or two in the right direction, and guide me towards the right approach ? Thank you ! — 79.113.194.160 (talk) 14:35, 2 June 2014 (UTC)

Polar coordinates will work. Write $x=\rho \cos \theta ,y=\rho \sin \theta$ in the defining equation and solve for $\rho$ as a function of $\theta$. The area is then ${\tfrac {1}{2}}\int _{0}^{2\pi }\rho ^{2}d\theta$. This is a (messy) integral of a rational function of sine and cosine. There are probably simplifications, but at worst it can be evaluated exactly using a tangent half jangle substitution and partial fractions. Sławomir Biały (talk) 15:12, 2 June 2014 (UTC)

Thank you! Your comments are always helpful, Mr. Slawomir! :-) The resulting expression does not seem to evaluate to a closed form, since we are left with integrating the fourth order root of an eighth-degree polynomial in sin and cos. I don't even think that it can be expressed in terms of elliptic integrals. However, the result is numerically correct, and at least now I know of a general method to approach this type of expressions! — 79.113.194.160 (talk) 20:35, 2 June 2014 (UTC)

Resolved

Note that the symmetry gives you the area

$\int _{0}^{2\pi }{\frac {\rho ^{2}}{2}}d\theta =8\int _{0}^{\pi /8}\rho ^{2}d\theta$.

Bo Jacoby (talk) 04:23, 3 June 2014 (UTC).

The equation is

$0.3(x^{8}+y^{8})-3.3x^{2}y^{2}(x^{4}+y^{4})+10x^{4}y^{4}=2^{8}$

Integer coefficients:

$3(x^{8}+y^{8})-33x^{2}y^{2}(x^{4}+y^{4})+100x^{4}y^{4}=2560$

Introduce polar coordinates

$x=\rho \cos \theta$

$y=\rho \sin \theta$

and simplify using wolframalpha [1]

$\rho ^{8}{\frac {45+8\cos(4\theta )+43\cos(8\theta )}{32}}=2560$

The area is

$8\int _{0}^{\pi /8}\rho ^{2}d\theta =8\cdot (32\cdot 2560)^{1/4}\int _{0}^{\pi /8}(45+8\cos(4\theta )+43\cos(8\theta ))^{-1/4}d\theta$

=23.01145712526269417506543107713910396038016657560966307867403... according to wolframalpha [2]

Bo Jacoby (talk) 10:08, 3 June 2014 (UTC).

Already on it ! :-) — 79.113.236.111 (talk) 13:26, 3 June 2014 (UTC)

Observe that the number 12.0165220075768590 is not the area, as ${\sqrt {\frac {12.02}{\pi }}}<2$ and the r=2 circle is completely inside the figure. Bo Jacoby (talk) 15:32, 3 June 2014 (UTC).

It is exactly half of the total area, since I omitted a factor of 2. — 79.113.236.111 (talk) 18:13, 3 June 2014 (UTC)

Our result are not compatible. If you chose A=1/3, B=−10/3, C=10, r=2 then the equation

$A(x^{8}+y^{8})+B(x^{6}y^{2}+x^{2}y^{6})+C~x^{4}y^{4}=r^{8},$

is

$x^{8}+y^{8}-10(x^{6}y^{2}+x^{2}y^{6})+30x^{4}y^{4}=3\cdot 2^{8}=768$

or, in polar coordinates [3]

$\rho ^{8}{\frac {4\cos(4\theta )+13\cos(8\theta )+15}{32}}=768$

Then the area is

$8\cdot (32\cdot 768)^{1/4}\int _{0}^{\pi /8}(15+4\cos(4\theta )+13\cos(8\theta ))^{-1/4}d\theta$

which is [4] 21.06422987184598836624721527569046268741543998487210947421617...

Still not in accordance with your result! Bo Jacoby (talk) 11:52, 4 June 2014 (UTC).

Actually, both your results are symbolically correct, and in accordance with my own two results, the only difference being that both Maple and Mathematica give a different numerical value than Wolfram Alpha. E.g., about 24.0330440 in the first case, and 22.1466460 in the second. The precision used was 50 decimals. — 79.113.245.159 (talk) 12:33, 4 June 2014 (UTC)

I believe that Wolfram Alpha uses Mathematica for calculations. The difference between the results is on the second digitposition!!!! Something is really wrong. Bo Jacoby (talk) 15:30, 4 June 2014 (UTC).