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[[Image:Kappa curve with asymptotes - by Pt.png|frame|The kappa curve has two vertical [[asymptote]]s.]]
In [[geometry]], the '''kappa curve''' or '''Gutschoven's curve''' is a two-dimensional [[algebraic curve]] resembling the [[Greek alphabet|Greek letter]] [[Kappa (letter)|ϰ (kappa)]].
The kappa curve was first studied by [[Gérard van Gutschoven]] around 1662.
In the history of mathematics, it is remembered as one of the first examples of [[Isaac Barrow]]'s application of rudimentary calculus methods to determine the [[tangent]] of a curve.  [[Isaac Newton]] and [[Johann Bernoulli]] continued the studies of this curve subsequently.


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Using the [[Cartesian coordinate system]] it can be expressed as:
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:<math>x^2(x^2 + y^2)=a^2y^2</math>
<ul>
or, using [[parametric equation]]s:
 
:<math>
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\begin{matrix}
 
x&=&a\sin t\\
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y&=&a\sin t\tan t
 
\end{matrix}
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In [[polar coordinates]] its equation is even simpler:
:<math>r=a\tan\theta</math>


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It has two vertical [[asymptote]]s at <math>x=\pm a</math>, shown as dashed blue lines in the figure at right.
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The kappa curve's [[curvature]]:
:<math>\kappa(\theta)={8\left(3-\sin^2\theta\right)\sin^4\theta\over a\left[\sin^2(2\theta)+4\right]^{3\over2}}</math>


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[[Tangent]]ial angle:
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:<math>\phi(\theta)=-\arctan\left[{1\over2}\sin(2\theta)\right]</math>
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==Tangents via infinitessimals==


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The tangent lines of the kappa curve can also be determined geometrically using [[Differential (infinitesimal)|differentials]] and the elementary rules
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of [[infinitessimal]] arithmetic. Suppose x and y are variables, while a is taken as a constantFrom
<ul>
the definition of the kappa curve,
 
:<math> x^2(x^2 + y^2)-a^2y^2 = 0 </math>
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Now, an infinitessimal change in our location must also change the value
 
of the left hand side, so
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:<math>d (x^2(x^2 + y^2)-a^2y^2) = 0 </math>
 
Distributing the differential and applying [[Differential_(calculus)|appropriate rules]],
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:<math>d (x^2(x^2 + y^2))-d (a^2y^2) = 0 </math>


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:<math> (2 x dx ) (x^2+y^2) + x^2 (2xdx + 2y dy) - a^2 2y dy = 0 </math>
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:<math> x ( 2 x^2 + y^2 ) dx  + y (x^2 - a^2) dy = 0 </math>
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:<math> \frac{ x ( 2 x^2 + y^2 ) }{ y (a^2 - x^2)} = \frac{dy}{dx} </math>


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*{{MathWorld|title=Kappa curve|urlname=KappaCurve}}
*[http://www-groups.dcs.st-and.ac.uk/~history/Java/Kappa.html A Java applet for playing with the curve]
*{{MacTutor|class=Curves|id=Kappa|title=Kappa Curve}}


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Revision as of 07:01, 4 July 2013

The kappa curve has two vertical asymptotes.

In geometry, the kappa curve or Gutschoven's curve is a two-dimensional algebraic curve resembling the Greek letter ϰ (kappa). The kappa curve was first studied by Gérard van Gutschoven around 1662. In the history of mathematics, it is remembered as one of the first examples of Isaac Barrow's application of rudimentary calculus methods to determine the tangent of a curve. Isaac Newton and Johann Bernoulli continued the studies of this curve subsequently.

Using the Cartesian coordinate system it can be expressed as:

x2(x2+y2)=a2y2

or, using parametric equations:

x=asinty=asinttant

In polar coordinates its equation is even simpler:

r=atanθ

It has two vertical asymptotes at x=±a, shown as dashed blue lines in the figure at right.

The kappa curve's curvature:

κ(θ)=8(3sin2θ)sin4θa[sin2(2θ)+4]32

Tangential angle:

ϕ(θ)=arctan[12sin(2θ)]

Tangents via infinitessimals

The tangent lines of the kappa curve can also be determined geometrically using differentials and the elementary rules of infinitessimal arithmetic. Suppose x and y are variables, while a is taken as a constant. From the definition of the kappa curve,

x2(x2+y2)a2y2=0

Now, an infinitessimal change in our location must also change the value of the left hand side, so

d(x2(x2+y2)a2y2)=0

Distributing the differential and applying appropriate rules,

d(x2(x2+y2))d(a2y2)=0
(2xdx)(x2+y2)+x2(2xdx+2ydy)a22ydy=0
(4x3+2xy2)dx+(2yx22a2y)dy=0
x(2x2+y2)dx+y(x2a2)dy=0
x(2x2+y2)y(a2x2)=dydx

Derivative

If we use the modern concept of a functional relationship y(x) and apply implicit differentiation, the slope of a tangent line to the kappa curve at a point (x,y) is :

2x(x2+y2)+x2(2x+2ydydx)=2a2ydydx
2x(x2+y2)+x2(2x+2ydydx)=2a2ydydx
2x3+2xy2+2x3=2a2ydydx2x2ydydx
4x3+2xy2=(2a2y2x2y)dydx
2x3+xy2a2yx2y=dydx

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  • Can a new property agent make $eight,seven-hundred inside her first 2.5 months? (See bolded part above) A couple of yr ago, I joined one of the top actual property firms in Singapore, and happened to be recruited beneath one of many high-selling divisions. I later left the business for personal causes, but I still receive work related sales figures, and I can honestly say that yes it is completely doable. I've seen the figures.

    Solely once you pass both paper in the RES exams, can you apply for the salesperson license with Council of Estate Agency through Knight Frank. An annual charge of $299.60 is payable to CEA for this license by means of Knight Frank. Create a Monetary Finances.is important given the up and down of this risky market place. Your monetary price range ought to plan for your advertising costs, any extra prices resembling training and your forecasted revenue. Fascinating, fascinated or still uncertain? Call Leon Low @ 9369 5588 and yow will discover out more on how he and his Staff members can assist you on this Actual Estate Business. See you actual quickly! By visitor contributor Getty Goh, Director of Ascendant Belongings , a real estate research and funding consultancy firm.

    One to One Mentoring Free Caveat Listings Complete In-Home Programs Paxel-On The Job Coaching (Zero value to agent) Be taught and Earn Prog. State-Of-The Artwork Residential and Commercial Property Information Mission new condo launches singapore Launches - Native and Overseas These are high 6 ways that you can do to claim your PIC grant as knowledgeable property agent. There are some necessary documents that you have to undergo the IRAS, so your application could be authorized quickly. This PIC grant allows you to develop your property agency company from the tax deduction or money payout choices. KS Tan, property agent (HSR International/Knight Frank), 90060063 (Starhub) Vincent Ong , property agent with Knight Frank, 98531741 (Starhub) and 81876384 LLB (Honours), College of London

    Resale Levy for Second-Timer Applicants – Previously second timers aren't required to pay a levy. This is applicable to only new EC land gross sales which are launched on or after 9th December 2013 Revision of Mortgage Loan Terms – From a earlier mortgage servicing ratio (MSR) stage of 60% to now 30% of a borrower's gross monthly revenue. The MSR cap will apply to EC purchases from at this time onwards. Property looking process is busy and thrilling, I am here to verify the expertise goes as smoothly as attainable by providing you with all of the provide help to need under one roof and to ensure your new home meets all of your expectations. Try to discover a companion to share an apartment through the next expatriate chat rooms. - Set Up A Digital Tour And Digital Open House Chesney Actual Property

    All potential consumers might be fastidiously screened by a single agent together with your requirement and scheduling in thoughts thus ensuring a peace of mind advertising effort. An Unique Agent is authorised to barter on your behalf because you each have a customized advertising and marketing strategy, thus leading to a sooner closure at the very best price. For properties that include each residential and non-residential portions, solely the non-residential portion is subject to GST. If you received the rental payment. Whole rental of the furnished flat = $four,500 per thirty days He wished to take up a business course in a polytechnic, however his grades did not qualify him to do so. One month's rental for the goodfaith deposit Rental Period of 4 years (48 months) or MUCH LESS Total rental x zero.004

    It is the sellers obligation to hand over the property upon authorized completion as agreed in the contract. Within the case of a HDB flat, vacant possession is a requirement. Plan your move early to prevent a final minute rush. Remove all your belongings and discard any unwanted furnishings and rubbish. Be sure that all fixtures and items within the inventory record are left behind. Invite the customer (as required by HDB) for an inspection earlier than completion. In the case of private properties, they are often sold with present tenancy or vacant possession. The gross sales contract will specify your authorized obligations. Examine with your agent in case you are in doubt.

    Moreover dedicating himself to the IP profession, Moo Kwang additionally contributes to the coaching and growth of expectant IP professionals by acting as a mentor and providing practical guidance to patent agent trainees. Moo Kwang is presently an adjunct lecturer at the Nationwide University of Singapore. He lectures on intellectual capital management and IP rights to engineering undergraduates. In a separate statement, trade promotion agency IE Singapore mentioned the nation's key exports rose 28 per cent in the second quarter from the 12 months earlier than, accelerating from a 23 per cent rise within the first quarter. - CNA/vm The aged are easy targets who may be coaxed into promoting and buying their property. Member of Institute of Estate Agent Housing Builders Guidelines