|
:<math>y = r \sin \varphi \,</math>
[[Sistem koordinat Kartesius|Koordinat KartesianKartesius]] ''x'' dan ''y'' dapat dikonversi ke dalam koordinat polar ''r'' dan ''φ'' dengan ''r'' ≥ 0 dan ''φ'' dalam interval (−π, π] dengan:<ref>{{Cite book|first=Bruce Follett|last=Torrence|author2=Eve Torrence|title=The Student's Introduction to Mathematica|year=1999|publisher=Cambridge University Press|isbn=0-521-59461-8}}</ref>
:<math>r = \sqrt{x^2 + y^2} \quad</math> (sebagaimana dalam [[teorema Pythagoras]] atau [[EuclideanNorma normEuklides]]), dan
:<math>\varphi = \operatorname{atan2}(y, x) \quad</math>,
di mana [[atan2]] merupakan variasi umum pada fungsi [[arctangent]] yang didefinisikan sebagai
\end{cases}</math>
Nilai ''φ'' di atas adalah [[principalnilai valuepokok]] dari fungsi [[bilangan kompleks]] [[:en:argumentargumen (complexanalisis analysiskompleks)|arg]] yang diterapkan pada ''x''+''iy''. Suatu sudut dalam rentang [0, 2π) dapat diperoleh dengan menambahkan 2π pada nilai sudut itu jika nilainya negatif.
== Persamaan kutub dari sebuah kurva ==
==Polar equation of a curve==
ThePersamaan equationyang defining anmenentukan [[algebraickurva curvealjabar]] expressedyang in polardinyatakan coordinatesdalam iskoordinat knownkutub asdikenal asebagai '' persamaan polar equation''. InDalam manybanyak caseskasus, suchpersamaan anseperti equationitu candapat simplydengan bemudah specifiedditentukan bydengan definingmendefinisikan '' r '' as asebagai [[functionFungsi (mathematicsmatematika)|functionfungsi]] ofdari ''φ''. TheKurva resultingyang curvedihasilkan thenkemudian consiststerdiri ofdari pointstitik-titik of the formbentuk (''r''(''φ''), ''φ'') anddan candapat bedianggap regarded as thesebagai [[graphgrafik ofsuatu a functionfungsi|graphgrafik]] ofdari the polarfungsi functionkutub ''r''.
DifferentBerbagai forms ofbentuk [[symmetrysimetri]] candapat bedisimpulkan deduceddari frompersamaan thefungsi equation of a polar functionkutub '' r ''. IfBila {{nowrap|''r''(−''φ'') {{=}} ''r''(''φ'')}} thekurva curveakan willsimetris be symmetrical about thetentang horizontal (0°/180°) pada ray, ifbila {{nowrap|''r''(π − ''φ'') {{=}} ''r''(''φ'')}} ititu willakan besimetris symmetricterhadap aboutsinar the verticalvertikal (90°/270°) ray, anddan ifbila {{nowrap|''r''(''φ'' − α) {{=}} ''r''(''φ'')}} itmaka willhal beitu akan menjadi [[rotationalsimetri symmetryrotasi|rotationallysimetris symmetricrotasi]] byoleh α [[clockwisesearah jarum jam|counterclockwiseberlawanan arah jarum jam]] aboutdi thesekitar polekutub.
BecauseKarena ofsistem thekoordinat circular nature of the polarkutub coordinatebersifat systemmelingkar, manybanyak curveskurva candapat bedijelaskan describeddengan bypersamaan akutub ratheryang simpleagak polar equationsederhana, whereassedangkan theirbentuk CartesianCartesiannya formjauh islebih much more intricaterumit. AmongDi theantara bestyang knownpaling ofterkenal thesedari curveskurva areini theadalah [[RoseMawar (mathematicsmatematika)|polarmawar rosepolar]], [[Archimedean spiral Archimedean]], [[Lemniscate of Bernoulli|lemniscate]], [[limaçon]], anddan [[cardioid]].
ForUntuk the circlelingkaran, linegaris, anddan polarmawar rosekutub belowdi bawahnya, itdipahami isbahwa understoodtidak thatada therebatasan are no restrictions on thepada domain anddan range of the curvekurva.
===Circle Lingkaran ===
[[ImageGambar:circle r=1.svg|thumb|right|ALingkaran circle withdengan equationpersamaan {{nowrap|''r''(''φ'') {{=}} 1}}]]
ThePersamaan generalumum equationuntuk forlingkaran a circle with adengan centerpusat atdi {{nowrap|(''r''<sub>0</sub>, <math>\gamma</math>)}} anddan radius '' a '' isadalah
:<math>r^2 - 2 r r_0 \cos(\varphi - \gamma) + r_0^2 = a^2.\, </math>
Ini dapat disederhanakan dengan berbagai cara, untuk menyesuaikan dengan kasus yang lebih spesifik, seperti persamaan
This can be simplified in various ways, to conform to more specific cases, such as the equation
:<math>r(\varphi)=a \,</math>
foruntuk alingkaran circledengan withpusat adi centerkutub atdan the pole and radiusjari-jari ''a''.<ref name="ping">{{Cite web| first=Johan| last=Claeys| url=http://www.ping.be/~ping1339/polar.htm| title=Polar coordinates| accessdate=2006-05-25}}</ref>
WhenKapan {{math|''r''}}<sub>0</sub> = {{math|a}}, oratau whenketika thetitik originasal liesterletak onpada the circlelingkaran, the equationpersamaan becomesmenjadi
:<math>r = 2 a\cos(\varphi - \gamma)</math>.
InDalam thekasus general caseumum, the equation can bepersamaan solveddapat fordiselesaikan {{math|''r''}}, givingmemberi
:<math>r = r_0 \cos(\varphi - \gamma) + \sqrt{a^2 - r_0^2 \sin^2(\varphi - \gamma)}</math>,
the solution with a minus sign in front of the square root gives the same curve.
===Line Garis ===
[[ImageGambar:Rose 2sin(4theta).svg|thumb|right|AMawar polarkutub rosedengan with equationpersamaan {{nowrap|''r''(''φ'') {{=}} 2 sin 4''φ''}}]]
''Garis radial'' (yang melewati kutub) diwakili oleh persamaan
''Radial'' lines (those running through the pole) are represented by the equation
:<math>\varphi = \gamma \,</math>,
wheredimana ɣ isadalah thesudut angleelevasi ofgaris; elevationmaka of the line;hal thatitu isadalah, {{nowrap|ɣ {{=}} arctan ''m''}} wheredi mana '' m '' is theadalah [[slopekemiringan]] ofgaris thedalam linesistem inkoordinat the Cartesian coordinate systemKartesius. TheGaris non- radial lineyang melintasi that crosses thegaris radial line {{nowrap|''φ'' {{=}} ɣ}} [[perpendiculartegak lurus]]ly at thepada pointtitik (''r<sub>0</sub>'', ɣ) has thememiliki equationpersamaan
:<math>r(\varphi) = {r_0}\sec(\varphi-\gamma). \,</math>
OtherwiseDinyatakan statedsebaliknya (''r<sub>0</sub>'', ɣ) isadalah thetitik pointdi inmana whichgaris thesinggung tangentmemotong intersectslingkaran theimajiner imaginary circle of radiusjari-jari ''r<sub>0</sub>''.
=== Polar rosemawar ===
[[mawar (matematika)|Polar mawar]] adalah kurva matematika terkenal yang terlihat seperti kelopak bunga, dan dapat diekspresikan sebagai persamaan kutub sederhana,
A [[rose (mathematics)|polar rose]] is a famous mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation,
:<math>r(\varphi) = a \cos (k\varphi + \gamma_0)\,</math>
for any constant ɣ<sub>0</sub> (including 0). If ''k'' is an integer, these equations will produce a ''k''-petaled rose if ''k'' is [[even and odd numbers|odd]], or a 2''k''-petaled rose if ''k'' is even. If ''k'' israsional rationaltetapi butbukan notbilangan an integerbulat, abentuk rose-likeseperti shapemawar maydapat formterbentuk buttetapi withdengan overlappingkelopak petalsyang tumpang tindih. NotePerhatikan thatbahwa thesepersamaan equationsini nevertidak definepernah amendefinisikan rosemawar withdengan kelopak 2, 6, 10, 14, etcdll. petals. The [[variableVariabel (mathmatematika) |variable variabel]] '' a '' represents the length of the petalsmewakili ofpanjang thekelopak rosemawar.
{{-}}
=== Spiral Archimedean spiral===
[[FileBerkas:Spiral of Archimedes.svg|thumb|right|OneSatu armlengan of anspiral Archimedean spiral withdengan equationpersamaan {{nowrap|''r''(''φ'') {{=}} ''φ'' / 2π }} foruntuk {{nowrap|0 < ''φ'' < 6π}}]]
The [[Spiral Archimedean spiral]] is a famousadalah spiral thatterkenal wasyang discoveredditemukan byoleh [[Archimedes]], whichyang canjuga alsodapat bedinyatakan expressedsebagai aspersamaan akutub simple polar equationsederhana. It is representedItu bydiwakili theoleh equationpersamaan
:<math>r(\varphi) = a+b\varphi. \,</math>
Changing theMengubah parameter '' a '' will turnakan thememutar spiral, whilesedangkan '' b '' controlsmengontrol thejarak distanceantar between the armslengan, which for ayang givenuntuk spiral istertentu alwaysselalu constantkonstan. TheSpiral Archimedean spiral hasmemiliki twodua armslengan, onesatu foruntuk {{nowrap|''φ'' > 0}} anddan onesatu foruntuk {{nowrap|''φ'' < 0}}. TheKedua twolengan armsterhubung aredengan smoothlymulus connecteddi at the poletiang. TakingMengambil thebayangan mirrorcermin imagedari ofsatu onelengan armmelintasi across thegaris 90°/270° lineakan willmenghasilkan yieldlengan the other armlainnya. ThisKurva curveini isterkenal notablesebagai assalah onesatu ofkurva the first curvespertama, after thesetelah [[conicbagian sectionkerucut]]s, tountuk bedijelaskan describeddalam inrisalah a mathematical treatisematematika, anddan assebagai beingcontoh autama primedari examplekurva ofyang apaling curvebaik thatdidefinisikan is best defined by adengan polarpersamaan equationkutub.
{{-}}
===Conic sectionsBagian kerucut ===
[[ImageGambar:Elps-slr.svg|thumb|right|250px|EllipseElips, showingmenunjukkan rektum semi-latus rectum]]
Sebuah [[bagian kerucut]] dengan satu fokus pada kutub dan yang lainnya pada suatu tempat pada sinar 0° (sehingga [[sumbu semi-mayor|sumbu mayor]] kerucut terletak di sepanjang sumbu kutub) diberikan oleh:
A [[conic section]] with one focus on the pole and the other somewhere on the 0° ray (so that the conic's [[semi-major axis|major axis]] lies along the polar axis) is given by:
: <math>r = { \ell\over {1 - e \cos \varphi}}</math>
wheredi mana '' e '' is theadalah [[eccentricityeksentrisitas (mathematicsmatematika) |eccentricity eksentrisitas]] anddan <math>\ell</math> is theadalah [[rektum semi-latus rectum]] (thejarak perpendiculartegak distancelurus atpada afokus focusdari fromsumbu theutama majorke axis to the curvekurva). IfBila {{nowrap|''e'' > 1}}, thispersamaan equationini defines amendefinisikan [[hyperbolahiperbola]]; ifbila {{nowrap|''e'' {{=}} 1}}, ititu defines amendefinisikan [[parabola]]; anddan ifbila {{nowrap|''e'' < 1}}, ititu defines anmendefinisikan [[ellipseelips]]. The specialKasus casekhusus {{nowrap|''e'' {{=}} 0}} ofhasil theterakhir latterdalam resultslingkaran in a circle of radiusjari-jari <math>\ell</math>.
{{-}}
==Intersection ofPerpotongan twodua kurva polar curves==
TheGrafik graphsdua of twofungsi polar functions <math>r=f(\theta)</math> anddan <math>r=g(\theta)</math> havememiliki possiblekemungkinan intersectionspersimpangan indalam 3 caseskasus:
# InDi theasal originbila if the equationspersamaan <math>f(\theta)=0</math> anddan <math>g(\theta)=0</math> have atmasing-masing leastmemiliki onesetidaknya solutionsatu eachsolusi.
# AllSemua the pointspoin <math>[g(\theta_i),\theta_i]</math> wheredimana <math>\theta_i</math> are theadalah solutionssolusi tountuk thepersamaan equationtersebut <math>f(\theta)=g(\theta)</math>.
# AllSemua the pointspoin <math>[g(\theta_i),\theta_i]</math> wheredimana <math>\theta_i</math> are theadalah solutionssolusi tountuk thepersamaan equationtersebut <math>f(\theta+(2k+1)\pi)=-g(\theta)</math> wheredimana <math>k</math> isadalah anbilangan integerbulat.
{{-}}
== Bilangan kompleks ==
==Complex numbers==
[[ImageGambar:Imaginarynumber2.svg|thumb|right|265px|AnIlustrasi illustrationbilangan of a complex numberkompleks '' z '' plottedyang ondiplot thepada complexbidang planekompleks]]
[[ImageGambar:Euler's formula.svg|thumb|right|265px|AnIlustrasi illustrationbilangan ofkompleks ayang complexdiplot numberpada plottedbidang onkompleks the complex plane usingmenggunakan [[rumus Euler's formula]]]]
Setiap [[bilangan kompleks]] dapat direpresentasikan sebagai sebuah titik dalam [[bidang kompleks]], dan oleh karena itu dapat diekspresikan dengan menentukan koordinat Kartesius titik tersebut (disebut bentuk persegi panjang atau kartesius) atau koordinat kutub titik (disebut bentuk polar). Bilangan kompleks '' z '' dapat direpresentasikan dalam bentuk persegi panjang sebagai
Every [[complex number]] can be represented as a point in the [[complex plane]], and can therefore be expressed by specifying either the point's Cartesian coordinates (called rectangular or Cartesian form) or the point's polar coordinates (called polar form). The complex number ''z'' can be represented in rectangular form as
: <math>z = x + iy\,</math>
di mana '' i '' adalah [[unit imajiner]], atau dapat juga ditulis dalam bentuk kutub (melalui rumus konversi yang diberikan [[#Konversi antara koordinat kutub dan Kartesius|di atas]]) sebagai
where ''i'' is the [[imaginary unit]], or can alternatively be written in polar form (via the conversion formulae given [[#Converting between polar and Cartesian coordinates|above]]) as
:<math>z = r\cdot(\cos\varphi+i\sin\varphi)</math>
and from there as
: <math>z = re^{i\varphi} \,</math>
wheredi mana '' e '' isadalah [[e (mathematicalkonstanta constantmatematika)|Euler'sbilangan numberEuler]], whichyang aresetara equivalentdengan asyang shownditunjukkan byoleh [[rumus Euler's formula]].<ref>
{{Cite book| last = Smith| first = Julius O.| title = Mathematics of the Discrete Fourier Transform (DFT)| accessdate = 2006-09-22| year = 2003| publisher = W3K Publishing| isbn = 0-9745607-0-7| chapter = Euler's Identity| chapterurl = http://ccrma-www.stanford.edu/~jos/mdft/Euler_s_Identity.html}}</ref> (NotePerhatikan thatbahwa thisrumus formulaini, likeseperti allsemua thoserumus involvingyang exponentialsmelibatkan ofsudut angleseksponensial, assumesmengasumsikan thatbahwa the anglesudut '' φ '' isdinyatakan expressed indalam [[radian]]s.) ToUntuk convertmengonversi betweenantara thebentuk rectangularpersegi andpanjang polardan formskutub ofdari asebuah complexbilangan numberkompleks, therumus conversionkonversi formulaeyang givendiberikan [[#ConvertingMengubah betweenkoordinat polar anddan CartesianKartesius|di coordinates|aboveatas]] can bedapat useddigunakan.
Untuk operasi [[perkalian]], [[pembagian (matematika)|pembagian]], dan [[eksponen]] bilangan kompleks, it umumnya jauh lebih sederhana untuk bekerja dengan bilangan kompleks yang diekspresikan dalam bentuk kutub daripada persegi panjang. Dari hukum eksponen:
For the operations of [[multiplication]], [[division (mathematics)|division]], and [[exponentiation]] of complex numbers, it is generally much simpler to work with complex numbers expressed in polar form rather than rectangular form. From the laws of exponentiation:
*Perkalian:
*Multiplication:
:: <math>r_0 e^{i\varphi_0} \cdot r_1 e^{i\varphi_1}=r_0 r_1 e^{i(\varphi_0 + \varphi_1)} \,</math>
*Divisi:
*Division:
:: <math>\frac{r_0 e^{i\varphi_0}}{r_1 e^{i\varphi_1}}=\frac{r_0}{r_1}e^{i(\varphi_0 - \varphi_1)} \,</math>
*ExponentiationEksponensial ([[rumus De Moivre's formula]]):
:: <math>(re^{i\varphi})^n=r^ne^{in\varphi} \,</math>
==Calculus Kalkulus ==
[[CalculusKalkulus]] candapat bediterapkan appliedpada topersamaan equationsyang expresseddinyatakan indalam polarkoordinat coordinatespolar.<ref>{{Cite web|url=http://archives.math.utk.edu/visual.calculus/5/polar.1/index.html|title=Areas Bounded by Polar Curves|author=Husch, Lawrence S.|accessdate=2006-11-25}}</ref><ref>{{Cite web|url=http://archives.math.utk.edu/visual.calculus/3/polar.1/index.html|title=Tangent Lines to Polar Graphs|author=Lawrence S. Husch|accessdate=2006-11-25}}</ref>
Koordinat sudut '' φ '' dinyatakan dalam radian di sepanjang bagian ini, yang merupakan pilihan konvensional saat mengerjakan kalkulus.
The angular coordinate ''φ'' is expressed in radians throughout this section, which is the conventional choice when doing calculus.
=== Kalkulus diferensial ===
===Differential calculus===
UsingMenggunakan {{nowrap|''x'' {{=}} ''r'' cos ''φ'' }} anddan {{nowrap|''y'' {{=}} ''r'' sin ''φ'' }}, oneseseorang candapat derivememperoleh ahubungan relationshipantara betweenturunan derivatives indi Cartesian anddan polarkoordinat coordinateskutub. ForUntuk afungsi given functiontertentu, ''u''(''x'',''y''), itMaka follows thatitu (bydengan computing itsmenghitung [[turunan total derivative]]s)
:<math>r \frac{\partial u}{\partial r} = r \frac{\partial u}{\partial x}\frac{\partial x}{\partial r} + r \frac{\partial u}{\partial y}\frac{\partial y}{\partial r},</math>
:<math>\frac{\partial u}{\partial \varphi} = \frac{\partial u}{\partial x}\frac{\partial x}{\partial \varphi} + \frac{\partial u}{\partial y}\frac{\partial y}{\partial \varphi},</math>
atau
or
:<math>r \frac{\partial u}{\partial r} = r \frac{\partial u}{\partial x} \cos \varphi + r \frac{\partial u}{\partial y} \sin \varphi = x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y},</math>
:<math>\frac{\partial u}{\partial \varphi} = - \frac{\partial u}{\partial x} r \sin \varphi + \frac{\partial u}{\partial y} r \cos \varphi = -y \frac{\partial u}{\partial x} + x \frac{\partial u}{\partial y}.</math>
Karenanya, kami memiliki rumus berikut:
Hence, we have the following formulae:
:<math>r \frac{\partial}{\partial r}= x \frac{\partial}{\partial x} + y \frac{\partial}{\partial y} \,</math>
:<math>\frac{\partial}{\partial \varphi} = -y \frac{\partial}{\partial x} + x \frac{\partial}{\partial y} .</math>
Menggunakan transformasi koordinat terbalik, hubungan timbal balik analog dapat diturunkan antara turunannya. Diberikan fungsi ''u''(''r'',''φ''), maka hal ini mengikuti
Using the inverse coordinates transformation, an analogous reciprocal relationship can be derived between the derivatives. Given a function ''u''(''r'',''φ''), it follows that
:<math>\frac{\partial u}{\partial x} = \frac{\partial u}{\partial r}\frac{\partial r}{\partial x} + \frac{\partial u}{\partial \varphi}\frac{\partial \varphi}{\partial x},</math>
:<math>\frac{\partial u}{\partial y} = \frac{\partial u}{\partial r}\frac{\partial r}{\partial y} + \frac{\partial u}{\partial \varphi}\frac{\partial \varphi}{\partial y},</math>
atau
or
:<math>\frac{\partial u}{\partial x} = \frac{\partial u}{\partial r}\frac{x}{\sqrt{x^2+y^2}} - \frac{\partial u}{\partial \varphi}\frac{y}{x^2+y^2} = \cos \varphi \frac{\partial u}{\partial r} - \frac{1}{r} \sin \varphi \frac{\partial u}{\partial \varphi},</math>
:<math>\frac{\partial u}{\partial y} = \frac{\partial u}{\partial r}\frac{y}{\sqrt{x^2+y^2}} + \frac{\partial u}{\partial \varphi}\frac{x}{x^2+y^2} = \sin \varphi \frac{\partial u}{\partial r} + \frac{1}{r} \cos \varphi \frac{\partial u}{\partial \varphi}.</math>
Karenanya, kami memiliki rumus berikut:
Hence, we have the following formulae:
:<math>\frac{\partial}{\partial x} = \cos \varphi \frac{\partial}{\partial r} - \frac{1}{r} \sin \varphi \frac{\partial}{\partial \varphi} \,</math>
:<math>\frac{\partial}{\partial y} = \sin \varphi \frac{\partial}{\partial r} + \frac{1}{r} \cos \varphi \frac{\partial}{\partial \varphi}.</math>
ToUntuk findmencari thekemiringan Cartesian slopedari ofgaris thesinggung tangentke line to akurva polar curve ''r''(''φ'') atpada anytitik given pointtertentu, the curve is firstkurva expressedpertama askali adinyatakan systemsebagai ofsistem [[parametricpersamaan equationsparametrik]].
:<math>x=r(\varphi)\cos\varphi \,</math>
:<math>y=r(\varphi)\sin\varphi \,</math>
[[Turunan|Diferensiasi]] kedua persamaan sehubungan dengan hasil '' φ ''
[[Derivative|Differentiating]] both equations with respect to ''φ'' yields
:<math>\frac{dx}{d\varphi}=r'(\varphi)\cos\varphi-r(\varphi)\sin\varphi \,</math>
:<math>\frac{dy}{d\varphi}=r'(\varphi)\sin\varphi+r(\varphi)\cos\varphi. \,</math>
DividingMembagi thepersamaan secondkedua equationdengan bypersamaan thepertama firstmenghasilkan yieldskemiringan theKartesius Cartesiandari slopegaris ofsinggung theke tangentkurva linepada totitik the curve at the pointtersebut. {{nowrap|(''r''(''φ''), ''φ'')}}:
:<math>\frac{dy}{dx}=\frac{r'(\varphi)\sin\varphi+r(\varphi)\cos\varphi}{r'(\varphi)\cos\varphi-r(\varphi)\sin\varphi}.</math>
ForUntuk otherrumus usefulberguna formulaslainnya includingtermasuk divergencedivergensi, gradientgradien, anddan Laplacian indalam polarkoordinat coordinatespolar, seelihat [[curvilinearkoordinat coordinateslengkung]].
===Integral calculusKalkulus integral (arcpanjang lengthbusur) ===
ThePanjang arc lengthbusur (lengthpanjang ofsegmen a line segmentgaris) defined by ayang polarditentukan functionoleh isfungsi foundkutub byditentukan theoleh integrationintegrasi overdi theatas curvekurva ''r''(''φ''). LetContohnya '' L '' denotemenunjukkan thispanjang lengthini alongsepanjang thekurva curvemulai startingdari from pointstitik '' A '' throughhingga to pointtitik '' B '', wheredi mana thesetitik-titik pointsini correspondsesuai todengan ''φ'' = ''a'' anddan ''φ'' = ''b'' suchseperti thatyang {{nowrap|0 < ''b'' − ''a'' < 2π}}. The length ofPanjang '' L '' isdiberikan givenoleh byintegral the following integralberikut
:<math>L = \int_a^b \sqrt{ \left[r(\varphi)\right]^2 + \left[ {\tfrac{dr(\varphi) }{ d\varphi }} \right] ^2 } d\varphi</math>
===Integral calculusKalkulus integral (arealuas) ===
[[ImageGambar:Polar coordinates integration region.svg|thumb|The integration region ''R'' is bounded by the curve ''r''(''φ'') and the rays ''φ'' = ''a'' and ''φ'' = ''b''.]]
Let ''R'' denote the region enclosed by a curve ''r''(''φ'') and the rays ''φ'' = ''a'' and ''φ'' = ''b'', where {{nowrap|0 < ''b'' − ''a'' ≤ 2π}}. Then, the area of ''R'' is
This comparison, plus the recognition that by the definition of the co-rotating frame at time ''t'' it has a rate of rotation Ω = ''dφ''/''dt'', shows that we can interpret the terms in the acceleration (multiplied by the mass of the particle) as found in the inertial frame as the negative of the centrifugal and Coriolis forces that would be seen in the instantaneous, non-inertial co-rotating frame.
For general motion of a particle (as opposed to simple circular motion), the centrifugal and Coriolis forces in a particle's frame of reference commonly are referred to the instantaneous [[osculating circle]] of its motion, not to a fixed center of polar coordinates. For more detail, see [[Centripetal force#Local coordinates|centripetal force]].-->
== Koneksi ke koordinat bola dan tabung ==
==Connection to spherical and cylindrical coordinates==
Sistem koordinat kutub diperluas menjadi tiga dimensi dengan dua sistem koordinat yang berbeda, [[sistem koordinat tabung|tabung]] dan [[sistem koordinat bola]].
The polar coordinate system is extended into three dimensions with two different coordinate systems, the [[cylindrical coordinate system|cylindrical]] and [[spherical coordinate system]].
==Aplikasi ==
Koordinat polar adalah dua dimensi dan karenanya hanya dapat digunakan jika posisi titik terletak pada bidang dua dimensi tunggal. Mereka paling sesuai dalam konteks apa pun di mana fenomena yang sedang dipertimbangkan secara inheren terkait dengan arah dan panjang dari titik pusat. Contohnya, Contoh di atas menunjukkan bagaimana persamaan kutub elementer cukup untuk mendefinisikan kurva, seperti spiral Archimedean yang persamaannya dalam sistem koordinat Cartesian akan jauh lebih rumit. Selain itu, banyak sistem fisik — seperti yang berkaitan dengan benda yang bergerak di sekitar titik pusat atau dengan fenomena yang berasal dari titik pusat lebih sederhana dan lebih intuitif untuk dimodelkan menggunakan polat. Motivasi awal untuk pengenalan sistem kutub adalah mempelajari [[gerakan melingkar|melingkar]] dan [[gerakan orbital]].
Polar coordinates are two-dimensional and thus they can be used only where point positions lie on a single two-dimensional plane. They are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point. For instance, the examples above show how elementary polar equations suffice to define curves—such as the Archimedean spiral—whose equation in the Cartesian coordinate system would be much more intricate. Moreover, many physical systems—such as those concerned with bodies moving around a central point or with phenomena originating from a central point—are simpler and more intuitive to model using polar coordinates. The initial motivation for the introduction of the polar system was the study of [[circular motion|circular]] and [[orbital motion]].
===Posisi dan navigasi ===
PolarKoordinat coordinateskutub aresering useddigunakan often indalam [[navigationnavigasi]], askarena thetujuan destinationatau orarah directionperjalanan ofdapat traveldiberikan cansebagai besudut givendan asjarak andari angleobjek andyang distance from the object being considereddipertimbangkan. For instanceContohnya, [[aircraftpesawat]] usemenggunakan aversi slightlyyang modifiedsedikit versiondimodifikasi ofdari thekoordinat polar coordinatesuntuk for navigationnavigasi. InDalam thissistem systemini, theyang oneumumnya generallydigunakan useduntuk forsegala anyjenis sort of navigationnavigasi, thesinar 0 ° rayumumnya is generally calleddisebut heading 360, anddan thesudutnya anglesberlanjut continueke in aarah [[clockwisesearah jarum jam]] direction, ratherbukan berlawanan arah thanjarum counterclockwisejam, asseperti indalam thesistem mathematical systemmatematika. HeadingJudul 360 correspondsberkaitan todengan [[magneticmagnet northutara]], whilesedangkan headingsjudul 90, 180, anddan 270 correspondmasing-masing toterkait magneticdengan east,magnet southtimur, and westselatan, respectivelydan barat.<ref>{{Cite web|url=http://www.thaitechnics.com/nav/adf.html|title=Aircraft Navigation System|accessdate=2006-11-26|first=Sumrit|last=Santhi}}</ref> ThusDengan demikian, ansebuah pesawat terbang aircraftyang travelingmenempuh 5 nauticalmil mileslaut dueke eastarah willtimur beakan travelingmenempuh 5 unitsunit atpada headingpos 90 (readbaca [[ICAOAlfabet spellingejaan alphabetICAO|zeronol-niner-zeronol]] byoleh [[airkontrol lalu trafficlintas controludara]]).<ref>{{Cite web|url=http://www.faa.gov/library/manuals/aircraft/airplane_handbook/media/faa-h-8083-3a-7of7.pdf|title=Emergency Procedures|format=PDF|accessdate=2007-01-15}}</ref>
=== Pemodelan ===
Sistem yang menampilkan [[simetri radial]] memberikan pengaturan alami untuk sistem koordinat kutub, dengan titik pusat bertindak sebagai kutub. Contoh utama dari penggunaan ini adalah [[persamaan aliran air tanah]]. Sistem dengan [[gaya pusat | gaya radial]] juga merupakan kandidat yang baik untuk penggunaan sistem koordinat polar. Sistem ini mencakup [[gravitasi|medan gravitasi]], yang mematuhi [[hukum kuadrat terbalik]], serta sistem dengan [[sumber titik]], seperti [[antena (radio)|antena radio]].
Sistem asimetris radial juga dapat dimodelkan dengan koordinat polar. Contohnya, [[Mikrofon#Pola kutub mikrofon|pola pengambilan]] [[mikrofon]] mengilustrasikan respons proporsionalnya terhadap suara yang masuk dari arah tertentu, dan pola ini dapat diulang. Kurva untuk mikrofon cardioid standar, mikrofon searah yang paling umum, dapat direpresentasikan sebagai {{nowrap|''r'' {{=}} 0.5 + 0.5sin(''φ'')}} pada frekuensi desain targetnya.<ref>{{Cite book|last=Eargle|first=John|authorlink=John M. Eargle|title=Handbook of Recording Engineering|year=2005|edition=Fourth|publisher=Springer|isbn = 0-387-28470-2}}</ref> Pola bergeser ke arah omnidirectionality pada frekuensi yang lebih rendah.
===Modeling===
Systems displaying [[radial symmetry]] provide natural settings for the polar coordinate system, with the central point acting as the pole. A prime example of this usage is the [[groundwater flow equation]] when applied to radially symmetric wells. Systems with a [[central force|radial force]] are also good candidates for the use of the polar coordinate system. These systems include [[gravitation|gravitational fields]], which obey the [[inverse-square law]], as well as systems with [[point source]]s, such as [[antenna (radio)|radio antennas]].
Radially asymmetric systems may also be modeled with polar coordinates. For example, a [[microphone]]'s [[Microphone#Microphone polar patterns|pickup pattern]] illustrates its proportional response to an incoming sound from a given direction, and these patterns can be represented as polar curves. The curve for a standard cardioid microphone, the most common unidirectional microphone, can be represented as {{nowrap|''r'' {{=}} 0.5 + 0.5sin(''φ'')}} at its target design frequency.<ref>{{Cite book|last=Eargle|first=John|authorlink=John M. Eargle|title=Handbook of Recording Engineering|year=2005|edition=Fourth|publisher=Springer|isbn = 0-387-28470-2}}</ref> The pattern shifts toward omnidirectionality at lower frequencies.
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== Lihat pula ==
* [[Koordinat kurvilinear]]
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