about equations in mathematics the chemistry term chemical equation NOTOC Image First Equation Ever.png thumb right 300px The first equation to ever be written in symbolic notation, by Robert Recorde in 1557 . In modern notation, the equation reads math 14x 15 71 math . An equation is a mathematics mathematical ... expressions . ref cite web url http dictionary.reference.com browse equation title Equation work Dictionary.com ... 3 isbn 0 691 11822 1 ref In this case, they can be equation solving solved to find the values that satisfy the equality. For example, consider the following. math x 2 x 0 ,. math The equation is true only for two values of x , the solutions of the equation. In this case, the solutions are math x 0 math and math x 1 math . Many mathematicians ref name Nahin reserve the term equation exclusively for the second ... x 1 , math is an equation with solutions math x 0 math and math x 1 math . Whether a statement is meant to be an identity or an equation can usually be determined from its context. In some cases, a distinction is made between the equality sign math math for an equation and the equivalence symbol math ... , a convention initiated by Ren Descartes Descartes . Properties If an equation in elementary algebra algebra is known to be true, the following operations may be used to produce another true equation ... solution s. For example, the equation y x x has 2 solutions y 1 and y 0. Dividing both sides by x simplifies the equation to y 1, but the second solution is lost. The algebraic properties ... numbers , which is an example of a field. However, if the equation were based on the natural ... sides of a true equation, then the resulting equation will still be true, but it may be less ... count 2 webkit column count 2 Cubic equation Differential equation Diophantine equation Formula editor Functional equation Indeterminate equation Inequality mathematics Inequality Inequation Integral equation Linear equation List of equations Quadratic equation Quartic equation Quintic equation Parametric ... more details
Bernoulli equation may refer to Bernoulli differential equation Bernoulli s equation , in fluid dynamics. Euler Bernoulli beam equation , in solid mechanics disambig zh ... more details
Equation editor may refer to Formula editor Read this for the comparison chart for major mathematical equation editors Microsoft Equation Editor MathType MathMagic equation editor Category Formula editors dab A long comment added to the page to prevent it being listed on Special Shortpages. Generated via Template Longcomment. ... more details
In mathematics, the term exact equation can refer either of the following Exact differential equation Closed and exact differential forms Exact differential form disambig ... more details
HH equation may refer to Henderson Hasselbach equation Hodgkin Huxley model disambig Long comment to avoid being listed on short pages ... more details
Stokes equation may refer to the Airy equation the equations of Stokes flow , a linearised form of the Navier Stokes equations in the limit of small Reynolds number Stokes law disambig ... more details
In mathematics , a summation equation or discrete integral equation is an equation in which an unknown function mathematics function appears under a summation sign. The theories of summation equations and integral equation s can be unified as integral equations on time scales ref http web.maths.unsw.edu.au cct tis tomasia IJDE rev.pdf Volterra integral equations on time scales Basic qualitative and quantitative results with applications to initial value problems on unbounded domains , Tomasia Kulik, Christopher C. Tisdell, September 3, 2007 ref using time scale calculus . A summation equation compares to a difference equation as an integral equation compares to a differential equation . The Volterra summation equation is math x t f t sum i m n k t, s, x s math where x is the unknown function, and s, a, t are integers, and f, k are known functions. References references http scholar.google.com scholar?q 22discrete integral equations 22 OR 22summation equations 22 OR 22discrete integral equation 22 OR 22summation equation 22 Summation equations or discrete integral equations Category Integral equations ... more details
An adjoint equation is a linear differential equation , usually derived from its primal equation using integration by parts . Gradient values with respect to a particular quantity of interest can be efficiently calculated by solving the adjoint equation. Methods based on solution of adjoint equations are used in wing shape optimization , flow control and uncertainty quantification . References reflist cite journal last Jameson first Antony title Aerodynamic Optimization via Control Theory journal Journal of Scientific Computing volume 3 issue 3 year 1988 DEFAULTSORT Adjoint Equation Category Differential calculus ru ... more details
This article is about the Hill differential equation, for the equation used in biochemistry see Hill equation In mathematics , the Hill s equation or Hill differential equation harvtxt Hill 1886 is the second order linear ordinary differential equation math frac d 2y dt 2 f t y 0, math where f t is a periodic function. Dimensional analysis can be used to transform this equation so the period is always 2 . Using a Fourier series representation, this may be rewritten as math frac d 2y dt 2 left theta 0 2 sum n 1 infty theta n cos 2nt right y 0, math where the s are Coefficient constants . Important special cases of Hill s equation include the Mathieu Equation and the Meissner Equation. Hill s equation is an important example in our understanding of oscillatorily forced systems. Depending on the exact shape of f t , solutions may stay bounded for all time, or the amplitude of the oscillations in solutions may grow exponentially. See also Mathieu equation References Heinrich Guggenheimer 1977 Applicable Geometry , pages 73&ndash 98, Krieger, Huntington ISBN 0882753681 . citation doi 10.1007 BF02417081 authorlink George William Hill first G.W. last Hill title On the Part of the Motion of Lunar Perigee Which is a Function of the Mean Motions of the Sun and Moon journal Acta Math. volume 8 issue 1 pages 1 36 year 1886 Magnus, Wilhelm and Winkler, Stanley. http books.google.com books?id ML5wm T4RVQC&dq Hill 27s equation&printsec frontcover&source bl&ots kXKt76K3v0&sig weeo6n5znT9hjDQvWWZNOmKtnk&hl en&ei MKbSSsbODdCTlAeVnOyoCg&sa X&oi book result&ct result&resnum 6&ved 0CBcQ6AEwBQ v onepage&q &f false Hills Equation Dover Books. dlmf first G. last Wolf id 28.29 title Mathieu Functions and Hill s Equation External links mathworld urlname HillsDifferentialEquation title Hill s Differential Equation Category Ordinary differential equations mathapplied stub it Equazione di Hill matematica km ru sl Hillova ena ba matematika ... more details
Dispersionless or quasi classical limits of integrable partial differential equations PDE arise in various problems of mathematics and physics and are intensively studied in the recent literature see, f.i., 1 5 . Examples Dispersionless KP equation The dispersionless Kadomtsev Petviashvili equation dKPE has the form math u t uu x x u yy 0, qquad 1 math It arises from the commutation math L 1, L 2 0. qquad 2 math of the following pair of 1 parameter families of vector fields math L 1 partial y lambda partial x u x partial lambda , qquad 3a math math L 2 partial t lambda 2 u partial x lambda u x u y partial lambda , qquad 3b math where math lambda math is a spectral parameter. The dKPE is the math x math dispersionless limit of the celebrated Kadomtsev Petviashvili equation . Dispersionless Korteweg de Vries equation The dispersionless Korteweg de Vries equation dKdVE reads as math u t frac 3 2 uu x . qquad 4 math It is the dispersionless or quasiclassical limit of the Korteweg de Vries equation . Dispersionless Davey Stewartson equation Dispersionless Novikov Veselov equation Dispersionless Hirota equation See also Integrable systems Nonlinear Schr dinger equation Nonlinear systems Davey Stewartson equation Dispersive partial differential equation Kadomtsev Petviashvili equation Korteweg de Vries equation References Kodam Y., Gibbons J. Integrability of the dispersionless KP hierarchy Zakharov V.E. Dispersionless limit of integrable systems in 2 1 dimensions Takasaki K. , Takebe T. Rev. Math. Phys., 7, 743 1995 Konopelchenko B.G. Quasiclassical generalized Weierstrass representation and dispersionless DS equation , ArXiv 0709.4148 Dunajski M. Interpolating integrable system . ArXiv 0804.1234 External links http tosio.math.toronto.edu wiki index.php Ishimori system Ishimori system at the dispersive equations wiki Category Partial differential equations ... more details
Unreferenced date December 2009 In mathematics , LHS is informal shorthand for the left hand side of an equation . Similarly, RHS is the right hand side . Each is solely a name for a term as part of an expression and they are in practice interchangeable, since equality mathematics equality is equivalence relation symmetric . This abbreviation is seldom if ever used in print it is very informal. More generally, these terms may apply to an inequation or inequality mathematics inequality . In the inequality case , there is no symmetry. The right hand side is everything on the right side of a test operator in an Expression mathematics expression . Conversely, the left hand side is everything on the left side. Some examples The equation on the right side right part of the sign is the right side of the equation and the left of the is the left side left part of equation. br br Take x 5 y 8 where x 5 would be the left hand side and y 8 would be the right hand side Homogeneous and inhomogeneous equations In solving mathematical equations, particularly linear simultaneous equations , differential equation s and integral equation s, the terminology homogeneous is often used for equations with the RHS set equal to zero. The corresponding inhomogeneous or nonhomogeneous equation then has the RHS ... operator L , with the difference being that between the equation Lf 0, to be solved for a function f , and the equation Lf g , with g a fixed function, to solve again for f . The point of the terminology appears for L a linear operator . Then any solution of the inhomogeneous equation may have a solution of the homogeneous equation added to it, and still remain a solution. For example in mathematical physics , the homogeneous equation may correspond to a physical theory formulated in empty space , while the inhomogeneous equation asks for more realistic solutions with some matter, or charged ..., though. See also equal sign DEFAULTSORT Sides Of An Equation Category Mathematical terminology es ... more details
functions in one variable using Felix Klein s approach to solving the quintic equation . ref name Mathworld Sextic Equation See also Cubic function Septic equation References references Polynomials DEFAULTSORT Sextic Equation Category Equations Category Galois theory Category Polynomials mathematics ... more details
Refimprove date January 2010 In mathematics , an algebraic equation , also called polynomial equation over a given Field mathematics field is an equation of the form math P Q math where P and Q are possibly Multivariate polynomial multivariate polynomial s over that field. For example math y 4 frac xy 2 frac x 3 3 xy 2 y 2 frac 1 7 math is an algebraic equation over the rationals. Two equations are equivalent if they have the same set of Equation solutions . In particular the equation math P Q math is equivalent with math P Q 0 math . It follows that the study of algebraic equations is equivalent to the study of polynomials. An algebraic equation over the rationals can always be converted to an equivalent ... 3 7 and grouping its terms in the first member, the algebraic equation above becomes the algebraic equation math 42y 4 21xy 14x 3 42xy 2 42y 2 6 0 math Although the equation math e T x 2 frac 1 T xy sin T z 2 0 math is not an algebraic equation in four variables x , y , z and T over the rational numbers because sine , exponentiation and 1 T are not polynomial functions . It is an algebraic equation ... T 3 3 frac T 5 5 frac T 7 7 cdots math 1 T and 2 are all elements of Q T . The solutions of an equation ... to precise in which field the solutions are looked for. For example, for an equation over the rationals one may look for the integer of rational solutions. In this case the equation is a diophantine equation . One may also want for the complex solutions and this relies to fundamental theorem of algebra ... function equation of degree 3 and Lodovico Ferrari has solved the Quartic function equation of degree 4 . Finally Niels Henrik Abel has proved in 1824 that the quintic equationequation of degree ... after variste Galois , were introduced to give criteria deciding if an equation is solvable using radicals. References MathWorld title Algebraic Equation urlname AlgebraicEquation See also Algebraic ... DEFAULTSORT Algebraic Equation Category Polynomials Category Equations ar de Algebraische ... more details
An independent equation is an equation in a system of simultaneous equations which cannot be derived algebraically from the other equations. See also Linear algebra Indeterminate system References Unreferenced date June 2008 Category Linear algebra Algebra stub ... more details
An indeterminate equation , in mathematics , is an equation for which there is an infinite set of solutions for example, 2x y is a simple indeterminate equation. Indeterminate equations cannot be directly solved from the given information. For example, the equations math ax by c math math x 2 Py 2 1 math where a, b, c, and P are given integers provided that P is not a square number , are indeterminate equations. Equations of the second form are named Pell s equation s. See also Indeterminate system Indeterminate variable Linear algebra References Unreferenced date August 2008 Category Algebra math stub es Ecuaci n indeterminada ko nl Onbepaalde vergelijking ... more details
In physics, the diffusion equation with drift term is often called Smoluchowski equation after Marian von Smoluchowski ref M. v. Smoluchowski, ber Brownsche Molekularbewegung unter Einwirkung u erer Kr fte und den Zusammenhang mit der verallgemeinerten Diffusionsgleichung, Ann. Phys. 353 4. Folge 48 , 1103 1112 1915 http matwbn.icm.edu.pl ksiazki pms pms2 pms2132.pdf ref . The equation Let w r , t be a density , D a diffusion constant , a friction coefficient , and U r , t a potential. Then the Smoluchowski equation reads math frac partial w partial t D nabla 2w zeta 1 nabla w nabla U . math It is a stochastic differential equation . It is formally identical to the Fokker Planck equation , the only difference being the physical meaning of w a spatial distribution for the Smoluchowski equation, a velocity distribution for the Fokker Planck equation. Other equations named for Smoluchowski In the first half of the 20th century, a number of different equations were referred to as Smoluchowski s. In an influential review, ref S. Chandrasekhar, Rev. Mod. Phys. 15, 1 1943 , equation 312 . ref Chandrasekhar s stated in 1943 that the diffusion equation with drift term is sometimes called Smoluchowski equation . Since then, this has become the standard nomenclature. Other equations due to Smoluchowski are now usually given more specific names Einstein relation kinetic theory Einstein Smoluchowski relation Smoluchowski coagulation equation References references Category Diffusion Category Stochastic differential equations ... more details
Noref date November 2009 The Prony equation is a historically important equation in hydraulics , used to calculate the head loss due to friction within a given run of pipe. It is an empirical equation developed by France Frenchman Gaspard de Prony in the 19th century math h f frac L D aV bV 2 math where h sub f sub is the head loss due to friction, calculated from the ratio of the length to diameter of the pipe L D , the velocity of the flow V , and two empirical factors a and b to account for friction. This equation has been supplanted in modern hydraulics by the Darcy Weisbach equation , which used it as a starting point. Category Equations of fluid dynamics mathapplied stub ca Equaci de Prony es Ecuaci n de Prony fr quation de Prony pt Equa o de Prony ru sl Pronyjeva ena ba ... more details
Unreferenced date December 2009 Context date October 2009 In mathematical physics , the Marchenko equation is a solution of the inverse scattering problem , when the energy of the input particle is a variable, and the impact parameter b is constant. The main equation is math K r,r prime g r,r prime int r infty K r,r prime prime g r prime prime ,r prime mathrm d r prime prime 0 math where math g r,r prime , math is a symmetric kernel , so that math g r,r prime g r prime,r . , math This equation is derived from the Gel fand Levitan integral equation , using the Povzner Levitan representation . DEFAULTSORT Marchenko Equation Category Integral equations Category Scattering theory ... more details
In geometry , the Ces ro equation of a plane curve is an equation relating curvature math kappa math to arc length math s math . It may also be given as an equation relating the radius of curvature math R math to arc length . These are equivalent because math R 1 kappa math . Two congruence geometry congruent curves will have the same Ces ro equation. It is named for Ernesto Ces ro . Some curves have an especially simple representation by a Ces ro equation. Some examples are line geometry Line math kappa 0 math . Circle math kappa 1 alpha math , where math alpha math is the radius. Logarithmic spiral math kappa C s math , where math C math is a constant. Involute Circle involute math kappa C sqrt s math , where math C math is a constant. Cornu spiral math kappa Cs math , where math C math is a constant. Catenary math kappa frac a s 2 a 2 math . The Ces ro equation of a curve is related to its Whewell equation in the following way, if the Whewell equation is math varphi f s math then the Ces ro equation is math kappa f s math . References cite book title The Mathematics Teacher year 1908 publisher National Council of Teachers of Mathematics pages 402 cite book author Edward Kasner title The Present Problems of Geometry publisher Congress of Arts and Science Universal Exposition, St. Louis year 1904 pages 574 cite book author J. Dennis Lawrence title A catalog of special plane curves publisher Dover Publications year 1972 isbn 0 486 60288 5 pages 1 5 External links MathWorld title Ces ro Equation urlname CesaroEquation MathWorld title Natural Equation urlname NaturalEquation http www.2dcurves.com derived curvature.html curvature Curvature Curves at 2dcurves.com. Category Curves eo Ekvacio de Ces ro ... more details
A transcendental equation is an equation containing a transcendental function . Examples of such an equation are math x e x math math x sin x math Solution methods Some methods of finding solutions to a transcendental equation use graphical or numerical solution numerical methods. For a graphical solution, one method is to set each side of a single variable transcendental equation equal to a dependent variable for example, y and plot the two graphs, using their intersecting points to find solutions. The numerical solution extends from finding the point at which the intersections occur using some kind of numerical calculations calculator or math software . Approximations can also be made by truncating the Taylor series if the variable is considered to be small. Additionally, Newton s method could be used to solve the equation. Often special functions can be used to write the solutions to transcendental equations in closed form . In particular, the first equation has a solution in terms of the Lambert W Function . See also Transcendental number Lambert W Function Category Equations mathanalysis stub cs Transcendentn rovnice es Ecuaci n trascendente it Equazione trascendente pl R wnanie przest pne pt Equa o transcendente ru uk zh ... more details
In the mathematics mathematical theory of partial differential equations , a Monge equation , named after Gaspard Monge , is a first order partial differential equation for an unknown function u in the independent variables x sub 1 sub ,..., x sub n sub math F left u,x 1,x 2, dots,x n, frac partial u partial x 1 , dots, frac partial u partial x n right 0 math that is a polynomial in the partial derivatives of u . Any Monge equation has a Monge cone . Classically, putting u x sub 0 sub , a Monge equation of degree k is written in the form math sum i 0 cdots i n k P i 0 dots i n x 0,x 1, dots,x k , dx 0 i 0 , dx 1 i 1 cdots dx n i n 0 math and expresses a relation between the differential of a function differentials dx sub k sub . The Monge cone at a given point x sub 0 sub , ..., x sub n sub is the zero locus of the equation in the tangent space at the point. The Monge equation is unrelated to the second order Monge Amp re equation . math stub Category Partial differential equations ... more details
In mathematics , more specifically in the study of dynamical system s and differential equation s, a Li nard equation ref Li nard, A. 1928 Etude des oscillations entretenues, Revue g n rale de l lectricit 23 , pp. 901 912 and 946 954. ref is a second order differential equation, named after the French physicist Alfred Marie Li nard . During the development of radio and vacuum tube technology, Li nard equations were intensely studied as they can be used to model oscillating circuit s. Under certain additional assumptions Li nard s theorem guarantees the uniqueness and existence of a limit cycle for such a system. Definition Let f and g be two continuously differentiable functions on R , with g an odd function and f an even function then the second order ordinary differential equation of the form math d 2x over dt 2 f x dx over dt g x 0 math is called the Li nard equation . Properties The equation can be transformed into an equivalent two dimensional system of ordinary differential equation s. We define math F x int 0 x f xi d xi math math x 1 x , math math x 2 dx over dt F x math then math ... bmatrix math is called a Li nard system . Alternatively, since Li nard equation itself also belongs to autonomous differential equation , the substitution math v dx over dt math leads the Li nard equation math d 2x over dt 2 f x dx over dt g x 0 math to a first order differential equation math v dv over dx f x v g x 0 math , which belongs to Abel equation of the second kind . ref http eqworld.ipmnet.ru en solutions ode ode0317.pdf Li nard equation at eqworld . ref ref http eqworld.ipmnet.ru en solutions ode ode0125.pdf Abel equation of the second kind at eqworld . ref Example The Van der Pol oscillator math d 2x over dt 2 mu 1 x 2 dx over dt x 0 math is a Li nard equation. Li nard s theorem ... x p and F x 0 and monotonic for x p . See also Autonomous differential equation Abel equation of the second ... DEFAULTSORT Lienard equation Category Dynamical systems Category Differential equations Category Mathematical ... more details
The Abel equation , named after Niels Henrik Abel , is special case of functional equation s which can be written in the form math f h x h x 1 , math or math alpha f x alpha x 1 math and shows non trivial properties at the iteration. Equivalence These equations are equivalent. Assuming that is an invertible function , the second equation can be written as math alpha 1 alpha f x alpha 1 alpha x 1 , . math Taking math x alpha 1 y math , the equation can be written as math f alpha 1 y alpha 1 y 1 , . math For a function f x assumed to be known, the task is to solve the functional equation for the function sup 1 sup , possibly satisfying additional requirements, such as sup 1 sup 0 1. The change of variables s sup x sup x , for a real parameter s , brings Abel s equation into the celebrated Schr der s equation , f x s x . History Initially, the equation in the more general form ref name abel cite journal url author Abel, N.H. coauthors title http gdz.sub.uni goettingen.de ru dms load img ?PPN PPN243919689 0001&DMDID dmdlog6 Untersuchung der Functionen zweier unabh ngig ver nderlichen Gr en x und y, wie f x, y , welche die Eigenschaft haben, ... journal Crelle s Journal , Berlin Journal f r die reine und angewandte Mathematik volume 1 pages 11 15 year 1826 ref ref name s cite .... Then it happens that even in the case of single variable, the equation is not trivial, and requires ... j.nahs.2006.04.002 author Jitka Laitochov title Group iteration for Abel s functional equation abstract Studied is the Abel functional equation f x x 1 ref In the case of linear transfer function ... coauthor Yu. Lubish title The Abel equation and total solvability of linear functional equtions journal Studia Mathematica volume 127 year 1998 pages 81 89 ref Special cases Equation of tetration is special case of Abel s equation, with math f exp math . In the case of integer argument, the equation is just a recurrent procedure. See also Functional equation Abel function Schr der s equation References ... more details
A milepost equation , milepoint equation , or postmile equation is a place where milepost s on a linear feature, such as a highway or rail line , fail to increase normally, usually due to realignment or changes in planned alignment. In order to make mileposts consistent with the real mileage, every milepost beyond the equation would need to be moved. ref name ODOT Oregon Department of Transportation , http www.oregon.gov ODOT TD asset mgmt AssetMgmtTermsAndDefinitions.shtml ODOT Approved Terms & Definitions , accessed October 2007 ref ref Utah Department of Transportation , http www.udot.utah.gov mileposts info.html UDOT Milepost Project Project Plan , accessed October 2007 ref ref Federal Register , Volume 69, Number 190, October 1, 2004 http edocket.access.gpo.gov 2004 04 21983.htm STB Docket No. AB 33 Sub No. 220X ref For example, an equation of 7.6 back 9.2 ahead means that the feature does not have any section between mile 7.6 and mile 9.2, and the distance between mileposts 7 and 10 is only 1.4 miles. This would usually be caused by a relocation that shortened the distance by 1.6 miles. It is also possible for an equation to add mileage to what it would otherwise be the duplicated mileposts receive a special prefix, such as Z. ref name ODOT References reflist Category Transport infrastructure Category Scales Measurement stub US road stub ... more details
A simple mathematical representation of Brownian motion , the Wiener equation , named after Norbert Wiener , assumes the current velocity of a fluid particle fluctuates random ly math mathbf v frac d mathbf x dt g t , math where v is velocity , x is position, d dt is the time derivative , and g t may for instance be white noise . Since velocity changes instantly in this formalism, the Wiener equation is not suitable for short time scales. In those cases, the Langevin equation , which looks at particle acceleration , must be used. Category Stochastic processes Category Equations mathapplied stub eo Ekvacio de Wiener ... more details
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