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160-548: In mathematics and physics , Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace , who first studied its properties. This is often written as ∇ 2 f = 0 {\displaystyle \nabla ^{2}\!f=0} or Δ f = 0 , {\displaystyle \Delta f=0,} where Δ = ∇ ⋅ ∇ = ∇ 2 {\displaystyle \Delta =\nabla \cdot \nabla =\nabla ^{2}}

320-447: A 2 d u d r | r = a . {\displaystyle -1=\iiint _{V}\nabla \cdot \nabla u\,dV=\iint _{S}{\frac {du}{dr}}\,dS=\left.4\pi a^{2}{\frac {du}{dr}}\right|_{r=a}.} It follows that d u d r = − 1 4 π r 2 , {\displaystyle {\frac {du}{dr}}=-{\frac {1}{4\pi r^{2}}},} on

480-459: A 2 ρ . {\displaystyle \rho '={\frac {a^{2}}{\rho }}.\,} Note that if P is inside the sphere, then P′ will be outside the sphere. The Green's function is then given by 1 4 π R − a 4 π ρ R ′ , {\displaystyle {\frac {1}{4\pi R}}-{\frac {a}{4\pi \rho R'}},\,} where R denotes

640-886: A 2 + ρ 2 − 2 a ρ cos ⁡ Θ ) 3 2 d θ ′ d φ ′ {\displaystyle u(P)={\frac {1}{4\pi }}a^{3}\left(1-{\frac {\rho ^{2}}{a^{2}}}\right)\int _{0}^{2\pi }\int _{0}^{\pi }{\frac {g(\theta ',\varphi ')\sin \theta '}{(a^{2}+\rho ^{2}-2a\rho \cos \Theta )^{\frac {3}{2}}}}d\theta '\,d\varphi '} where cos ⁡ Θ = cos ⁡ θ cos ⁡ θ ′ + sin ⁡ θ sin ⁡ θ ′ cos ⁡ ( φ − φ ′ ) {\displaystyle \cos \Theta =\cos \theta \cos \theta '+\sin \theta \sin \theta '\cos(\varphi -\varphi ')}

800-498: A n r n sin ⁡ n θ + b n r n cos ⁡ n θ ] , {\displaystyle f(z)=\sum _{n=0}^{\infty }\left[a_{n}r^{n}\cos n\theta -b_{n}r^{n}\sin n\theta \right]+i\sum _{n=1}^{\infty }\left[a_{n}r^{n}\sin n\theta +b_{n}r^{n}\cos n\theta \right],} which is a Fourier series for f . These trigonometric functions can themselves be expanded, using multiple angle formulae . Let

960-440: A n + i b n . {\displaystyle c_{n}=a_{n}+ib_{n}.} Therefore f ( z ) = ∑ n = 0 ∞ [ a n r n cos ⁡ n θ − b n r n sin ⁡ n θ ] + i ∑ n = 1 ∞ [

1120-675: A Magnetic scalar potential , ψ , as H = − ∇ ψ . {\displaystyle \mathbf {H} =-\nabla \psi .} Mathematics Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as

1280-464: A fundamental solution famously known as the heat kernel . By integrating the differential form over the material's total surface S {\displaystyle S} , we arrive at the integral form of Fourier's law: where (including the SI units): The above differential equation , when integrated for a homogeneous material of 1-D geometry between two endpoints at constant temperature, gives

1440-854: A homogeneous polynomial that is harmonic (see below ), and so counting dimensions shows that there are 2 ℓ + 1 linearly independent such polynomials. The general solution to Laplace's equation in a ball centered at the origin is a linear combination of the spherical harmonic functions multiplied by the appropriate scale factor r , f ( r , θ , φ ) = ∑ ℓ = 0 ∞ ∑ m = − ℓ ℓ f ℓ m r ℓ Y ℓ m ( θ , φ ) , {\displaystyle f(r,\theta ,\varphi )=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }f_{\ell }^{m}r^{\ell }Y_{\ell }^{m}(\theta ,\varphi ),} where

1600-591: A set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra , as established by the influence and works of Emmy Noether . Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include: The study of types of algebraic structures as mathematical objects

1760-517: A barrier, it is sometimes important to consider the conductance of the thin film of fluid that remains stationary next to the barrier. This thin film of fluid is difficult to quantify because its characteristics depend upon complex conditions of turbulence and viscosity —but when dealing with thin high-conductance barriers it can sometimes be quite significant. The previous conductance equations, written in terms of extensive properties , can be reformulated in terms of intensive properties . Ideally,

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1920-718: A complex analytic function both satisfy the Laplace equation. That is, if z = x + iy , and if f ( z ) = u ( x , y ) + i v ( x , y ) , {\displaystyle f(z)=u(x,y)+iv(x,y),} then the necessary condition that f ( z ) be analytic is that u and v be differentiable and that the Cauchy–Riemann equations be satisfied: u x = v y , v x = − u y . {\displaystyle u_{x}=v_{y},\quad v_{x}=-u_{y}.} where u x

2080-616: A different sign convention for this equation than one typically does when defining fundamental solutions. This choice of sign is often convenient to work with because −Δ is a positive operator . The definition of the fundamental solution thus implies that, if the Laplacian of u is integrated over any volume that encloses the source point, then ∭ V ∇ ⋅ ∇ u d V = − 1. {\displaystyle \iiint _{V}\nabla \cdot \nabla u\,dV=-1.} The Laplace equation

2240-406: A fixed integer ℓ , every solution Y ( θ , φ ) of the eigenvalue problem r 2 ∇ 2 Y = − ℓ ( ℓ + 1 ) Y {\displaystyle r^{2}\nabla ^{2}Y=-\ell (\ell +1)Y} is a linear combination of Y ℓ . In fact, for any such solution, r Y ( θ , φ ) is the expression in spherical coordinates of

2400-614: A foundation for all mathematics). Mathematics involves the description and manipulation of abstract objects that consist of either abstractions from nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, called axioms . Mathematics uses pure reason to prove properties of objects, a proof consisting of a succession of applications of deductive rules to already established results. These results include previously proved theorems , axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of

2560-669: A fruitful interaction between mathematics and science , to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January ;2006 issue of the Bulletin of the American Mathematical Society , "The number of papers and books included in the Mathematical Reviews (MR) database since 1940 (the first year of operation of MR)

2720-408: A hot copper ball is dropped into oil at a low temperature. Here, the temperature field within the object begins to change as a function of time, as the heat is removed from the metal, and the interest lies in analyzing this spatial change of temperature within the object over time until all gradients disappear entirely (the ball has reached the same temperature as the oil). Mathematically, this condition

2880-404: A mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space . Today's subareas of geometry include: Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were

3040-422: A mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture . Through a series of rigorous arguments employing deductive reasoning , a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma . A proven instance that forms part of

3200-402: A more general finding is termed a corollary . Numerous technical terms used in mathematics are neologisms , such as polynomial and homeomorphism . Other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning. For example, in mathematics, " or " means "one, the other or both", while, in common language, it

3360-647: A multilayer partition, the following formula is usually used: Δ Q Δ t = A ( − Δ T ) Δ x 1 k 1 + Δ x 2 k 2 + Δ x 3 k 3 + ⋯ . {\displaystyle {\frac {\Delta Q}{\Delta t}}={\frac {A\,(-\Delta T)}{{\frac {\Delta x_{1}}{k_{1}}}+{\frac {\Delta x_{2}}{k_{2}}}+{\frac {\Delta x_{3}}{k_{3}}}+\cdots }}.} For heat conduction from one fluid to another through

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3520-422: A particular medium conducts, engineers employ the thermal conductivity , also known as the conductivity constant or conduction coefficient, k . In thermal conductivity , k is defined as "the quantity of heat, Q , transmitted in time ( t ) through a thickness ( L ), in a direction normal to a surface of area ( A ), due to a temperature difference (Δ T ) [...]". Thermal conductivity is a material property that

3680-535: A population mean with a given level of confidence. Because of its use of optimization , the mathematical theory of statistics overlaps with other decision sciences , such as operations research , control theory , and mathematical economics . Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory ; numerical analysis broadly includes

3840-868: A product of trigonometric functions , here represented as a complex exponential , and associated Legendre polynomials: Y ℓ m ( θ , φ ) = N e i m φ P ℓ m ( cos ⁡ θ ) {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )=Ne^{im\varphi }P_{\ell }^{m}(\cos {\theta })} which fulfill r 2 ∇ 2 Y ℓ m ( θ , φ ) = − ℓ ( ℓ + 1 ) Y ℓ m ( θ , φ ) . {\displaystyle r^{2}\nabla ^{2}Y_{\ell }^{m}(\theta ,\varphi )=-\ell (\ell +1)Y_{\ell }^{m}(\theta ,\varphi ).} Here Y ℓ

4000-403: A scalar function to another scalar function. If the right-hand side is specified as a given function, h ( x , y , z ) {\displaystyle h(x,y,z)} , we have Δ f = h {\displaystyle \Delta f=h} This is called Poisson's equation , a generalization of Laplace's equation. Laplace's equation and Poisson's equation are

4160-411: A separate branch of mathematics until the seventeenth century. At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas. Some of these areas correspond to the older division, as

4320-437: A simple 1-D steady heat conduction equation which is analogous to Ohm's law for a simple electric resistance : Δ T = R Q ˙ {\displaystyle \Delta T=R\,{\dot {Q}}} This law forms the basis for the derivation of the heat equation . Writing U = k Δ x , {\displaystyle U={\frac {k}{\Delta x}},} where U

4480-424: A single unknown , which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices , modular integers , and geometric transformations ), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of

4640-630: A sphere of radius r that is centered on the source point, and hence u = 1 4 π r . {\displaystyle u={\frac {1}{4\pi r}}.} Note that, with the opposite sign convention (used in physics ), this is the potential generated by a point particle , for an inverse-square law force, arising in the solution of Poisson equation . A similar argument shows that in two dimensions u = − log ⁡ ( r ) 2 π . {\displaystyle u=-{\frac {\log(r)}{2\pi }}.} where log( r ) denotes

4800-418: A statistical action, such as using a procedure in, for example, parameter estimation , hypothesis testing , and selecting the best . In these traditional areas of mathematical statistics , a statistical-decision problem is formulated by minimizing an objective function , like expected loss or cost , under specific constraints. For example, designing a survey often involves minimizing the cost of estimating

4960-730: A steady incompressible, irrotational, inviscid fluid flow in the plane. The real part is the velocity potential, and the imaginary part is the stream function. According to Maxwell's equations , an electric field ( u , v ) in two space dimensions that is independent of time satisfies ∇ × ( u , v , 0 ) = ( v x − u y ) k ^ = 0 , {\displaystyle \nabla \times (u,v,0)=(v_{x}-u_{y}){\hat {\mathbf {k} }}=\mathbf {0} ,} and ∇ ⋅ ( u , v ) = ρ , {\displaystyle \nabla \cdot (u,v)=\rho ,} where ρ

Laplace's equation - Misplaced Pages Continue

5120-402: A very high thermal conductivity . It is known as "second sound" because the wave motion of heat is similar to the propagation of sound in air.this is called Quantum conduction The law of heat conduction, also known as Fourier's law (compare Fourier's heat equation ), states that the rate of heat transfer through a material is proportional to the negative gradient in the temperature and to

5280-477: A wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory , type theory , computability theory and computational complexity theory . Although these aspects of mathematical logic were introduced before the rise of computers , their use in compiler design, formal verification , program analysis , proof assistants and other aspects of computer science , contributed in turn to

5440-703: Is Fermat's Last Theorem . This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles , who used tools including scheme theory from algebraic geometry , category theory , and homological algebra . Another example is Goldbach's conjecture , which asserts that every even integer greater than 2 is the sum of two prime numbers . Stated in 1742 by Christian Goldbach , it remains unproven despite considerable effort. Number theory includes several subareas, including analytic number theory , algebraic number theory , geometry of numbers (method oriented), diophantine equations , and transcendence theory (problem oriented). Geometry

5600-441: Is flat " and "a field is always a ring ". Heat conduction Every process involving heat transfer takes place by only three methods: A region with greater thermal energy (heat) corresponds with greater molecular agitation. Thus when a hot object touches a cooler surface, the highly agitated molecules from the hot object bump the calm molecules of the cooler surface, transferring the microscopic kinetic energy and causing

5760-490: Is a Sturm–Liouville problem that forces the parameter λ to be of the form λ = ℓ ( ℓ + 1) for some non-negative integer with ℓ ≥ | m | ; this is also explained below in terms of the orbital angular momentum . Furthermore, a change of variables t = cos θ transforms this equation into the Legendre equation , whose solution is a multiple of the associated Legendre polynomial P ℓ (cos θ ) . Finally,

5920-408: Is a complex constant, but because Φ must be a periodic function whose period evenly divides 2 π , m is necessarily an integer and Φ is a linear combination of the complex exponentials e . The solution function Y ( θ , φ ) is regular at the poles of the sphere, where θ = 0, π . Imposing this regularity in the solution Θ of the second equation at the boundary points of the domain

6080-691: Is additive when several conducting layers lie between the hot and cool regions, because A and Q are the same for all layers. In a multilayer partition, the total conductance is related to the conductance of its layers by: R = R 1 + R 2 + R 3 + ⋯ {\displaystyle R=R_{1}+R_{2}+R_{3}+\cdots } or equivalently 1 U = 1 U 1 + 1 U 2 + 1 U 3 + ⋯ {\displaystyle {\frac {1}{U}}={\frac {1}{U_{1}}}+{\frac {1}{U_{2}}}+{\frac {1}{U_{3}}}+\cdots } So, when dealing with

6240-481: Is also approached exponentially; in theory, it takes infinite time, but in practice, it is over, for all intents and purposes, in a much shorter period. At the end of this process with no heat sink but the internal parts of the ball (which are finite), there is no steady-state heat conduction to reach. Such a state never occurs in this situation, but rather the end of the process is when there is no heat conduction at all. The analysis of non-steady-state conduction systems

6400-806: Is also known as the electrostatic condition. ∇ ⋅ E = ∇ ⋅ ( − ∇ V ) = − ∇ 2 V {\displaystyle \nabla \cdot \mathbf {E} =\nabla \cdot (-\nabla V)=-\nabla ^{2}V} ∇ 2 V = − ∇ ⋅ E {\displaystyle \nabla ^{2}V=-\nabla \cdot \mathbf {E} } Plugging this relation into Gauss's law, we obtain Poisson's equation for electricity, ∇ 2 V = − ρ ε 0 . {\displaystyle \nabla ^{2}V=-{\frac {\rho }{\varepsilon _{0}}}.} In

6560-474: Is an engine starting in an automobile. In this case, the transient thermal conduction phase for the entire machine is over, and the steady-state phase appears, as soon as the engine reaches steady-state operating temperature . In this state of steady-state equilibrium, temperatures vary greatly from the engine cylinders to other parts of the automobile, but at no point in space within the automobile does temperature increase or decrease. After establishing this state,

Laplace's equation - Misplaced Pages Continue

6720-468: Is an intimate connection between power series and Fourier series . If we expand a function f in a power series inside a circle of radius R , this means that f ( z ) = ∑ n = 0 ∞ c n z n , {\displaystyle f(z)=\sum _{n=0}^{\infty }c_{n}z^{n},} with suitably defined coefficients whose real and imaginary parts are given by c n =

6880-1088: Is any solution of the Poisson equation in V : ∇ ⋅ ∇ u = − f , {\displaystyle \nabla \cdot \nabla u=-f,} and u assumes the boundary values g on S , then we may apply Green's identity , (a consequence of the divergence theorem) which states that ∭ V [ G ∇ ⋅ ∇ u − u ∇ ⋅ ∇ G ] d V = ∭ V ∇ ⋅ [ G ∇ u − u ∇ G ] d V = ∬ S [ G u n − u G n ] d S . {\displaystyle \iiint _{V}\left[G\,\nabla \cdot \nabla u-u\,\nabla \cdot \nabla G\right]\,dV=\iiint _{V}\nabla \cdot \left[G\nabla u-u\nabla G\right]\,dV=\iint _{S}\left[Gu_{n}-uG_{n}\right]\,dS.\,} The notations u n and G n denote normal derivatives on S . In view of

7040-628: Is called a spherical harmonic function of degree ℓ and order m , P ℓ is an associated Legendre polynomial , N is a normalization constant, and θ and φ represent colatitude and longitude, respectively. In particular, the colatitude θ , or polar angle, ranges from 0 at the North Pole, to π /2 at the Equator, to π at the South Pole, and the longitude φ , or azimuth , may assume all values with 0 ≤ φ < 2 π . For

7200-403: Is commonly used for advanced parts. Analysis is further subdivided into real analysis , where variables represent real numbers , and complex analysis , where variables represent complex numbers . Analysis includes many subareas shared by other areas of mathematics which include: Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example

7360-513: Is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic , the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. This mathematical abstraction from reality is embodied in the modern philosophy of formalism , as founded by David Hilbert around 1910. The "nature" of

7520-428: Is due to the way that metals bond chemically: metallic bonds (as opposed to covalent or ionic bonds ) have free-moving electrons that transfer thermal energy rapidly through the metal. The electron fluid of a conductive metallic solid conducts most of the heat flux through the solid. Phonon flux is still present but carries less of the energy. Electrons also conduct electric current through conductive solids, and

7680-407: Is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called " exclusive or "). Finally, many mathematical terms are common words that are used with a completely different meaning. This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every free module

7840-456: Is ended, although steady-state conduction may continue if heat flow continues. If changes in external temperatures or internal heat generation changes are too rapid for the equilibrium of temperatures in space to take place, then the system never reaches a state of unchanging temperature distribution in time, and the system remains in a transient state. An example of a new source of heat "turning on" within an object, causing transient conduction,

8000-493: Is in Babylonian mathematics that elementary arithmetic ( addition , subtraction , multiplication , and division ) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time. In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as

8160-402: Is more complex than that of steady-state systems. If the conducting body has a simple shape, then exact analytical mathematical expressions and solutions may be possible (see heat equation for the analytical approach). However, most often, because of complicated shapes with varying thermal conductivities within the shape (i.e., most complex objects, mechanisms or machines in engineering) often

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8320-586: Is mostly used for numerical calculations . Number theory dates back to ancient Babylon and probably China . Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria. The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler . The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss . Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example

8480-404: Is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results ( theorems ) and a few basic statements. The basic statements are not subject to proof because they are self-evident ( postulates ), or are part of the definition of the subject of study ( axioms ). This principle, foundational for all mathematics,

8640-1192: Is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs." Mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations , unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas. More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and often include subscripts . Operation and relations are generally represented by specific symbols or glyphs , such as + ( plus ), × ( multiplication ), ∫ {\textstyle \int } ( integral ), = ( equal ), and < ( less than ). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of

8800-547: Is often held to be Archimedes ( c.  287  – c.  212 BC ) of Syracuse . He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series , in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are conic sections ( Apollonius of Perga , 3rd century BC), trigonometry ( Hipparchus of Nicaea , 2nd century BC), and

8960-433: Is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines , angles and circles , which were developed mainly for the needs of surveying and architecture , but has since blossomed out into many other subfields. A fundamental innovation was the ancient Greeks' introduction of the concept of proofs , which require that every assertion must be proved . For example, it

9120-414: Is primarily dependent on the medium's phase , temperature, density, and molecular bonding. Thermal effusivity is a quantity derived from conductivity, which is a measure of its ability to exchange thermal energy with its surroundings. Steady-state conduction is the form of conduction that happens when the temperature difference(s) driving the conduction are constant, so that (after an equilibration time),

9280-399: Is resistivity, x is length, and A is cross-sectional area, we have G = k A / x {\displaystyle G=kA/x\,\!} , where G is conductance, k is conductivity, x is length, and A is cross-sectional area. For heat, U = k A Δ x , {\displaystyle U={\frac {kA}{\Delta x}},} where U

9440-403: Is said to be a result of a thermal contact resistance existing between the contacting surfaces. Interfacial thermal resistance is a measure of an interface's resistance to thermal flow. This thermal resistance differs from contact resistance, as it exists even at atomically perfect interfaces. Understanding the thermal resistance at the interface between two materials is of primary significance in

9600-567: Is sometimes mistranslated as a condemnation of mathematicians. The apparent plural form in English goes back to the Latin neuter plural mathematica ( Cicero ), based on the Greek plural ta mathēmatiká ( τὰ μαθηματικά ) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after

9760-487: Is surrounded by a conducting material with a specified charge density ρ {\displaystyle \rho } , and if the total charge Q {\displaystyle Q} is known, then V {\displaystyle V} is also unique. For the magnetic field, when there is no free current, ∇ × H = 0 , {\displaystyle \nabla \times \mathbf {H} =\mathbf {0} ,} . We can thus define

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9920-502: Is the Laplace operator , ∇ ⋅ {\displaystyle \nabla \cdot } is the divergence operator (also symbolized "div"), ∇ {\displaystyle \nabla } is the gradient operator (also symbolized "grad"), and f ( x , y , z ) {\displaystyle f(x,y,z)} is a twice-differentiable real-valued function. The Laplace operator therefore maps

10080-637: Is the Poisson equation . The Laplace equation can be used in three-dimensional problems in electrostatics and fluid flow just as in two dimensions. A fundamental solution of Laplace's equation satisfies Δ u = u x x + u y y + u z z = − δ ( x − x ′ , y − y ′ , z − z ′ ) , {\displaystyle \Delta u=u_{xx}+u_{yy}+u_{zz}=-\delta (x-x',y-y',z-z'),} where

10240-468: Is the Euclidean metric tensor relative to the new coordinates and Γ denotes its Christoffel symbols . The Dirichlet problem for Laplace's equation consists of finding a solution φ on some domain D such that φ on the boundary of D is equal to some given function. Since the Laplace operator appears in the heat equation , one physical interpretation of this problem is as follows: fix

10400-437: Is the amount of energy that flows through a unit area per unit time. q = − k ∇ T , {\displaystyle \mathbf {q} =-k\nabla T,} where (including the SI units) The thermal conductivity k {\displaystyle k} is often treated as a constant, though this is not always true. While the thermal conductivity of a material generally varies with temperature,

10560-736: Is the charge density. The first Maxwell equation is the integrability condition for the differential d φ = − u d x − v d y , {\displaystyle d\varphi =-u\,dx-v\,dy,} so the electric potential φ may be constructed to satisfy φ x = − u , φ y = − v . {\displaystyle \varphi _{x}=-u,\quad \varphi _{y}=-v.} The second of Maxwell's equations then implies that φ x x + φ y y = − ρ , {\displaystyle \varphi _{xx}+\varphi _{yy}=-\rho ,} which

10720-779: Is the conductance, in W/(m K). Fourier's law can also be stated as: Δ Q Δ t = U A ( − Δ T ) . {\displaystyle {\frac {\Delta Q}{\Delta t}}=UA\,(-\Delta T).} The reciprocal of conductance is resistance, R {\displaystyle {\big .}R} is given by: R = 1 U = Δ x k = A ( − Δ T ) Δ Q Δ t . {\displaystyle R={\frac {1}{U}}={\frac {\Delta x}{k}}={\frac {A\,(-\Delta T)}{\frac {\Delta Q}{\Delta t}}}.} Resistance

10880-747: Is the conductance. Fourier's law can also be stated as: Q ˙ = U Δ T , {\displaystyle {\dot {Q}}=U\,\Delta T,} analogous to Ohm's law, I = V / R {\displaystyle I=V/R} or I = V G . {\displaystyle I=VG.} The reciprocal of conductance is resistance, R , given by: R = Δ T Q ˙ , {\displaystyle R={\frac {\Delta T}{\dot {Q}}},} analogous to Ohm's law, R = V / I . {\displaystyle R=V/I.} The rules for combining resistances and conductances (in series and parallel) are

11040-1431: Is the cosine of the angle between ( θ , φ ) and ( θ ′, φ ′) . A simple consequence of this formula is that if u is a harmonic function, then the value of u at the center of the sphere is the mean value of its values on the sphere. This mean value property immediately implies that a non-constant harmonic function cannot assume its maximum value at an interior point. Laplace's equation in spherical coordinates is: ∇ 2 f = 1 r 2 ∂ ∂ r ( r 2 ∂ f ∂ r ) + 1 r 2 sin ⁡ θ ∂ ∂ θ ( sin ⁡ θ ∂ f ∂ θ ) + 1 r 2 sin 2 ⁡ θ ∂ 2 f ∂ φ 2 = 0. {\displaystyle \nabla ^{2}f={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}=0.} Consider

11200-508: Is the electrical analogue of Fourier's law and Fick's laws of diffusion is its chemical analogue. The differential form of Fourier's law of thermal conduction shows that the local heat flux density q {\displaystyle \mathbf {q} } is equal to the product of thermal conductivity k {\displaystyle k} and the negative local temperature gradient − ∇ T {\displaystyle -\nabla T} . The heat flux density

11360-470: Is the first partial derivative of u with respect to x . It follows that u y y = ( − v x ) y = − ( v y ) x = − ( u x ) x . {\displaystyle u_{yy}=(-v_{x})_{y}=-(v_{y})_{x}=-(u_{x})_{x}.} Therefore u satisfies the Laplace equation. A similar calculation shows that v also satisfies

11520-418: Is the purpose of universal algebra and category theory . The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces ; this particular area of application is called algebraic topology . Calculus, formerly called infinitesimal calculus,

11680-405: Is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithms —especially their implementation and computational complexity —play a major role in discrete mathematics. The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of

11840-508: Is true regarding number theory (the modern name for higher arithmetic ) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations . Number theory began with

12000-537: Is unchanged under a rotation of coordinates, and hence we can expect that a fundamental solution may be obtained among solutions that only depend upon the distance r from the source point. If we choose the volume to be a ball of radius a around the source point, then Gauss's divergence theorem implies that − 1 = ∭ V ∇ ⋅ ∇ u d V = ∬ S d u d r d S = 4 π

12160-839: Is used in its one-dimensional form, for example, in the x direction: q x = − k d T d x . {\displaystyle q_{x}=-k{\frac {dT}{dx}}.} In an isotropic medium, Fourier's law leads to the heat equation ∂ T ∂ t = α ( ∂ 2 T ∂ x 2 + ∂ 2 T ∂ y 2 + ∂ 2 T ∂ z 2 ) {\displaystyle {\frac {\partial T}{\partial t}}=\alpha \left({\frac {\partial ^{2}T}{\partial x^{2}}}+{\frac {\partial ^{2}T}{\partial y^{2}}}+{\frac {\partial ^{2}T}{\partial z^{2}}}\right)} with

12320-2363: The ( r , θ , φ ) {\displaystyle (r,\theta ,\varphi )} convention, ∇ 2 f = 1 r 2 ∂ ∂ r ( r 2 ∂ f ∂ r ) + 1 r 2 sin ⁡ θ ∂ ∂ θ ( sin ⁡ θ ∂ f ∂ θ ) + 1 r 2 sin 2 ⁡ θ ∂ 2 f ∂ φ 2 = 0. {\displaystyle \nabla ^{2}f={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}=0.} More generally, in arbitrary curvilinear coordinates (ξ) , ∇ 2 f = ∂ ∂ ξ j ( ∂ f ∂ ξ k g k j ) + ∂ f ∂ ξ j g j m Γ m n n = 0 , {\displaystyle \nabla ^{2}f={\frac {\partial }{\partial \xi ^{j}}}\left({\frac {\partial f}{\partial \xi ^{k}}}g^{kj}\right)+{\frac {\partial f}{\partial \xi ^{j}}}g^{jm}\Gamma _{mn}^{n}=0,} or ∇ 2 f = 1 | g | ∂ ∂ ξ i ( | g | g i j ∂ f ∂ ξ j ) = 0 , ( g = det { g i j } ) {\displaystyle \nabla ^{2}f={\frac {1}{\sqrt {|g|}}}{\frac {\partial }{\partial \xi ^{i}}}\!\left({\sqrt {|g|}}g^{ij}{\frac {\partial f}{\partial \xi ^{j}}}\right)=0,\qquad (g=\det\{g_{ij}\})} where g ij

12480-523: The f ℓ are constants and the factors r Y ℓ are known as solid harmonics . Such an expansion is valid in the ball r < R = 1 lim sup ℓ → ∞ | f ℓ m | 1 / ℓ . {\displaystyle r<R={\frac {1}{\limsup _{\ell \to \infty }|f_{\ell }^{m}|^{{1}/{\ell }}}}.} For r > R {\displaystyle r>R} ,

12640-586: The Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It

12800-451: The Dirac delta function δ denotes a unit source concentrated at the point ( x ′, y ′, z ′) . No function has this property: in fact it is a distribution rather than a function; but it can be thought of as a limit of functions whose integrals over space are unity, and whose support (the region where the function is non-zero) shrinks to a point (see weak solution ). It is common to take

12960-768: The Golden Age of Islam , especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra . Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarizmi , Omar Khayyam and Sharaf al-Dīn al-Ṭūsī . The Greek and Arabic mathematical texts were in turn translated to Latin during

13120-511: The Pythagoreans appeared to have considered it a subject in its own right. Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof. His book, Elements , is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity

13280-536: The Renaissance , mathematics was divided into two main areas: arithmetic , regarding the manipulation of numbers, and geometry , regarding the study of shapes. Some types of pseudoscience , such as numerology and astrology , were not then clearly distinguished from mathematics. During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of

13440-446: The controversy over Cantor's set theory . In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour . This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory . Roughly speaking, each mathematical object

13600-1163: The natural logarithm . Note that, with the opposite sign convention, this is the potential generated by a pointlike sink (see point particle ), which is the solution of the Euler equations in two-dimensional incompressible flow . A Green's function is a fundamental solution that also satisfies a suitable condition on the boundary S of a volume V . For instance, G ( x , y , z ; x ′ , y ′ , z ′ ) {\displaystyle G(x,y,z;x',y',z')} may satisfy ∇ ⋅ ∇ G = − δ ( x − x ′ , y − y ′ , z − z ′ ) in  V , {\displaystyle \nabla \cdot \nabla G=-\delta (x-x',y-y',z-z')\qquad {\text{in }}V,} G = 0 if ( x , y , z ) on  S . {\displaystyle G=0\quad {\text{if}}\quad (x,y,z)\qquad {\text{on }}S.} Now if u

13760-483: The thermal and electrical conductivities of most metals have about the same ratio. A good electrical conductor, such as copper , also conducts heat well. Thermoelectricity is caused by the interaction of heat flux and electric current. Heat conduction within a solid is directly analogous to diffusion of particles within a fluid, in the situation where there are no fluid currents. In gases, heat transfer occurs through collisions of gas molecules with one another. In

13920-400: The 17th century, when René Descartes introduced what is now called Cartesian coordinates . This constituted a major change of paradigm : Instead of defining real numbers as lengths of line segments (see number line ), it allowed the representation of points using their coordinates , which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry

14080-405: The 19th century, mathematicians discovered non-Euclidean geometries , which do not follow the parallel postulate . By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics . This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not

14240-532: The 20th century. The P versus NP problem , which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems. Discrete mathematics includes: The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic , although used for mathematical proofs, belonged to philosophy and

14400-551: The Laplace equation are called conjugate harmonic functions . This construction is only valid locally, or provided that the path does not loop around a singularity. For example, if r and θ are polar coordinates and φ = log ⁡ r , {\displaystyle \varphi =\log r,} then a corresponding analytic function is f ( z ) = log ⁡ z = log ⁡ r + i θ . {\displaystyle f(z)=\log z=\log r+i\theta .} However,

14560-1004: The Laplace equation. Conversely, given a harmonic function, it is the real part of an analytic function, f ( z ) (at least locally). If a trial form is f ( z ) = φ ( x , y ) + i ψ ( x , y ) , {\displaystyle f(z)=\varphi (x,y)+i\psi (x,y),} then the Cauchy–Riemann equations will be satisfied if we set ψ x = − φ y , ψ y = φ x . {\displaystyle \psi _{x}=-\varphi _{y},\quad \psi _{y}=\varphi _{x}.} This relation does not determine ψ , but only its increments: d ψ = − φ y d x + φ x d y . {\displaystyle d\psi =-\varphi _{y}\,dx+\varphi _{x}\,dy.} The Laplace equation for φ implies that

14720-637: The Middle Ages and made available in Europe. During the early modern period , mathematics began to develop at an accelerating pace in Western Europe , with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation ,

14880-456: The absence of convection, which relates to a moving fluid or gas phase, thermal conduction through a gas phase is highly dependent on the composition and pressure of this phase, and in particular, the mean free path of gas molecules relative to the size of the gas gap, as given by the Knudsen number K n {\displaystyle K_{n}} . To quantify the ease with which

15040-442: The amount of heat coming out (if this were not so, the temperature would be rising or falling, as thermal energy was tapped or trapped in a region). For example, a bar may be cold at one end and hot at the other, but after a state of steady-state conduction is reached, the spatial gradient of temperatures along the bar does not change any further, as time proceeds. Instead, the temperature remains constant at any given cross-section of

15200-449: The angle θ is single-valued only in a region that does not enclose the origin. The close connection between the Laplace equation and analytic functions implies that any solution of the Laplace equation has derivatives of all orders, and can be expanded in a power series , at least inside a circle that does not enclose a singularity. This is in sharp contrast to solutions of the wave equation , which generally have less regularity. There

15360-420: The application of approximate theories is required, and/or numerical analysis by computer. One popular graphical method involves the use of Heisler Charts . Occasionally, transient conduction problems may be considerably simplified if regions of the object being heated or cooled can be identified, for which thermal conductivity is very much greater than that for heat paths leading into the region. In this case,

15520-401: The area, at right angles to that gradient, through which the heat flows. We can state this law in two equivalent forms: the integral form, in which we look at the amount of energy flowing into or out of a body as a whole, and the differential form, in which we look at the flow rates or fluxes of energy locally. Newton's law of cooling is a discrete analogue of Fourier's law, while Ohm's law

15680-920: The assumption that Y has the form Y ( θ , φ ) = Θ( θ ) Φ( φ ) . Applying separation of variables again to the second equation gives way to the pair of differential equations 1 Φ d 2 Φ d φ 2 = − m 2 {\displaystyle {\frac {1}{\Phi }}{\frac {d^{2}\Phi }{d\varphi ^{2}}}=-m^{2}} λ sin 2 ⁡ θ + sin ⁡ θ Θ d d θ ( sin ⁡ θ d Θ d θ ) = m 2 {\displaystyle \lambda \sin ^{2}\theta +{\frac {\sin \theta }{\Theta }}{\frac {d}{d\theta }}\left(\sin \theta {\frac {d\Theta }{d\theta }}\right)=m^{2}} for some number m . A priori, m

15840-469: The bar is finally set up, and this gradient then stays constant in time. Typically, such a new steady-state gradient is approached exponentially with time after a new temperature-or-heat source or sink, has been introduced. When a "transient conduction" phase is over, heat flow may continue at high power, so long as temperatures do not change. An example of transient conduction that does not end with steady-state conduction, but rather no conduction, occurs when

16000-583: The beginnings of algebra (Diophantus, 3rd century AD). The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics . Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine , and an early form of infinite series . During

16160-498: The colder part or object to heat up. Mathematically, thermal conduction works just like diffusion. As temperature difference goes up, the distance traveled gets shorter or the area goes up thermal conduction increases: Where: Conduction is the main mode of heat transfer for solid materials because the strong inter-molecular forces allow the vibrations of particles to be easily transmitted, in comparison to liquids and gases. Liquids have weaker inter-molecular forces and more space between

16320-511: The concept of a proof and its associated mathematical rigour first appeared in Greek mathematics , most notably in Euclid 's Elements . Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions ), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new fields. Since then,

16480-469: The conditions satisfied by u and G , this result simplifies to u ( x ′ , y ′ , z ′ ) = ∭ V G f d V + ∬ S G n g d S . {\displaystyle u(x',y',z')=\iiint _{V}Gf\,dV+\iint _{S}G_{n}g\,dS.\,} Thus the Green's function describes

16640-399: The current language, where expressions play the role of noun phrases and formulas play the role of clauses . Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is

16800-569: The derived expression mathēmatikḗ tékhnē ( μαθηματικὴ τέχνη ), meaning ' mathematical science ' . It entered the English language during the Late Middle English period through French and Latin. Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely

16960-569: The differential of a function ψ by d ψ = u d y − v d x , {\displaystyle d\psi =u\,dy-v\,dx,} then the continuity condition is the integrability condition for this differential: the resulting function is called the stream function because it is constant along flow lines . The first derivatives of ψ are given by ψ x = − v , ψ y = u , {\displaystyle \psi _{x}=-v,\quad \psi _{y}=u,} and

17120-501: The distance to the source point P and R ′ denotes the distance to the reflected point P ′. A consequence of this expression for the Green's function is the Poisson integral formula . Let ρ , θ , and φ be spherical coordinates for the source point P . Here θ denotes the angle with the vertical axis, which is contrary to the usual American mathematical notation, but agrees with standard European and physical practice. Then

17280-453: The domain where the equation is satisfied. If any two functions are solutions to Laplace's equation (or any linear homogeneous differential equation), their sum (or any linear combination) is also a solution. This property, called the principle of superposition , is very useful. For example, solutions to complex problems can be constructed by summing simple solutions. Laplace's equation in two independent variables in rectangular coordinates has

17440-480: The electric field can be expressed as the negative gradient of the electric potential V {\displaystyle V} , E = − ∇ V , {\displaystyle \mathbf {E} =-\nabla V,} if the field is irrotational, ∇ × E = 0 {\displaystyle \nabla \times \mathbf {E} =\mathbf {0} } . The irrotationality of E {\displaystyle \mathbf {E} }

17600-518: The electric field, ρ {\displaystyle \rho } be the electric charge density, and ε 0 {\displaystyle \varepsilon _{0}} be the permittivity of free space. Then Gauss's law for electricity (Maxwell's first equation) in differential form states ∇ ⋅ E = ρ ε 0 . {\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}.} Now,

17760-405: The equation for R has solutions of the form R ( r ) = A r + B r ; requiring the solution to be regular throughout R forces B = 0 . Here the solution was assumed to have the special form Y ( θ , φ ) = Θ( θ ) Φ( φ ) . For a given value of ℓ , there are 2 ℓ + 1 independent solutions of this form, one for each integer m with − ℓ ≤ m ≤ ℓ . These angular solutions are

17920-428: The expansion of these logical theories. The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory . Statisticians generate data with random sampling or randomized experiments . Statistical theory studies decision problems such as minimizing the risk ( expected loss ) of

18080-567: The first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established. In Latin and English, until around 1700, the term mathematics more commonly meant " astrology " (or sometimes " astronomy ") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine 's warning that Christians should beware of mathematici , meaning "astrologers",

18240-470: The form ∂ 2 ψ ∂ x 2 + ∂ 2 ψ ∂ y 2 ≡ ψ x x + ψ y y = 0. {\displaystyle {\frac {\partial ^{2}\psi }{\partial x^{2}}}+{\frac {\partial ^{2}\psi }{\partial y^{2}}}\equiv \psi _{xx}+\psi _{yy}=0.} The real and imaginary parts of

18400-455: The formulae for conductance should produce a quantity with dimensions independent of distance, like Ohm's law for electrical resistance, R = V / I {\displaystyle R=V/I\,\!} , and conductance, G = I / V {\displaystyle G=I/V\,\!} . From the electrical formula: R = ρ x / A {\displaystyle R=\rho x/A} , where ρ

18560-492: The function φ itself on the boundary of D but its normal derivative . Physically, this corresponds to the construction of a potential for a vector field whose effect is known at the boundary of D alone. For the example of the heat equation it amounts to prescribing the heat flux through the boundary. In particular, at an adiabatic boundary, the normal derivative of φ is zero. Solutions of Laplace's equation are called harmonic functions ; they are all analytic within

18720-403: The heat flow rate as Q = − k A Δ t L Δ T , {\displaystyle Q=-k{\frac {A\Delta t}{L}}\Delta T,} where One can define the (macroscopic) thermal resistance of the 1-D homogeneous material: R = 1 k L A {\displaystyle R={\frac {1}{k}}{\frac {L}{A}}} With

18880-395: The influence at ( x ′, y ′, z ′) of the data f and g . For the case of the interior of a sphere of radius a , the Green's function may be obtained by means of a reflection ( Sommerfeld 1949 ): the source point P at distance ρ from the center of the sphere is reflected along its radial line to a point P' that is at a distance ρ ′ =

19040-1014: The inner and outer wall, T 2 − T 1 {\displaystyle T_{2}-T_{1}} . The surface area of the cylinder is A r = 2 π r ℓ {\displaystyle A_{r}=2\pi r\ell } When Fourier's equation is applied: Q ˙ = − k A r d T d r = − 2 k π r ℓ d T d r {\displaystyle {\dot {Q}}=-kA_{r}{\frac {dT}{dr}}=-2k\pi r\ell {\frac {dT}{dr}}} and rearranged: Q ˙ ∫ r 1 r 2 1 r d r = − 2 k π ℓ ∫ T 1 T 2 d T {\displaystyle {\dot {Q}}\int _{r_{1}}^{r_{2}}{\frac {1}{r}}\,dr=-2k\pi \ell \int _{T_{1}}^{T_{2}}dT} then

19200-413: The integrability condition for ψ is satisfied: ψ x y = ψ y x , {\displaystyle \psi _{xy}=\psi _{yx},} and thus ψ may be defined by a line integral. The integrability condition and Stokes' theorem implies that the value of the line integral connecting two points is independent of the path. The resulting pair of solutions of

19360-491: The interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method , which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics. Before

19520-400: The introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and

19680-510: The irrotationality condition implies that ψ satisfies the Laplace equation. The harmonic function φ that is conjugate to ψ is called the velocity potential . The Cauchy–Riemann equations imply that φ x = ψ y = u , φ y = − ψ x = v . {\displaystyle \varphi _{x}=\psi _{y}=u,\quad \varphi _{y}=-\psi _{x}=v.} Thus every analytic function corresponds to

19840-409: The manipulation of numbers , that is, natural numbers ( N ) , {\displaystyle (\mathbb {N} ),} and later expanded to integers ( Z ) {\displaystyle (\mathbb {Z} )} and rational numbers ( Q ) . {\displaystyle (\mathbb {Q} ).} Number theory was once called arithmetic, but nowadays this term

20000-454: The mode of thermal energy flow is termed transient conduction. Another term is "non-steady-state" conduction, referring to the time-dependence of temperature fields in an object. Non-steady-state situations appear after an imposed change in temperature at a boundary of an object. They may also occur with temperature changes inside an object, as a result of a new source or sink of heat suddenly introduced within an object, causing temperatures near

20160-400: The natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer , who promoted intuitionistic logic , which explicitly lacks the law of excluded middle . These problems and debates led to

20320-536: The objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains

20480-439: The origin would be felt at infinity instantaneously. The speed of information propagation is faster than the speed of light in vacuum, which is physically inadmissible within the framework of relativity. Second sound is a quantum mechanical phenomenon in which heat transfer occurs by wave -like motion, rather than by the more usual mechanism of diffusion . Heat takes the place of pressure in normal sound waves. This leads to

20640-407: The particles, which makes the vibrations of particles harder to transmit. Gases have even more space, and therefore infrequent particle collisions. This makes liquids and gases poor conductors of heat. Thermal contact conductance is the study of heat conduction between solid bodies in contact. A temperature drop is often observed at the interface between the two surfaces in contact. This phenomenon

20800-477: The particular case of a source-free region, ρ = 0 {\displaystyle \rho =0} and Poisson's equation reduces to Laplace's equation for the electric potential. If the electrostatic potential V {\displaystyle V} is specified on the boundary of a region R {\displaystyle {\mathcal {R}}} , then it is uniquely determined. If R {\displaystyle {\mathcal {R}}}

20960-521: The pattern of physics and metaphysics , inherited from Greek. In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math . In addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years. Evidence for more complex mathematics does not appear until around 3000  BC , when

21120-1177: The problem of finding solutions of the form f ( r , θ , φ ) = R ( r ) Y ( θ , φ ) . By separation of variables , two differential equations result by imposing Laplace's equation: 1 R d d r ( r 2 d R d r ) = λ , 1 Y 1 sin ⁡ θ ∂ ∂ θ ( sin ⁡ θ ∂ Y ∂ θ ) + 1 Y 1 sin 2 ⁡ θ ∂ 2 Y ∂ φ 2 = − λ . {\displaystyle {\frac {1}{R}}{\frac {d}{dr}}\left(r^{2}{\frac {dR}{dr}}\right)=\lambda ,\qquad {\frac {1}{Y}}{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial Y}{\partial \theta }}\right)+{\frac {1}{Y}}{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}Y}{\partial \varphi ^{2}}}=-\lambda .} The second equation can be simplified under

21280-658: The proof of numerous theorems. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss , who made numerous contributions to fields such as algebra, analysis, differential geometry , matrix theory , number theory, and statistics . In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems , which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved. Mathematics has since been greatly extended, and there has been

21440-578: The quantities u and v be the horizontal and vertical components of the velocity field of a steady incompressible, irrotational flow in two dimensions. The continuity condition for an incompressible flow is that u x + v y = 0 , {\displaystyle u_{x}+v_{y}=0,} and the condition that the flow be irrotational is that ∇ × V = v x − u y = 0. {\displaystyle \nabla \times \mathbf {V} =v_{x}-u_{y}=0.} If we define

21600-1249: The rate of heat transfer is: Q ˙ = 2 k π ℓ T 1 − T 2 ln ⁡ ( r 2 / r 1 ) {\displaystyle {\dot {Q}}=2k\pi \ell {\frac {T_{1}-T_{2}}{\ln(r_{2}/r_{1})}}} the thermal resistance is: R c = Δ T Q ˙ = ln ⁡ ( r 2 / r 1 ) 2 π k ℓ {\displaystyle R_{c}={\frac {\Delta T}{\dot {Q}}}={\frac {\ln(r_{2}/r_{1})}{2\pi k\ell }}} and Q ˙ = 2 π k ℓ r m T 1 − T 2 r 2 − r 1 {\textstyle {\dot {Q}}=2\pi k\ell r_{m}{\frac {T_{1}-T_{2}}{r_{2}-r_{1}}}} , where r m = r 2 − r 1 ln ⁡ ( r 2 / r 1 ) {\textstyle r_{m}={\frac {r_{2}-r_{1}}{\ln(r_{2}/r_{1})}}} . It

21760-433: The region with high conductivity can often be treated in the lumped capacitance model , as a "lump" of material with a simple thermal capacitance consisting of its aggregate heat capacity . Such regions warm or cool, but show no significant temperature variation across their extent, during the process (as compared to the rest of the system). This is due to their far higher conductance. During transient conduction, therefore,

21920-408: The rod normal to the direction of heat transfer, and this temperature varies linearly in space in the case where there is no heat generation in the rod. In steady-state conduction, all the laws of direct current electrical conduction can be applied to "heat currents". In such cases, it is possible to take "thermal resistances" as the analog to electrical resistances . In such cases, temperature plays

22080-470: The role of the resistor in the circuit. The theory of relativistic heat conduction is a model that is compatible with the theory of special relativity. For most of the last century, it was recognized that the Fourier equation is in contradiction with the theory of relativity because it admits an infinite speed of propagation of heat signals. For example, according to the Fourier equation, a pulse of heat at

22240-422: The role of voltage, and heat transferred per unit time (heat power) is the analog of electric current. Steady-state systems can be modeled by networks of such thermal resistances in series and parallel, in exact analogy to electrical networks of resistors. See purely resistive thermal circuits for an example of such a network. During any period in which temperatures changes in time at any place within an object,

22400-405: The same for both heat flow and electric current. Conduction through cylindrical shells (e.g. pipes) can be calculated from the internal radius, r 1 {\displaystyle r_{1}} , the external radius, r 2 {\displaystyle r_{2}} , the length, ℓ {\displaystyle \ell } , and the temperature difference between

22560-497: The simplest examples of elliptic partial differential equations . Laplace's equation is also a special case of the Helmholtz equation . The general theory of solutions to Laplace's equation is known as potential theory . The twice continuously differentiable solutions of Laplace's equation are the harmonic functions , which are important in multiple branches of physics, notably electrostatics, gravitation, and fluid dynamics . In

22720-609: The solid harmonics with negative powers of r {\displaystyle r} are chosen instead. In that case, one needs to expand the solution of known regions in Laurent series (about r = ∞ {\displaystyle r=\infty } ), instead of Taylor series (about r = 0 {\displaystyle r=0} ), to match the terms and find f ℓ m {\displaystyle f_{\ell }^{m}} . Let E {\displaystyle \mathbf {E} } be

22880-549: The solution of the Laplace equation with Dirichlet boundary values g inside the sphere is given by( Zachmanoglou & Thoe 1986 , p. 228) u ( P ) = 1 4 π a 3 ( 1 − ρ 2 a 2 ) ∫ 0 2 π ∫ 0 π g ( θ ′ , φ ′ ) sin ⁡ θ ′ (

23040-420: The source or sink to change in time. When a new perturbation of temperature of this type happens, temperatures within the system change in time toward a new equilibrium with the new conditions, provided that these do not change. After equilibrium, heat flow into the system once again equals the heat flow out, and temperatures at each point inside the system no longer change. Once this happens, transient conduction

23200-399: The spatial distribution of temperatures (temperature field) in the conducting object does not change any further. Thus, all partial derivatives of temperature concerning space may either be zero or have nonzero values, but all derivatives of temperature at any point concerning time are uniformly zero. In steady-state conduction, the amount of heat entering any region of an object is equal to

23360-657: The study and the manipulation of formulas . Calculus , consisting of the two subfields differential calculus and integral calculus , is the study of continuous functions , which model the typically nonlinear relationships between varying quantities, as represented by variables . This division into four main areas—arithmetic, geometry, algebra, and calculus —endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics. The subject of combinatorics has been studied for much of recorded history, yet did not become

23520-568: The study of approximation and discretization with special focus on rounding errors . Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic- matrix -and- graph theory . Other areas of computational mathematics include computer algebra and symbolic computation . The word mathematics comes from the Ancient Greek word máthēma ( μάθημα ), meaning ' something learned, knowledge, mathematics ' , and

23680-1419: The study of heat conduction , the Laplace equation is the steady-state heat equation . In general, Laplace's equation describes situations of equilibrium, or those that do not depend explicitly on time. In rectangular coordinates , ∇ 2 f = ∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 + ∂ 2 f ∂ z 2 = 0. {\displaystyle \nabla ^{2}f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}=0.} In cylindrical coordinates , ∇ 2 f = 1 r ∂ ∂ r ( r ∂ f ∂ r ) + 1 r 2 ∂ 2 f ∂ ϕ 2 + ∂ 2 f ∂ z 2 = 0. {\displaystyle \nabla ^{2}f={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \phi ^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}=0.} In spherical coordinates , using

23840-484: The study of its thermal properties. Interfaces often contribute significantly to the observed properties of the materials. The inter-molecular transfer of energy could be primarily by elastic impact, as in fluids, or by free-electron diffusion, as in metals, or phonon vibration , as in insulators. In insulators , the heat flux is carried almost entirely by phonon vibrations. Metals (e.g., copper, platinum, gold, etc.) are usually good conductors of thermal energy. This

24000-446: The temperature across their conductive regions changes uniformly in space, and as a simple exponential in time. An example of such systems is those that follow Newton's law of cooling during transient cooling (or the reverse during heating). The equivalent thermal circuit consists of a simple capacitor in series with a resistor. In such cases, the remainder of the system with a high thermal resistance (comparatively low conductivity) plays

24160-433: The temperature on the boundary of the domain according to the given specification of the boundary condition. Allow heat to flow until a stationary state is reached in which the temperature at each point on the domain does not change anymore. The temperature distribution in the interior will then be given by the solution to the corresponding Dirichlet problem. The Neumann boundary conditions for Laplace's equation specify not

24320-672: The theory under consideration. Mathematics is essential in the natural sciences , engineering , medicine , finance , computer science , and the social sciences . Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation. Some areas of mathematics, such as statistics and game theory , are developed in close correlation with their applications and are often grouped under applied mathematics . Other areas are developed independently from any application (and are therefore called pure mathematics ) but often later find practical applications. Historically,

24480-487: The title of his main treatise . Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas . Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra ), and polynomial equations in

24640-409: The transient conduction phase of heat transfer is over. New external conditions also cause this process: for example, the copper bar in the example steady-state conduction experiences transient conduction as soon as one end is subjected to a different temperature from the other. Over time, the field of temperatures inside the bar reaches a new steady-state, in which a constant temperature gradient along

24800-508: The two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in

24960-432: The variation can be small over a significant range of temperatures for some common materials. In anisotropic materials, the thermal conductivity typically varies with orientation; in this case k {\displaystyle k} is represented by a second-order tensor . In non-uniform materials, k {\displaystyle k} varies with spatial location. For many simple applications, Fourier's law

25120-462: Was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements . The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane ( plane geometry ) and the three-dimensional Euclidean space . Euclidean geometry was developed without change of methods or scope until

25280-414: Was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz . It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis"

25440-437: Was not specifically studied by mathematicians. Before Cantor 's study of infinite sets , mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration . Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument . This led to

25600-571: Was split into two new subfields: synthetic geometry , which uses purely geometrical methods, and analytic geometry , which uses coordinates systemically. Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions , the study of which led to differential geometry . They can also be defined as implicit equations , often polynomial equations (which spawned algebraic geometry ). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions. In

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