Misplaced Pages

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70-410: (Redirected from Le Gendre ) Legendre , LeGendre or Le Gendre is a French surname. It may refer to: Adrien-Marie Legendre (1752–1833), French mathematician Associated Legendre polynomials Legendre's equation Legendre polynomials Legendre symbol Legendre transformation Legendre (crater) , a lunar impact crater located near

140-1017: A convex function ; then the Legendre transform of f {\displaystyle f} is the function f ∗ : I ∗ → R {\displaystyle f^{*}:I^{*}\to \mathbb {R} } defined by f ∗ ( x ∗ ) = sup x ∈ I ( x ∗ x − f ( x ) ) ,         I ∗ = { x ∗ ∈ R : f ∗ ( x ∗ ) < ∞ }   {\displaystyle f^{*}(x^{*})=\sup _{x\in I}(x^{*}x-f(x)),\ \ \ \ I^{*}=\left\{x^{*}\in \mathbb {R} :f^{*}(x^{*})<\infty \right\}~} where sup {\textstyle \sup } denotes

210-638: A Quebec politician Robert LeGendre (1898–1931), American athlete Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Legendre . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Legendre&oldid=1016000117 " Categories : Disambiguation pages Disambiguation pages with surname-holder lists Surnames of French origin Hidden categories: Short description

280-567: A book along with contemporary mathematicians such as Lagrange. The only known portrait of Legendre, rediscovered in 2008, is found in the 1820 book Album de 73 portraits-charge aquarellés des membres de I'Institut , a book of caricatures of seventy-three members of the Institut de France in Paris by the French artist Julien-Léopold Boilly as shown below: Legendre transformation In physical problems,

350-1182: A convex continuous function that is not everywhere differentiable, consider f ( x ) = | x | {\displaystyle f(x)=|x|} . This gives f ∗ ( x ∗ ) = sup x ( x x ∗ − | x | ) = max ( sup x ≥ 0 x ( x ∗ − 1 ) , sup x ≤ 0 x ( x ∗ + 1 ) ) , {\displaystyle f^{*}(x^{*})=\sup _{x}(xx^{*}-|x|)=\max \left(\sup _{x\geq 0}x(x^{*}-1),\,\sup _{x\leq 0}x(x^{*}+1)\right),} and thus f ∗ ( x ∗ ) = 0 {\displaystyle f^{*}(x^{*})=0} on its domain I ∗ = [ − 1 , 1 ] {\displaystyle I^{*}=[-1,1]} . Let f ( x ) = ⟨ x , A x ⟩ + c {\displaystyle f(x)=\langle x,Ax\rangle +c} be defined on X = R , where A

420-445: A convex function on the real line is differentiable and x ¯ {\displaystyle {\overline {x}}} is a critical point of the function of x ↦ p ⋅ x − f ( x ) {\displaystyle x\mapsto p\cdot x-f(x)} , then the supremum is achieved at x ¯ {\textstyle {\overline {x}}} (by convexity, see

490-426: A differentiable manifold, and d f , d x i , d p i {\displaystyle \mathrm {d} f,\mathrm {d} x_{i},\mathrm {d} p_{i}} their differentials (which are treated as cotangent vector field in the context of differentiable manifold). This definition is equivalent to the modern mathematicians' definition as long as f {\displaystyle f}

560-413: A function of two independent variables x and y , with the differential d f = ∂ f ∂ x d x + ∂ f ∂ y d y = p d x + v d y . {\displaystyle df={\frac {\partial f}{\partial x}}\,dx+{\frac {\partial f}{\partial y}}\,dy=p\,dx+v\,dy.} Assume that

630-403: A non-standard requirement is used, amounting to an alternative definition of f * with a minus sign , f ( x ) − f ∗ ( p ) = x p . {\displaystyle f(x)-f^{*}(p)=xp.} In analytical mechanics and thermodynamics, Legendre transformation is usually defined as follows: suppose f {\displaystyle f}

700-628: A set of tangent lines specified by their slope and intercept values. For a differentiable convex function f {\displaystyle f} on the real line with the first derivative f ′ {\displaystyle f'} and its inverse ( f ′ ) − 1 {\displaystyle (f')^{-1}} , the Legendre transform of f {\displaystyle f} , f ∗ {\displaystyle f^{*}} , can be specified, up to an additive constant, by

770-605: A spy during World War II Jacques Legendre (disambiguation) , several people Kevin Le Gendre , British journalist and broadcaster Louis Legendre (1752–1797), French politician of the Revolution period Louis Legendre (oceanographer) , Canadian oceanographer Pierre Legendre (historian) (born 1930), French historian of law and psychoanalyst Pierre Legendre (ecologist) , Canadian numerical ecologist Richard Legendre (born 1953), former professional tennis player and

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840-861: A variable x {\displaystyle x} is intentionally used as the argument of the function f ∗ ∗ {\displaystyle f^{**}} to show the involution property of the Legendre transform as f ∗ ∗ = f {\displaystyle f^{**}=f} . we compute 0 = d d x ∗ ( x x ∗ − x ∗ ( ln ⁡ ( x ∗ ) − 1 ) ) = x − ln ⁡ ( x ∗ ) {\displaystyle {\begin{aligned}0&={\frac {d}{dx^{*}}}{\big (}xx^{*}-x^{*}(\ln(x^{*})-1){\big )}=x-\ln(x^{*})\end{aligned}}} thus

910-1008: Is a fixed constant. For x * fixed, the function of x , x * x − f ( x ) = x * x − cx has the first derivative x * − 2 cx and second derivative −2 c ; there is one stationary point at x = x */2 c , which is always a maximum. Thus, I * = R and f ∗ ( x ∗ ) = x ∗ 2 4 c   . {\displaystyle f^{*}(x^{*})={\frac {{x^{*}}^{2}}{4c}}~.} The first derivatives of f , 2 cx , and of f * , x */(2 c ) , are inverse functions to each other. Clearly, furthermore, f ∗ ∗ ( x ) = 1 4 ( 1 / 4 c ) x 2 = c x 2   , {\displaystyle f^{**}(x)={\frac {1}{4(1/4c)}}x^{2}=cx^{2}~,} namely f ** = f . Let f ( x ) = x for x ∈ ( I = [2, 3]) . For x * fixed, x * x − f ( x )

980-522: Is a function of n {\displaystyle n} variables x 1 , x 2 , ⋯ , x n {\displaystyle x_{1},x_{2},\cdots ,x_{n}} , then we can perform the Legendre transformation on each one or several variables: we have where p i = ∂ f ∂ x i . {\displaystyle p_{i}={\frac {\partial f}{\partial x_{i}}}.} Then if we want to perform

1050-912: Is a function of x {\displaystyle x} ; then we have Performing the Legendre transformation on this function means that we take p = d f d x {\displaystyle p={\frac {\mathrm {d} f}{\mathrm {d} x}}} as the independent variable, so that the above expression can be written as and according to Leibniz's rule d ( u v ) = u d v + v d u , {\displaystyle \mathrm {d} (uv)=u\mathrm {d} v+v\mathrm {d} u,} we then have and taking f ∗ = x p − f , {\displaystyle f^{*}=xp-f,} we have d f ∗ = x d p , {\displaystyle \mathrm {d} f^{*}=x\mathrm {d} p,} which means When f {\displaystyle f}

1120-439: Is a point in x maximizing or making p x − f ( x , y ) {\displaystyle px-f(x,y)} bounded for given p and y ). Since the new independent variable of the transform with respect to f is p , the differentials dx and dy in df devolve to dp and dy in the differential of the transform, i.e., we build another function with its differential expressed in terms of

1190-427: Is a real, positive definite matrix. Then f is convex, and ⟨ p , x ⟩ − f ( x ) = ⟨ p , x ⟩ − ⟨ x , A x ⟩ − c , {\displaystyle \langle p,x\rangle -f(x)=\langle p,x\rangle -\langle x,Ax\rangle -c,} has gradient p − 2 Ax and Hessian −2 A , which

1260-448: Is always bounded as a function of x *∈{ c } , hence I ** = R . Then, for all x one has sup x ∗ ∈ { c } ( x x ∗ − f ∗ ( x ∗ ) ) = x c , {\displaystyle \sup _{x^{*}\in \{c\}}(xx^{*}-f^{*}(x^{*}))=xc,} and hence f **( x ) = cx = f ( x ) . As an example of

1330-966: Is an operator of differentiation, ⋅ {\displaystyle \cdot } represents an argument or input to the associated function, ( ϕ ) − 1 ( ⋅ ) {\displaystyle (\phi )^{-1}(\cdot )} is an inverse function such that ( ϕ ) − 1 ( ϕ ( x ) ) = x {\displaystyle (\phi )^{-1}(\phi (x))=x} , or equivalently, as f ′ ( f ∗ ′ ( x ∗ ) ) = x ∗ {\displaystyle f'(f^{*\prime }(x^{*}))=x^{*}} and f ∗ ′ ( f ′ ( x ) ) = x {\displaystyle f^{*\prime }(f'(x))=x} in Lagrange's notation . The generalization of

1400-449: Is continuous on I compact , hence it always takes a finite maximum on it; it follows that the domain of the Legendre transform of f {\displaystyle f} is I * = R . The stationary point at x = x */2 (found by setting that the first derivative of x * x − f ( x ) with respect to x {\displaystyle x} equal to zero) is in the domain [2, 3] if and only if 4 ≤ x * ≤ 6 . Otherwise

1470-476: Is convex, for every x (strict convexity is not required for the Legendre transformation to be well defined). Clearly x * x − f ( x ) = ( x * − c ) x is never bounded from above as a function of x , unless x * − c = 0 . Hence f * is defined on I * = { c } and f *( c ) = 0 . ( The definition of the Legendre transform requires the existence of the supremum , that requires upper bounds.) One may check involutivity: of course, x * x − f *( x *)

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1540-665: Is defined by f ∗ ( x ∗ ) = sup x ∈ X ( ⟨ x ∗ , x ⟩ − f ( x ) ) , x ∗ ∈ X ∗   , {\displaystyle f^{*}(x^{*})=\sup _{x\in X}(\langle x^{*},x\rangle -f(x)),\quad x^{*}\in X^{*}~,} where ⟨ x ∗ , x ⟩ {\displaystyle \langle x^{*},x\rangle } denotes

1610-413: Is different from Wikidata All article disambiguation pages All disambiguation pages Adrien-Marie Legendre Adrien-Marie Legendre ( / l ə ˈ ʒ ɑː n d ər , - ˈ ʒ ɑː n d / ; French: [adʁiɛ̃ maʁi ləʒɑ̃dʁ] ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as

1680-586: Is differentiable and convex for the variables x 1 , x 2 , ⋯ , x n . {\displaystyle x_{1},x_{2},\cdots ,x_{n}.} As shown above , for a convex function f ( x ) {\displaystyle f(x)} , with x = x ¯ {\displaystyle x={\bar {x}}} maximizing or making p x − f ( x ) {\displaystyle px-f(x)} bounded at each p {\displaystyle p} to define

1750-769: Is invertible and let the inverse be g = ( f ′ ) − 1 {\displaystyle g=(f')^{-1}} . Then for each p {\textstyle p} , the point g ( p ) {\displaystyle g(p)} is the unique critical point x ¯ {\textstyle {\overline {x}}} of the function x ↦ p x − f ( x ) {\displaystyle x\mapsto px-f(x)} (i.e., x ¯ = g ( p ) {\displaystyle {\overline {x}}=g(p)} ) because f ′ ( g ( p ) ) = p {\displaystyle f'(g(p))=p} and

1820-818: Is negative everywhere, so the maximal value is achieved at x = ln ⁡ ( x ∗ ) {\displaystyle x=\ln(x^{*})} . Thus, the Legendre transform is f ∗ ( x ∗ ) = x ∗ ln ⁡ ( x ∗ ) − e ln ⁡ ( x ∗ ) = x ∗ ( ln ⁡ ( x ∗ ) − 1 ) {\displaystyle f^{*}(x^{*})=x^{*}\ln(x^{*})-e^{\ln(x^{*})}=x^{*}(\ln(x^{*})-1)} and has domain I ∗ = ( 0 , ∞ ) . {\displaystyle I^{*}=(0,\infty ).} This illustrates that

1890-466: Is negative; hence the stationary point x = A p /2 is a maximum. We have X * = R , and f ∗ ( p ) = 1 4 ⟨ p , A − 1 p ⟩ − c . {\displaystyle f^{*}(p)={\frac {1}{4}}\langle p,A^{-1}p\rangle -c.} The Legendre transform is linked to integration by parts , p dx = d ( px ) − x dp . Let f ( x , y ) be

1960-927: Is obtained at x = 2 {\displaystyle x=2} while for x ∗ > 6 {\displaystyle x^{*}>6} it becomes the maximum at x = 3 {\displaystyle x=3} . Thus, it follows that f ∗ ( x ∗ ) = { 2 x ∗ − 4 , x ∗ < 4 x ∗ 2 4 , 4 ≤ x ∗ ≤ 6 , 3 x ∗ − 9 , x ∗ > 6. {\displaystyle f^{*}(x^{*})={\begin{cases}2x^{*}-4,&x^{*}<4\\{\frac {{x^{*}}^{2}}{4}},&4\leq x^{*}\leq 6,\\3x^{*}-9,&x^{*}>6.\end{cases}}} The function f ( x ) = cx

2030-477: Is one of the 72 names inscribed on the Eiffel Tower . Abel 's work on elliptic functions was built on Legendre's, and some of Gauss 's work in statistics and number theory completed that of Legendre. He developed, and first communicated to his contemporaries before Gauss, the least squares method which has broad application in linear regression , signal processing , statistics, and curve fitting ; this

2100-600: Is straightforward: f ∗ : X ∗ → R {\displaystyle f^{*}:X^{*}\to \mathbb {R} } has domain X ∗ = { x ∗ ∈ R n : sup x ∈ X ( ⟨ x ∗ , x ⟩ − f ( x ) ) < ∞ } {\displaystyle X^{*}=\left\{x^{*}\in \mathbb {R} ^{n}:\sup _{x\in X}(\langle x^{*},x\rangle -f(x))<\infty \right\}} and

2170-533: Is such that x ∗ x − f ( x ) {\displaystyle x^{*}x-f(x)} has a bounded value throughout I {\textstyle I} (e.g., when f ( x ) {\displaystyle f(x)} is a linear function). The function f ∗ {\displaystyle f^{*}} is called the convex conjugate function of f {\displaystyle f} . For historical reasons (rooted in analytic mechanics),

Legendre - Misplaced Pages Continue

2240-583: The Legendre polynomials and Legendre transformation are named after him. He is also known for his contributions to the method of least squares , and was the first to officially publish on it, though Carl Friedrich Gauss had discovered it before him. Adrien-Marie Legendre was born in Paris on 18 September 1752 to a wealthy family. He received his education at the Collège Mazarin in Paris, and defended his thesis in physics and mathematics in 1770. He taught at

2310-639: The domains of a function and its Legendre transform can be different. To find the Legendre transformation of the Legendre transformation of f {\displaystyle f} , f ∗ ∗ ( x ) = sup x ∗ ∈ R ( x x ∗ − x ∗ ( ln ⁡ ( x ∗ ) − 1 ) ) , x ∈ I , {\displaystyle f^{**}(x)=\sup _{x^{*}\in \mathbb {R} }(xx^{*}-x^{*}(\ln(x^{*})-1)),\quad x\in I,} where

2380-461: The dot product of x ∗ {\displaystyle x^{*}} and x {\displaystyle x} . The Legendre transformation is an application of the duality relationship between points and lines. The functional relationship specified by f {\displaystyle f} can be represented equally well as a set of ( x , y ) {\displaystyle (x,y)} points, or as

2450-792: The exponential function f ( x ) = e x , {\displaystyle f(x)=e^{x},} which has the domain I = R {\displaystyle I=\mathbb {R} } . From the definition, the Legendre transform is f ∗ ( x ∗ ) = sup x ∈ R ( x ∗ x − e x ) , x ∗ ∈ I ∗ {\displaystyle f^{*}(x^{*})=\sup _{x\in \mathbb {R} }(x^{*}x-e^{x}),\quad x^{*}\in I^{*}} where I ∗ {\displaystyle I^{*}} remains to be determined. To evaluate

2520-432: The internal energy . He is also the namesake of the Legendre polynomials , solutions to Legendre's differential equation, which occur frequently in physics and engineering applications, such as electrostatics . Legendre is best known as the author of Éléments de géométrie , which was published in 1794 and was the leading elementary text on the topic for around 100 years. This text greatly rearranged and simplified many of

2590-442: The quadratic reciprocity law, subsequently proved by Gauss; in connection to this, the Legendre symbol is named after him. He also did pioneering work on the distribution of primes , and on the application of analysis to number theory. His 1798 conjecture of the prime number theorem was rigorously proved by Hadamard and de la Vallée-Poussin in 1896. Legendre did an impressive amount of work on elliptic functions , including

2660-446: The supremum over I {\displaystyle I} , e.g., x {\textstyle x} in I {\textstyle I} is chosen such that x ∗ x − f ( x ) {\textstyle x^{*}x-f(x)} is maximized at each x ∗ {\textstyle x^{*}} , or x ∗ {\textstyle x^{*}}

2730-588: The supremum , compute the derivative of x ∗ x − e x {\displaystyle x^{*}x-e^{x}} with respect to x {\displaystyle x} and set equal to zero: d d x ( x ∗ x − e x ) = x ∗ − e x = 0. {\displaystyle {\frac {d}{dx}}(x^{*}x-e^{x})=x^{*}-e^{x}=0.} The second derivative − e x {\displaystyle -e^{x}}

2800-541: The École Militaire in Paris from 1775 to 1780 and at the École Normale from 1795. At the same time, he was associated with the Bureau des Longitudes . In 1782, the Berlin Academy awarded Legendre a prize for his treatise on projectiles in resistant media. This treatise also brought him to the attention of Lagrange . The Académie des sciences made Legendre an adjoint member in 1783 and an associate in 1785. In 1789, he

2870-782: The Hamiltonian: H ( q 1 , ⋯ , q n , p 1 , ⋯ , p n ) = ∑ i = 1 n p i q ˙ i − L ( q 1 , ⋯ , q n , q ˙ 1 ⋯ , q ˙ n ) . {\displaystyle H(q_{1},\cdots ,q_{n},p_{1},\cdots ,p_{n})=\sum _{i=1}^{n}p_{i}{\dot {q}}_{i}-L(q_{1},\cdots ,q_{n},{\dot {q}}_{1}\cdots ,{\dot {q}}_{n}).} In thermodynamics, people perform this transformation on variables according to

Legendre - Misplaced Pages Continue

2940-525: The Institut National, and Legendre became a member of the Geometry section. From 1799 to 1812, Legendre served as mathematics examiner for graduating artillery students at the École Militaire and from 1799 to 1815 he served as permanent mathematics examiner for the École Polytechnique . In 1824, Legendre's pension from the École Militaire was stopped because he refused to vote for the government candidate at

3010-466: The Institut National. In 1831, he was made an officer of the Légion d'Honneur . Legendre died in Paris on 9 January 1833, after a long and painful illness, and Legendre's widow carefully preserved his belongings to memorialize him. Upon her death in 1856, she was buried next to her husband in the village of Auteuil , where the couple had lived, and left their last country house to the village. Legendre's name

3080-419: The Legendre transform f ∗ ( p ) = p x ¯ − f ( x ¯ ) {\displaystyle f^{*}(p)=p{\bar {x}}-f({\bar {x}})} and with g ≡ ( f ′ ) − 1 {\displaystyle g\equiv (f')^{-1}} , the following identities hold. Consider

3150-533: The Legendre transform is used to convert functions of one quantity (such as position, pressure, or temperature) into functions of the conjugate quantity (momentum, volume, and entropy, respectively). In this way, it is commonly used in classical mechanics to derive the Hamiltonian formalism out of the Lagrangian formalism (or vice versa) and in thermodynamics to derive the thermodynamic potentials , as well as in

3220-624: The Legendre transformation is the one originally introduced by Legendre in his work in 1787, and is still applied by physicists nowadays. Indeed, this definition can be mathematically rigorous if we treat all the variables and functions defined above: for example, f , x 1 , ⋯ , x n , p 1 , ⋯ , p n , {\displaystyle f,x_{1},\cdots ,x_{n},p_{1},\cdots ,p_{n},} as differentiable functions defined on an open set of R n {\displaystyle \mathbb {R} ^{n}} or on

3290-963: The Legendre transformation on, e.g. x 1 {\displaystyle x_{1}} , then we take p 1 {\displaystyle p_{1}} together with x 2 , ⋯ , x n {\displaystyle x_{2},\cdots ,x_{n}} as independent variables, and with Leibniz's rule we have So for the function φ ( p 1 , x 2 , ⋯ , x n ) = f ( x 1 , x 2 , ⋯ , x n ) − x 1 p 1 , {\displaystyle \varphi (p_{1},x_{2},\cdots ,x_{n})=f(x_{1},x_{2},\cdots ,x_{n})-x_{1}p_{1},} we have We can also do this transformation for variables x 2 , ⋯ , x n {\displaystyle x_{2},\cdots ,x_{n}} . If we do it to all

3360-477: The Legendre transformation to affine spaces and non-convex functions is known as the convex conjugate (also called the Legendre–Fenchel transformation), which can be used to construct a function's convex hull . Let I ⊂ R {\displaystyle I\subset \mathbb {R} } be an interval , and f : I → R {\displaystyle f:I\to \mathbb {R} }

3430-480: The classification of elliptic integrals , but it took Abel 's stroke of genius to study the inverses of Jacobi 's functions and solve the problem completely. He is known for the Legendre transformation , which is used to go from the Lagrangian to the Hamiltonian formulation of classical mechanics . In thermodynamics it is also used to obtain the enthalpy and the Helmholtz and Gibbs (free) energies from

3500-526: The condition that the functions' first derivatives are inverse functions of each other, i.e., f ′ = ( ( f ∗ ) ′ ) − 1 {\displaystyle f'=((f^{*})')^{-1}} and ( f ∗ ) ′ = ( f ′ ) − 1 {\displaystyle (f^{*})'=(f')^{-1}} . To see this, first note that if f {\displaystyle f} as

3570-824: The conjugate variable is often denoted p {\displaystyle p} , instead of x ∗ {\displaystyle x^{*}} . If the convex function f {\displaystyle f} is defined on the whole line and is everywhere differentiable , then f ∗ ( p ) = sup x ∈ I ( p x − f ( x ) ) = ( p x − f ( x ) ) | x = ( f ′ ) − 1 ( p ) {\displaystyle f^{*}(p)=\sup _{x\in I}(px-f(x))=\left(px-f(x)\right)|_{x=(f')^{-1}(p)}} can be interpreted as

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3640-897: The domain of f ∗ ∗ {\displaystyle f^{**}} as I ∗ = ( 0 , ∞ ) . {\displaystyle I^{*}=(0,\infty ).} As a result, f ∗ ∗ {\displaystyle f^{**}} is found as f ∗ ∗ ( x ) = x e x − e x ( ln ⁡ ( e x ) − 1 ) = e x , {\displaystyle {\begin{aligned}f^{**}(x)&=xe^{x}-e^{x}(\ln(e^{x})-1)=e^{x},\end{aligned}}} thereby confirming that f = f ∗ ∗ , {\displaystyle f=f^{**},} as expected. Let f ( x ) = cx defined on R , where c > 0

3710-633: The eastern limb of the Moon 26950 Legendre , a main-belt asteroid discovered on May 11, 1997 Anne Legendre Armstrong (1927–2008), United States diplomat and politician Charles Le Gendre (1830–1899), French-born American general and diplomat François Legendre (1763–1853), surveyor, seigneur and political figure in Lower Canada Géraldine Legendre (born 1953), French-American cognitive scientist and linguist Gertrude Sanford Legendre (1902–2000), American socialite who served as

3780-459: The first figure in this Misplaced Pages page). Therefore, the Legendre transform of f {\displaystyle f} is f ∗ ( p ) = p ⋅ x ¯ − f ( x ¯ ) {\displaystyle f^{*}(p)=p\cdot {\overline {x}}-f({\overline {x}})} . Then, suppose that the first derivative f ′ {\displaystyle f'}

3850-454: The function f is convex in x for all y , so that one may perform the Legendre transform on f in x , with p the variable conjugate to x (for information, there is a relation ∂ f ∂ x | x ¯ = p {\displaystyle {\frac {\partial f}{\partial x}}|_{\bar {x}}=p} where x ¯ {\displaystyle {\bar {x}}}

3920-1751: The function's first derivative with respect to x {\displaystyle x} at g ( p ) {\displaystyle g(p)} is p − f ′ ( g ( p ) ) = 0 {\displaystyle p-f'(g(p))=0} . Hence we have f ∗ ( p ) = p ⋅ g ( p ) − f ( g ( p ) ) {\displaystyle f^{*}(p)=p\cdot g(p)-f(g(p))} for each p {\textstyle p} . By differentiating with respect to p {\textstyle p} , we find ( f ∗ ) ′ ( p ) = g ( p ) + p ⋅ g ′ ( p ) − f ′ ( g ( p ) ) ⋅ g ′ ( p ) . {\displaystyle (f^{*})'(p)=g(p)+p\cdot g'(p)-f'(g(p))\cdot g'(p).} Since f ′ ( g ( p ) ) = p {\displaystyle f'(g(p))=p} this simplifies to ( f ∗ ) ′ ( p ) = g ( p ) = ( f ′ ) − 1 ( p ) {\displaystyle (f^{*})'(p)=g(p)=(f')^{-1}(p)} . In other words, ( f ∗ ) ′ {\displaystyle (f^{*})'} and f ′ {\displaystyle f'} are inverses to each other . In general, if h ′ = ( f ′ ) − 1 {\displaystyle h'=(f')^{-1}} as

3990-497: The inverse of f ′ , {\displaystyle f',} then h ′ = ( f ∗ ) ′ {\displaystyle h'=(f^{*})'} so integration gives f ∗ = h + c . {\displaystyle f^{*}=h+c.} with a constant c . {\displaystyle c.} In practical terms, given f ( x ) , {\displaystyle f(x),}

4060-507: The maximum is taken either at x = 2 or x = 3 because the second derivative of x * x − f ( x ) with respect to x {\displaystyle x} is negative as − 2 {\displaystyle -2} ; for a part of the domain x ∗ < 4 {\displaystyle x^{*}<4} the maximum that x * x − f ( x ) can take with respect to x ∈ [ 2 , 3 ] {\displaystyle x\in [2,3]}

4130-457: The maximum occurs at x ∗ = e x {\displaystyle x^{*}=e^{x}} because the second derivative d 2 d x ∗ 2 f ∗ ∗ ( x ) = − 1 x ∗ < 0 {\displaystyle {\frac {d^{2}}{{dx^{*}}^{2}}}f^{**}(x)=-{\frac {1}{x^{*}}}<0} over

4200-477: The negative of the y {\displaystyle y} -intercept of the tangent line to the graph of f {\displaystyle f} that has slope p {\displaystyle p} . The generalization to convex functions f : X → R {\displaystyle f:X\to \mathbb {R} } on a convex set X ⊂ R n {\displaystyle X\subset \mathbb {R} ^{n}}

4270-633: The new basis dp and dy . We thus consider the function g ( p , y ) = f − px so that d g = d f − p d x − x d p = − x d p + v d y {\displaystyle dg=df-p\,dx-x\,dp=-x\,dp+v\,dy} x = − ∂ g ∂ p {\displaystyle x=-{\frac {\partial g}{\partial p}}} v = ∂ g ∂ y . {\displaystyle v={\frac {\partial g}{\partial y}}.} The function − g ( p , y )

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4340-463: The parametric plot of x f ′ ( x ) − f ( x ) {\displaystyle xf'(x)-f(x)} versus f ′ ( x ) {\displaystyle f'(x)} amounts to the graph of f ∗ ( p ) {\displaystyle f^{*}(p)} versus p . {\displaystyle p.} In some cases (e.g. thermodynamic potentials, below),

4410-459: The planet Uranus . Legendre lost his private fortune in 1793 during the French Revolution . That year, he also married Marguerite-Claudine Couhin, who helped him put his affairs in order. In 1795, Legendre became one of six members of the mathematics section of the reconstituted Académie des Sciences, renamed the Institut National des Sciences et des Arts. Later, in 1803, Napoleon reorganized

4480-413: The propositions from Euclid's Elements to create a more effective textbook. For two centuries, until the recent discovery of the error in 2005, books, paintings and articles have incorrectly shown a profile portrait of the obscure French politician Louis Legendre (1752–1797) as a portrait of the mathematician. The error arose from the fact that the sketch was labelled simply "Legendre" and appeared in

4550-793: The solution of differential equations of several variables. For sufficiently smooth functions on the real line, the Legendre transform f ∗ {\displaystyle f^{*}} of a function f {\displaystyle f} can be specified, up to an additive constant, by the condition that the functions' first derivatives are inverse functions of each other. This can be expressed in Euler's derivative notation as D f ( ⋅ ) = ( D f ∗ ) − 1 ( ⋅ )   , {\displaystyle Df(\cdot )=\left(Df^{*}\right)^{-1}(\cdot )~,} where D {\displaystyle D}

4620-418: The symbol Γ and normalizing it to Γ(n+1) = n!. Further results on the beta and gamma functions along with their applications to mechanics – such as the rotation of the earth, and the attraction of ellipsoids – appeared in the second volume. In 1830, he gave a proof of Fermat's Last Theorem for exponent n  = 5, which was also proven by Lejeune Dirichlet in 1828. In number theory , he conjectured

4690-426: The type of thermodynamic system they want; for example, starting from the cardinal function of state, the internal energy U ( S , V ) {\displaystyle U(S,V)} , we have so we can perform the Legendre transformation on either or both of S , V {\displaystyle S,V} to yield and each of these three expressions has a physical meaning. This definition of

4760-718: The variables, then we have In analytical mechanics, people perform this transformation on variables q ˙ 1 , q ˙ 2 , ⋯ , q ˙ n {\displaystyle {\dot {q}}_{1},{\dot {q}}_{2},\cdots ,{\dot {q}}_{n}} of the Lagrangian L ( q 1 , ⋯ , q n , q ˙ 1 , ⋯ , q ˙ n ) {\displaystyle L(q_{1},\cdots ,q_{n},{\dot {q}}_{1},\cdots ,{\dot {q}}_{n})} to get

4830-654: Was elected a Fellow of the Royal Society . He assisted with the Anglo-French Survey (1784–1790) to calculate the precise distance between the Paris Observatory and the Royal Greenwich Observatory by means of trigonometry . To this end in 1787 he visited Dover and London together with Dominique, comte de Cassini and Pierre Méchain . The three also visited William Herschel , the discoverer of

4900-423: Was published in 1806 as an appendix to his book on the paths of comets. Today, the term "least squares method" is used as a direct translation from the French "méthode des moindres carrés". His major work is Exercices de Calcul Intégral , published in three volumes in 1811, 1817 and 1819. In the first volume he introduced the basic properties of elliptic integrals, beta functions and gamma functions , introducing

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