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Legendre Transformation
In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions of one quantity (such as velocity, 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 the solution of differential equations of several variables. For sufficiently smooth functions on the real line, the Legendre transform f^* of a function 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 Df(\cdot) = \left( D f^* \r ...
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Legendre Transformation
In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions of one quantity (such as velocity, 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 the solution of differential equations of several variables. For sufficiently smooth functions on the real line, the Legendre transform f^* of a function 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 Df(\cdot) = \left( D f^* \r ...
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Interval (mathematics)
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other examples of intervals are the set of numbers such that , the set of all real numbers \R, the set of nonnegative real numbers, the set of positive real numbers, the empty set, and any singleton (set of one element). Real intervals play an important role in the theory of integration, because they are the simplest sets whose "length" (or "measure" or "size") is easy to define. The concept of measure can then be extended to more complicated sets of real numbers, leading to the Borel measure and eventually to the Lebesgue measure. Intervals are central to interval arithmetic, a general numerical computing technique that automatically provides guaranteed enclosures for arbitrary formulas, even in the presence of uncertainties, mathematical ...
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Involution (mathematics)
In mathematics, an involution, involutory function, or self-inverse function is a function that is its own inverse, : for all in the domain of . Equivalently, applying twice produces the original value. General properties Any involution is a bijection. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (x \mapsto -x), reciprocation (x \mapsto 1/x), and complex conjugation (z \mapsto \bar z) in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...; and reciprocal ciphers such as the ROT13 transformation and the Beaufort cipher, Beaufort polyalphabetic cipher. The Function composition, composition of two invol ...
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Chain Rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , then the chain rule is, in Lagrange's notation, :h'(x) = f'(g(x)) g'(x). or, equivalently, :h'=(f\circ g)'=(f'\circ g)\cdot g'. The chain rule may also be expressed in Leibniz's notation. If a variable depends on the variable , which itself depends on the variable (that is, and are dependent variables), then depends on as well, via the intermediate variable . In this case, the chain rule is expressed as :\frac = \frac \cdot \frac, and : \left.\frac\_ = \left.\frac\_ \cdot \left. \frac\_ , for indicating at which points the derivatives have to be evaluated. In integration, the counterpart to the chain rule is the substitution rule. Intuitive explanation Intuitively, the chain rule states that knowing the instantaneous rate of ...
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Product Rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + u \cdot v' or in Leibniz's notation as \frac (u\cdot v) = \frac \cdot v + u \cdot \frac. The rule may be extended or generalized to products of three or more functions, to a rule for higher-order derivatives of a product, and to other contexts. Discovery Discovery of this rule is credited to Gottfried Leibniz, who demonstrated it using differentials. (However, J. M. Child, a translator of Leibniz's papers, argues that it is due to Isaac Barrow.) Here is Leibniz's argument: Let ''u''(''x'') and ''v''(''x'') be two differentiable functions of ''x''. Then the differential of ''uv'' is : \begin d(u\cdot v) & = (u + du)\cdot (v + dv) - u\cdot v \\ & = u\cdot dv + v\cdot du + du\cdot dv. \end Since the term ''du''·''dv'' is "negli ...
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Inverse Functions And Differentiation
In calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function in terms of the derivative of . More precisely, if the inverse of f is denoted as f^, where f^(y) = x if and only if f(x) = y, then the inverse function rule is, in Lagrange's notation, :\left ^\right(a)=\frac. This formula holds in general whenever f is continuous and injective on an interval , with f being differentiable at f^(a)(\in I) and wheref'(f^(a)) \ne 0. The same formula is also equivalent to the expression :\mathcal\left ^\right\frac, where \mathcal denotes the unary derivative operator (on the space of functions) and \circ denotes function composition. Geometrically, a function and inverse function have graphs that are reflections, in the line y=x. This reflection operation turns the gradient of any line into its reciprocal. Assuming that f has an inverse in a neighbourhood of x and that its derivative at that point is non ...
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Critical Point (mathematics)
Critical point is a wide term used in many branches of mathematics. When dealing with functions of a real variable, a critical point is a point in the domain of the function where the function is either not differentiable or the derivative is equal to zero. When dealing with complex variables, a critical point is, similarly, a point in the function's domain where it is either not holomorphic or the derivative is equal to zero. Likewise, for a function of several real variables, a critical point is a value in its domain where the gradient is undefined or is equal to zero. The value of the function at a critical point is a critical value. This sort of definition extends to differentiable maps between and a critical point being, in this case, a point where the rank of the Jacobian matrix is not maximal. It extends further to differentiable maps between differentiable manifolds, as the points where the rank of the Jacobian matrix decreases. In this case, critical points ...
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Duality (projective Geometry)
In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and ( plane) duality is the formalization of this concept. There are two approaches to the subject of duality, one through language () and the other a more functional approach through special mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries that is called a ''duality''. Such a map can be constructed in many ways. The concept of plane duality readily extends to space duality and beyond that to duality in any finite-dimensional projective geometry. Principle of duality A projective plane may be defined axiomatically as an incidence structure, in terms of a set of ''points'', a set of ''lines'', and an incidence relation that determines which points lie on which l ...
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Graph Of A Function
In mathematics, the graph of a function f is the set of ordered pairs (x, y), where f(x) = y. In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in two-dimensional space and thus form a subset of this plane. In the case of functions of two variables, that is functions whose domain consists of pairs (x, y), the graph usually refers to the set of ordered triples (x, y, z) where f(x,y) = z, instead of the pairs ((x, y), z) as in the definition above. This set is a subset of three-dimensional space; for a continuous real-valued function of two real variables, it is a surface. In science, engineering, technology, finance, and other areas, graphs are tools used for many purposes. In the simplest case one variable is plotted as a function of another, typically using rectangular axes; see '' Plot (graphics)'' for details. A graph of a function is a special case of a relation. In the modern foundations of mathematics, and, ...
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Tangent Line
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve at a point if the line passes through the point on the curve and has slope , where ''f'' is the derivative of ''f''. A similar definition applies to space curves and curves in ''n''-dimensional Euclidean space. As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point. The tangent line to a point on a differentiable curve can also be thought of as a '' tangent line approximation'', the graph of the affine function that best approximates the original function at the given point. Similarly, ...
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Y-intercept
In analytic geometry, using the common convention that the horizontal axis represents a variable ''x'' and the vertical axis represents a variable ''y'', a ''y''-intercept or vertical intercept is a point where the graph of a function or relation intersects the ''y''-axis of the coordinate system. As such, these points satisfy ''x'' = 0. Using equations If the curve in question is given as y= f(x), the ''y''-coordinate of the ''y''-intercept is found by calculating f(0). Functions which are undefined at ''x'' = 0 have no ''y''-intercept. If the function is linear and is expressed in slope-intercept form as f(x)=a+bx, the constant term a is the ''y''-coordinate of the ''y''-intercept. Multiple y-intercepts Some 2-dimensional mathematical relationships such as circles, ellipses, and hyperbolas can have more than one ''y''-intercept. Because functions associate ''x'' values to no more than one ''y'' value as part of their definition, they can have at most one ''y''-intercept. ...
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Differentiable Function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If is an interior point in the domain of a function , then is said to be ''differentiable at'' if the derivative f'(x_0) exists. In other words, the graph of has a non-vertical tangent line at the point . is said to be differentiable on if it is differentiable at every point of . is said to be ''continuously differentiable'' if its derivative is also a continuous function over the domain of the function f. Generally speaking, is said to be of class if its first k derivatives f^(x), f^(x), \ldots, f^(x) exist and are continuous over the domain of t ...
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