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The function () is defined on the interval [,].For a given , the difference () takes the maximum at ′.Thus, the Legendre transformation of () is () = ′ (′).. In mathematics, the Legendre transformation (or Legendre transform), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface problem, [1] is an involutive transformation on real-valued functions that are ...
Transformation (function) A composition of four mappings coded in SVG, which transforms a rectangular repetitive pattern. into a rhombic pattern. The four transformations are linear. In mathematics, a transformation or self-map[1] is a function f, usually with some geometrical underpinning, that maps a set X to itself, i.e. f: X → X. [2][3][4 ...
In physics, chemistry, and other related fields like biology, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of matter: solid, liquid, and gas, and in rare cases, plasma.
Eigenvalues and eigenvectors. In linear algebra, an eigenvector (/ ˈaɪɡən -/ EYE-gən-) or characteristic vector is a vector that has its direction unchanged by a given linear transformation. More precisely, an eigenvector, , of a linear transformation, , is scaled by a constant factor, , when the linear transformation is applied to it: .
Phase transitions (phase changes) that help describe polymorphism include polymorphic transitions as well as melting and vaporization transitions. According to IUPAC, a polymorphic transition is "A reversible transition of a solid crystalline phase at a certain temperature and pressure (the inversion point) to another phase of the same chemical composition with a different crystal structure."
The exponential functions arise by definition as those limits, due to Euler, and can be understood physically and mathematically as follows. A net translation can be composed of many small translations, so to obtain the translation operator for a finite increment, replace Δ r by Δ r / N and Δ t by Δ t / N , where N is a positive non-zero ...
The phase diagram shows, in pressure–temperature space, the lines of equilibrium or phase boundaries between the three phases of solid, liquid, and gas. The curves on the phase diagram show the points where the free energy (and other derived properties) becomes non-analytic: their derivatives with respect to the coordinates (temperature and ...
The representing function g S is the impulse response of the transformation S. A more precise version of the theorem quoted above requires specifying the class of functions on which the convolution is defined, and also requires assuming in addition that S must be a continuous linear operator with respect to the appropriate topology.