In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions Pε, where ε is a small number, or vector of small quantities. The infinitesimal conditions are therefore the result of applying the approach of differential calculus to solving a problem with constraints. One can think of a structure that is not completely rigid, and that deforms slightly to accommodate forces applied from outside; this explains the name. ...more on Wikipedia about "Deformation theory"
In abstract algebra, a derivation on an algebra A over a ring or a field k is a linear map ...more on Wikipedia about "Derivation (abstract algebra)"
The machinery of differential Galois theory allows one to determine when an elementary function does or does not have an antiderivative which can be expressed as an elementary function. Differential Galois theory is a theory based on the model of Galois theory. Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential fields, i.e. fields which are equipped with a derivation, D. Much of the theory of differential Galois theory is parallel to algebraic Galois theory. One difference between the two constructions is that the Galois groups in differential Galois theory tend to be matrix Lie groups, as compared with the finite groups often encountered in algebraic Galois theory. ...more on Wikipedia about "Differential Galois theory"
In abstract algebra, the dual numbers are a particular two- dimensional commutative unital associative algebra over the real numbers, arising from the reals by adjoining one new element ε with the property ε2 = 0 (ε is nilpotent). Every dual number has the form z = a + bε with a and b uniquely determined real numbers. The plane of all dual numbers is an "alternative complex plane" that complements the ordinary complex number plane C and the plane of split-complex numbers. The "unit circle" of dual numbers consists of those with a = 1 or -1 since these satisfy z z * = 1 where z * = a - bε. However, note that exp(bε) = 1 + bε, so the exponential function applied to the ε-axis covers only half the "circle". ...more on Wikipedia about "Dual numbers"
In differential algebra, an elementary function is a function built from a finite number of exponentials, logarithms, constants, one variable, and roots of equations through composition and combinations using the four elementary operations (+ − × ÷). The trigonometric functions and their inverses are assumed to be included in the elementary functions by using complex variables ...more on Wikipedia about "Elementary function (differential algebra)"
In mathematics, the Kähler differentials are a universal construction Ω1S/R associated to a ring homomorphism of commutative rings, ...more on Wikipedia about "Kähler differential"
In mathematics, the Pincherle derivative of a linear operator T on the space of polynomials in x is another linear operator T′ defined by ...more on Wikipedia about "Pincherle derivative"
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The Risch algorithm is an algorithm for the calculus operation of indefinite integration (i.e. finding antiderivatives). The algorithm transforms the problem of integration into a problem in algebra. It is based on the form of the function being integrated and on methods for integrating rational functions, radicals, logarithms, and exponential functions. Robert Risch, who developed the algorithm in 1968, called it a decision procedure, because it is a method for deciding if a function has a simple-looking function as an indefinite integral; and also, if it does, determining it. The Risch-Norman algorithm, a faster but less powerful technique, was developed in 1976. ...more on Wikipedia about "Risch algorithm"
Symbolic integration is the application of computer software to solving problems in mathematics of find the integral of an expression, but finding an expression rather than a value. ...more on Wikipedia about "Symbolic integration"
In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space, at a point P on an algebraic variety V (and more generally). It does not use differential calculus, being based directly on abstract algebra, and in the most concrete cases just the theory of a system of linear equations. ...more on Wikipedia about "Zariski tangent space"
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