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In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus , which originally referred to the " infinity - eth " item in a sequence .
Originally called infinitesimal calculus or "the calculus of infinitesimals", it has two major branches, differential calculus and integral calculus. The former concerns instantaneous rates of change , and the slopes of curves , while the latter concerns accumulation of quantities, and areas under or between curves.
In 1655, John Wallis first used the notation for such a number in his De sectionibus conicis, [19] and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of . [20] But in Arithmetica infinitorum (1656), [21] he indicates infinite series, infinite products and infinite continued fractions by ...
The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers.
The matrices in the Lie algebra are not themselves rotations; the skew-symmetric matrices are derivatives, proportional differences of rotations. An actual "differential rotation", or infinitesimal rotation matrix has the form +, where dθ is vanishingly small and A ∈ so(n), for instance with A = L x,
In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally smaller) than any relevant dimension of the body; so that its geometry and the constitutive properties of the material (such as density and stiffness ...
An infinitesimal rotation matrix or differential rotation matrix is a matrix representing an infinitely small rotation.. While a rotation matrix is an orthogonal matrix = representing an element of () (the special orthogonal group), the differential of a rotation is a skew-symmetric matrix = in the tangent space (the special orthogonal Lie algebra), which is not itself a rotation matrix.
However, in the 1960s Abraham Robinson showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis. [10] Robinson developed his theory nonconstructively , using model theory ; however it is possible to proceed using only algebra and topology , and proving the transfer ...