This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources. Find sources: "Differential of the first kind" – news · newspapers · books · scholar · JSTOR(August 2022)
In mathematics, differential of the first kind is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic varieties), for everywhere-regular differential 1-forms. Given a complex manifold M, a differential of the first kind ω is therefore the same thing as a 1-form that is everywhere holomorphic; on an algebraic variety V that is non-singular it would be a global section of the coherent sheaf Ω1 of Kähler differentials. In either case the definition has its origins in the theory of abelian integrals.
The dimension of the space of differentials of the first kind, by means of this identification, is the Hodge number
h1,0.
The differentials of the first kind, when integrated along paths, give rise to integrals that generalise the elliptic integrals to all curves over the complex numbers. They include for example the hyperelliptic integrals of type
where Q is a square-free polynomial of any given degree > 4. The allowable power k has to be determined by analysis of the possible pole at the point at infinity on the corresponding hyperelliptic curve. When this is done, one finds that the condition is
k ≤ g − 1,
or in other words, k at most 1 for degree of Q 5 or 6, at most 2 for degree 7 or 8, and so on (as g = [(1+ deg Q)/2]).
Quite generally, as this example illustrates, for a compact Riemann surface or algebraic curve, the Hodge number is the genus g. For the case of algebraic surfaces, this is the quantity known classically as the irregularity q. It is also, in general, the dimension of the Albanese variety, which takes the place of the Jacobian variety.
and 26 Related for: Differential of the first kind information
In mathematics, differentialofthefirstkind is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic...
first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation...
In mathematics, an exact differential equation or total differential equation is a certain kindof ordinary differential equation which is widely used...
have broad applications in differential geometry. On complex manifolds, they are fundamental and serve as the basis for much of algebraic geometry, Kähler...
physical quantities, the derivatives represent their rates of change, and thedifferential equation defines a relationship between the two. Such relations...
indefinite integral of a differentialofthefirstkind. Suppose we are given a Riemann surface S {\displaystyle S} and on it a differential 1-form ω {\displaystyle...
geometry and the theory of complex manifolds, a logarithmic differential form is a differential form with poles of a certain kind. The concept was introduced...
therefore the same as an everywhere-regular differential form. Classically, these were called differentialsofthefirstkind. The degree ofthe canonical...
variety Abelian integral, a function related to the indefinite integral of a differentialofthefirstkind Abelian category, in category theory, a preabelian...
two groups referred to as the first and the second kind. A linear Volterra equation ofthefirstkind is f ( t ) = ∫ a t K ( t , s ) x ( s ) d s {\displaystyle...
mathematics, an Abel equation ofthefirstkind, named after Niels Henrik Abel, is any ordinary differential equation that is cubic in the unknown function. In...
Differential equations are prominent in many scientific areas. Nonlinear ones are of particular interest for their commonality in describing real-world...
differentials of the first kind on V {\displaystyle V} , which for surfaces is called the irregularity of a surface. In terms ofdifferential forms, any holomorphic...
functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation d 2 y d x 2 + ( a − 2 q cos ( 2 x ) ) y = 0 , {\displaystyle...
mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology...
calculus, thedifferential represents the principal part ofthe change in a function y = f ( x ) {\displaystyle y=f(x)} with respect to changes in the independent...
particular kindofdifferential equation under consideration. There is a well-developed theory for linear differential operators, due to Lars Gårding, in the context...
A differential amplifier is a type of electronic amplifier that amplifies the difference between two input voltages but suppresses any voltage common...
Differential scanning calorimetry (DSC) is a thermoanalytical technique in which the difference in the amount of heat required to increase the temperature...
partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function. The function...
Differential phase is a kindof linearity distortion which affects the color hue in TV broadcasting. Composite color video signal (CCVS) consists of three...
Global Information
