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In multivariate calculus, a differential or differential form is said to be exact or perfect (exact differential), as contrasted with an inexact differential, if it is equal to the general differential for some differentiable function in an orthogonal coordinate system (hence is a multivariable function whose variables are independent, as they are always expected to be when treated in ...
Given a simply connected and open subset D of and two functions I and J which are continuous on D, an implicit first-order ordinary differential equation of the form (,) + (,) =,is called an exact differential equation if there exists a continuously differentiable function F, called the potential function, [1] [2] so that
In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα = 0), and an exact form is a differential form, α, that is the exterior derivative of another differential form β. Thus, an exact form is in the image of d, and a closed form is in the kernel of ...
The differential was first introduced via an intuitive or heuristic definition by Isaac Newton and furthered by Gottfried Leibniz, who thought of the differential dy as an infinitely small (or infinitesimal) change in the value y of the function, corresponding to an infinitely small change dx in the function's argument x.
Differential forms can be multiplied together using the exterior product, and for any differential k-form α, there is a differential (k + 1)-form dα called the exterior derivative of α. Differential forms, the exterior product and the exterior derivative are independent of a choice of coordinates.
In mathematics, the Poincaré lemma gives a sufficient condition for a closed differential form to be exact (while an exact form is necessarily closed). Precisely, it states that every closed p-form on an open ball in R n is exact for p with 1 ≤ p ≤ n. [1] The lemma was introduced by Henri Poincaré in 1886. [2] [3]
Researchers studying the immune system response in humans have already determined that there are several components of the immune system that will not function without adequate hydration (3).
where is a function : [,), and the initial condition is a given vector. First-order means that only the first derivative of y appears in the equation, and higher derivatives are absent. Without loss of generality to higher-order systems, we restrict ourselves to first-order differential equations, because a higher-order ODE can be converted ...