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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 ...
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 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 ...
Some authors allow any real , [1] [2] whereas others require that not be 0 or 1. [ 3 ] [ 4 ] The equation was first discussed in a work of 1695 by Jacob Bernoulli , after whom it is named. The earliest solution, however, was offered by Gottfried Leibniz , who published his result in the same year and whose method is the one still used today.
Jan Kochanowski with his dead daughter in a painting by Jan Matejko inspired by the poet's Threnodies. A threnody is a wailing ode, song, hymn or poem of mourning composed or performed as a memorial to a dead person.
The ode begins with a priamel, where the rival distinctions of water and gold are introduced as a foil to the true prize, the celebration of victory in song. [7] Ring-composed, [8] Pindar returns in the final lines to the mutual dependency of victory and poetry, where "song needs deeds to celebrate, and success needs songs to make the areta last". [9]
In the 1730s, Handel wrote new musical scores for both "A Song for St. Cecilia's Day" and Dryden's second ode on the same theme, "Alexander's Feast" (1697). [6] In 1958, American composer Norman Dello Joio once again put the ode to music in his cantata for mixed voices and piano or brass instruments, and called it "To Saint Cecilia". [7]
"Ode-to-Napoleon" hexachord [1] in prime form [2] In music , the "Ode-to-Napoleon" hexachord (also magic hexachord [ 3 ] and hexatonic collection [ 4 ] or hexatonic set class ) [ 5 ] is the hexachord named after its use in the twelve-tone piece Ode to Napoleon Buonaparte Op. 41 (1942) by Arnold Schoenberg (setting a text by Byron ).