Dietrich ► diffusion I 545
nutrition and are most effective combined with exercise. Appetite sup¬ pressants may have dangerous side effects. Excessive weight loss may be a sign of anorexia nervosa.
Dietrich Vde-trik,\ English Vde-trikA, Marlene orig. Maria Magdalene Dietrich (b. Dec.
27, 1901, Berlin, Ger.—d. May 6,
1992, Paris, France) German-U.S. film actress and singer. After joining Max Reinhardt’s theatre company in 1922, she appeared in German films and became an international star as the destructive cabaret singer Lola- Lola in Josef von Sternberg’s The Blue Angel (1930). Sternberg took her to Hollywood, where they made many films together, including Morocco (1930), Shanghai Express (1932), and The Scarlet Empress (1934), which established her aura of glamorous sophistication and lan¬ guid sensuality. During World War II she made over 500 appearances before Allied troops. She also starred in films such as Destry Rides Again (1939), A Foreign Affair (1948), Witness for the Prosecution (1957), and Touch of Evil (1958). She toured widely as a nightclub performer into the 1960s, singing trademark songs such as “Falling in Love Again.”
Dietz \'dets\ A Howard (b. Sept. 9, 1896, New York, N.Y., U.S.—d. July 30, 1983, New York City) U.S. songwriter. He studied at Columbia University and later joined an advertising agency, where he designed the trademark roaring lion for Goldwyn Pictures (later MGM). He joined the film studio in 1919 and became director of advertising, a post he held until 1957. From 1923 he wrote lyrics in his spare time. In 1929 he met the composer Arthur Schwartz (1900-84); the duo established their repu¬ tation with The Little Show and went on to write such Broadway shows as Three’s a Crowd (1930), The Band Wagon (1931), and The Gay Life (1961). Dietz wrote about 500 songs; the Dietz-Schwartz collaborations include “Something to Remember You By,” “Dancing in the Dark,” and “You and the Night and the Music.”
Dievs \'de-9fs\ In the pre-Christian Baltic religion, the sky god. Along with the goddess Laima, he determined the course of the world and human destiny. He was pictured as a Baltic king who lived on a farmstead in the heavens, occasionally descending to earth on horseback or in a chariot to serve as the protector of farmers and their crops. His two sons were the morning and evening stars. In modern Baltic, dievs refers to the Chris¬ tian God.
Diez \'dets\, Friedrich Christian (b. March 15, 1794, Giessen, Hesse- Darmstadt—d. May 29, 1876, Bonn, Ger.) German linguist, regarded as the founder of Romance philology. He began his career as a scholar of medieval Provencal poetry and taught literature at the University of Bonn from 1823 to the end of his life. Diez applied the methodology of com¬ parative linguistics pioneered by Jacob Grimm and Franz Bopp to the Romance languages. In his Grammar of the Romance Languages (1836— 44) and Etymological Dictionary of the Romance Languages (1853), he demonstrated the relationship of “Vulgar” or Spoken Latin to Classical Latin and the evolution of Romance languages from Spoken Latin into their modern forms.
difference equation Equation involving differences between succes¬ sive values of a function of a discrete variable (i.e., one defined for a sequence of values that differ by the same amount, usually 1). A function of such a variable is a rule for assigning values in sequence to it. For example,/(x + 1) = x/(x) is a difference equation. Methods developed for solving such equations have much in common with methods for solving linear differential equations, which difference equations are often used to approximate.
differential In calculus, an expression based on the derivative of a func¬ tion, useful for approximating certain values of the function. The differ¬ ential of an independent variable x, written Ax, is an infinitesimal change in its value. The corresponding differential of its dependent variable y is given by Ay = /(x + Ax) - /(x). Because the derivative of the function /(x), f'(x), is equal to the ratio as Ax approaches zero (see limit), for
small values of Ax, Ay =/'(x)Ax. This formula often enables a quick and fairly accurate approximation to be made for what otherwise would be a tedious calculation.
differential calculus Branch of mathematical analysis, devised by Isaac Newton and G.W. Leibniz, and concerned with the problem of find¬ ing the rate of change of a function with respect to the variable on which it depends. Thus it involves calculating derivatives and using them to solve problems involving nonconstant rates of change. Typical applications include finding maximum and minimum values of functions in order to solve practical problems in optimization.
differential equation Mathematical statement that contains one or more derivatives. It states a relationship involving the rates of change of continuously changing quantities modeled by functions. Differential equa¬ tions are very common in physics, engineering, and all fields involving quantitative study of change. They are used whenever a rate of change is known but the process giving rise to it is not. The solution of a differen¬ tial equation is generally a function whose derivatives satisfy the equa¬ tion. Differential equations are classified into several broad categories. The most important are ordinary differential equations (ODEs), in which change depends on a single variable, and partial differential equations (PDEs), in which change depends on several valuables. See also differen¬ tiation.
differential gear In automotive mechanics, a gear arrangement that transmits power from the engine to a pair of driving wheels, dividing the force equally between them but permitting them to follow paths of dif¬ ferent lengths, as when turning a comer or traversing an uneven road. On a straight road the wheels rotate at the same speed; when turning a cor¬ ner the outside wheel has farther to go and would turn faster than the inner wheel if unrestrained. The automobile differential was invented in 1827; originally used on steam-driven vehicles, it was well known when internal-combustion engines finally appeared.
differential geometry Field of mathematics in which methods of cal¬ culus are applied to the local geometry of curves and surfaces (i.e., to a small portion of a surface or curve around a point). A simple example is finding the tangent line on a two-dimensional curve at a given point. Similar operations may be extended to calculate the curvature and length of a curve and to analogous properties of surfaces in any number of dimensions.
differential operator In mathematics, any combination of derivatives applied to a function. It takes the form of a polynomial of derivatives, such as D 2 XX - D 2 xy • D 2 yx , where D 2 is a second derivative and the subscripts indicate partial derivatives. Special differential operators include the gra¬ dient, divergence, curl, and Laplace operator (see Laplace's equation). Dif¬ ferential operators provide a generalized way to look at differentiation as a whole, as well as a framework for discussion of the theory of differen¬ tial EQUATIONS.
differentiation Mathematical process of finding the derivative of a function. Defined abstractly as a process involving limits, in practice it may be done using algebraic manipulations that rely on three basic formulas and four rules of operation. The formulas are: (1) the derivative of x" is nxP ~ 1 , (2) the derivative of sin x is cos x, and (3) the derivative of the exponential function e x is itself. The rules are: (1) ( af+ bg)' = af + bg\ (2) (fg)' =fg' + gf, (3) (fig)' = (gf -fg')/g 2 , and (4) (f(g))'=f\g)g\ where a and b are constants,/and g are functions, and a prime (') indi¬ cates the derivative. The last formula is called the chain rule. The deri¬ vation and exploration of these formulas and rules is the subject of DIFFERENTIAL CALCULUS. See also INTEGRATION.
diffraction Spreading of waves around obstacles. It occurs with water waves, sound, electromagnetic waves (see electromagnetic radiation), and small moving particles such as atoms, neutrons, and electrons, which show wavelike properties. When a beam of light falls on the edge of an object, it is bent slightly by the contact and causes a blur at the edge of the shadow of the object. Waves of long wavelength are diffracted more than those of short wavelength.