The Analysis of Curves

If a point r moves along a curve at arc length s from some fixed point, then t = dr/ds is a unit tangent vector to the curve at r. The normal vector n is perpendicular to the curve at the point and indicates the direction of the rate of change of t, i.e., the tendency of r to bend in the plane containing both r and t, and the binormal vector b is perpendicular to both t and n and indicates the tendency of the curve to twist out of the plane of t and n.

These three vectors are related by the three formulas of the French mathematician Jean Frédéric Frenet, which are fundamental to the study of space curves: dt/ds = κn; dn/ds = -κt + τb; db/ds = -τn, where the constants κ and τ are the curvature and the torsion of the curve, respectively. Of special interest are the curves called evolutes and involutes; the evolute of a curve is another curve whose tangents are the normals to the original curve, and an involute of a curve is a curve whose evolute is the given curve.

The Analysis of Surfaces

In the analysis of surfaces, points on a surface may be described not only with respect to the three-dimensional coordinates of the space in which the surface is considered but also with respect to an intrinsic coordinate system defined in terms of a system of curves on the surface itself. The curves on the surface that locally represent the shortest distances between points on the surface are called geodesics; geodesics on a plane are straight lines. Tangent and normal vectors are also defined for a surface, but the relationships between them are more complex than for a space curve (e.g., a surface has a whole circle of unit vectors tangent to it at a given point).

The results of the theory of surfaces are expressed most easily in the notation of tensors. It is found that the total, or Gaussian, curvature of a surface is a bending invariant, i.e., an intrinsic property of the surface itself, independent of the space in which the surface may be considered. Of particular importance are surfaces of constant curvature; planes, cylinders, cones, and other so-called developable surfaces have zero curvature, while the elliptic and hyperbolic planes of non-Euclidean geometry are surfaces of constant positive and negative curvature, respectively.

Development of Differential Geometry

Differential geometry was founded by Gaspard Monge and C. F. Gauss in the beginning of the 19th cent. Important contributions were made by many mathematicians during the 19th cent., including B. Riemann, E. B. Christoffel, and C. G. Ricci. This work was collected and systematized at the end of the century by J. G. Darboux and Luigi Bianchi. The importance of differential geometry may be seen from the fact that Einstein's general theory of relativity is formulated entirely in terms of the differential geometry, in tensor notation, of a four-dimensional manifold combining space and time.

In mathematics, an equation that contains partial derivatives, expressing a process of change that depends on more than one independent variable. It can be read as a statement about how a process evolves without specifying the formula defining the process. Given the initial state of the process (such as its size at time zero) and a description of how it is changing (i.e., the partial differential equation), its defining formula can be found by various methods, most based on integration. Important partial differential equations include the heat equation, the wave equation, and Laplace's equation, which are central to mathematical physics.

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Equation containing derivatives of a function of a single variable. Its order is the order of the highest derivative it contains (e.g., a first-order differential equation involves only the first derivative of the function). Because the derivative is a rate of change, such an equation states how a function changes but does not specify the function itself. Given sufficient initial conditions, however, such as a specific function value, the function can be found by various methods, most based on integration.

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Field of mathematics in which methods of calculus 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.

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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 different lengths, as when turning a corner or traversing an uneven road. On a straight road the wheels rotate at the same speed; when turning a corner 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.

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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 equations 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 differential equation is generally a function whose derivatives satisfy the equation. 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 variables. Seealso differentiation.

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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 math.x^math.n is math.nmath.x^{math.n − 1}, (2) the derivative of sin math.x is cos math.x, and (3) the derivative of the exponential function math.e^math.x is itself. The rules are: (1) (math.amath.f + math.bmath.g)' = math.amath.f' + math.bmath.g', (2) (math.fmath.g)' = math.fmath.g' + math.gmath.f', (3) (math.f/math.g)' = (math.gmath.f' − math.fmath.g')/math.g², and (4) (math.f(math.g))' = math.f'(math.g)math.g', where math.a and math.b are constants, math.f and math.g are functions, and a prime (') indicates the derivative. The last formula is called the chain rule. The derivation and exploration of these formulas and rules is the subject of differential calculus. Seealso integration.

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In calculus, an expression based on the derivative of a function, useful for approximating certain values of the function. The differential of an independent variable math.x, written Δmath.x, is an infinitesimal change in its value. The corresponding differential of its dependent variable math.y is given by Δmath.y = math.f(math.x + Δmath.x) − math.f(math.x). Because the derivative of the function math.f(math.x), math.f'(math.x), is equal to the ratio Δmath.yΔmath.x as Δmath.x approaches zero (see limit), for small values of Δmath.x, Δmath.y ≅ math.f'(math.x)Δmath.x. This formula often enables a quick and fairly accurate approximation to be made for what otherwise would be a tedious calculation.

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