DICT.TW Dictionary Taiwan
Cal·cu·lus n.; pl. Calculi
1. Med. Any solid concretion, formed in any part of the body, but most frequent in the organs that act as reservoirs, and in the passages connected with them; as, biliary calculi; urinary calculi, etc.
2. Math. A method of computation; any process of reasoning by the use of symbols; any branch of mathematics that may involve calculation.
Barycentric calculus, a method of treating geometry by defining a point as the center of gravity of certain other points to which coëfficients or weights are ascribed.
Calculus of functions, that branch of mathematics which treats of the forms of functions that shall satisfy given conditions.
Calculus of operations, that branch of mathematical logic that treats of all operations that satisfy given conditions.
Calculus of probabilities, the science that treats of the computation of the probabilities of events, or the application of numbers to chance.
Calculus of variations, a branch of mathematics in which the laws of dependence which bind the variable quantities together are themselves subject to change.
Differential calculus, a method of investigating mathematical questions by using the ratio of certain indefinitely small quantities called differentials. The problems are primarily of this form: to find how the change in some variable quantity alters at each instant the value of a quantity dependent upon it.
Exponential calculus, that part of algebra which treats of exponents.
Imaginary calculus, a method of investigating the relations of real or imaginary quantities by the use of the imaginary symbols and quantities of algebra.
Integral calculus, a method which in the reverse of the differential, the primary object of which is to learn from the known ratio of the indefinitely small changes of two or more magnitudes, the relation of the magnitudes themselves, or, in other words, from having the differential of an algebraic expression to find the expression itself.
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Func·tion n.
1. The act of executing or performing any duty, office, or calling; performance. “In the function of his public calling.”
2. Physiol. The appropriate action of any special organ or part of an animal or vegetable organism; as, the function of the heart or the limbs; the function of leaves, sap, roots, etc.; life is the sum of the functions of the various organs and parts of the body.
3. The natural or assigned action of any power or faculty, as of the soul, or of the intellect; the exertion of an energy of some determinate kind.
As the mind opens, and its functions spread. --Pope.
4. The course of action which peculiarly pertains to any public officer in church or state; the activity appropriate to any business or profession.
Tradesmen . . . going about their functions. --Shak.
The malady which made him incapable of performing his
regal functions. --Macaulay.
5. Math. A quantity so connected with another quantity, that if any alteration be made in the latter there will be a consequent alteration in the former. Each quantity is said to be a function of the other. Thus, the circumference of a circle is a function of the diameter. If x be a symbol to which different numerical values can be assigned, such expressions as x², 3ₓ, Log. x, and Sin. x, are all functions of x.
6. Eccl. A religious ceremony, esp. one particularly impressive and elaborate.
Every solemn ‘function' performed with the requirements of the liturgy. --Card. Wiseman.
7. A public or social ceremony or gathering; a festivity or entertainment, esp. one somewhat formal.
This function, which is our chief social event. --W. D. Howells.
Algebraic function, a quantity whose connection with the variable is expressed by an equation that involves only the algebraic operations of addition, subtraction, multiplication, division, raising to a given power, and extracting a given root; -- opposed to transcendental function.
Arbitrary function. See under Arbitrary.
Calculus of functions. See under Calculus.
Carnot's function Thermo-dynamics, a relation between the amount of heat given off by a source of heat, and the work which can be done by it. It is approximately equal to the mechanical equivalent of the thermal unit divided by the number expressing the temperature in degrees of the air thermometer, reckoned from its zero of expansion.
Circular functions. See Inverse trigonometrical functions (below). -- Continuous function, a quantity that has no interruption in the continuity of its real values, as the variable changes between any specified limits.
Discontinuous function. See under Discontinuous.
Elliptic functions, a large and important class of functions, so called because one of the forms expresses the relation of the arc of an ellipse to the straight lines connected therewith.
Explicit function, a quantity directly expressed in terms of the independently varying quantity; thus, in the equations y = 6x², y = 10 -x³, the quantity y is an explicit function of x.
Implicit function, a quantity whose relation to the variable is expressed indirectly by an equation; thus, y in the equation x² + y² = 100 is an implicit function of x.
Inverse trigonometrical functions, or Circular functions, the lengths of arcs relative to the sines, tangents, etc. Thus, AB is the arc whose sine is BD, and (if the length of BD is x) is written
One-valued function, a quantity that has one, and only one, value for each value of the variable.
Transcendental functions, a quantity whose connection with the variable cannot be expressed by algebraic operations; thus, y in the equation y = 10ₓ is a transcendental function of x. See Algebraic function (above).
Trigonometrical function, a quantity whose relation to the variable is the same as that of a certain straight line drawn in a circle whose radius is unity, to the length of a corresponding are of the circle. Let AB be an arc in a circle, whose radius OA is unity let AC be a quadrant, and let OC, DB, and AF be drawnpependicular to OA, and EB and CG parallel to OA, and let OB be produced to G and F. E Then BD is the sine of the arc AB; OD or EB is the cosine, AF is the tangent, CG is the cotangent, OF is the secant OG is the cosecant, AD is the versed sine, and CE is the coversed sine of the are AB. If the length of AB be represented by x (OA being unity) then the lengths of Functions. these lines (OA being unity) are the trigonometrical functions of x, and are written
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