calculus

calculus,

branch of mathematicsmathematics,
deductive study of numbers, geometry, and various abstract constructs, or structures; the latter often "abstract" the features common to several models derived from the empirical, or applied, sciences, although many emerge from purely mathematical or logical
..... Click the link for more information. that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limitlimit,
in mathematics, value approached by a sequence or a function as the index or independent variable approaches some value, possibly infinity. For example, the terms of the sequence 1-2, 1-4, 1-8, 1-16, … are obviously getting smaller and smaller; since, if enough
..... Click the link for more information. —the notion of tending toward, or approaching, an ultimate value. The English physicist Isaac NewtonNewton, Sir Isaac,
1642–1727, English mathematician and natural philosopher (physicist), who is considered by many the greatest scientist that ever lived. Early Life and Work
..... Click the link for more information. and the German mathematician G. W. LeibnizLeibniz or Leibnitz, Gottfried Wilhelm, Baron von
, 1646–1716, German philosopher and mathematician, b. Leipzig.
..... Click the link for more information. , working independently, developed the calculus during the 17th cent. The calculus and its basic tools of differentiation and integration serve as the foundation for the larger branch of mathematics known as analysisanalysis,
branch of mathematics that utilizes the concepts and methods of the calculus. It includes not only basic calculus, but also advanced calculus, in which such underlying concepts as that of a limit are subjected to rigorous examination; differential and integral
..... Click the link for more information. . The methods of calculus are essential to modern physics and to most other branches of modern science and engineering.

The Differential Calculus

The differential calculus arises from the study of the limit of a quotient, Δy/Δx, as the denominator Δx approaches zero, where x and y are variables. y may be expressed as some functionfunction,
in mathematics, a relation f that assigns to each member x of some set X a corresponding member y of some set Y; y is said to be a function of x, usually denoted f(x) (read "f of x ").
..... Click the link for more information. of x, or f(x), and Δy and Δx represent corresponding increments, or changes, in y and x. The limit of Δy/Δx is called the derivative of y with respect to x and is indicated by dy/dx or D_xy:

The symbols dy and dx are called differentials (they are single symbols, not products), and the process of finding the derivative of y=f(x) is called differentiation. The derivative dy/dx=df(x)/dx is also denoted by y′, or f′(x). The derivative f′(x) is itself a function of x and may be differentiated, the result being termed the second derivative of y with respect to x and denoted by y″, f″(x), or d²y/dx². This process can be continued to yield a third derivative, a fourth derivative, and so on. In practice formulas have been developed for finding the derivatives of all commonly encountered functions. For example, if y=xⁿ, then y′=nx^{n − 1}, and if y=sin x, then y′=cos x (see trigonometrytrigonometry
[Gr.,=measurement of triangles], a specialized area of geometry concerned with the properties of and relations among the parts of a triangle. Spherical trigonometry is concerned with the study of triangles on the surface of a sphere rather than in the plane; it is
..... Click the link for more information. ). In general, the derivative of y with respect to x expresses the rate of change in y for a change in x. In physical applications the independent variable (here x) is frequently time; e.g., if s=f(t) expresses the relationship between distance traveled, s, and time elapsed, t, then s′=f′(t) represents the rate of change of distance with time, i.e., the speed, or velocity.

Everyday calculations of velocity usually divide the distance traveled by the total time elapsed, yielding the average velocity. The derivative f′(t)=ds/dt, however, gives the velocity for any particular value of t, i.e., the instantaneous velocity. Geometrically, the derivative is interpreted as the slope of the line tangent to a curve at a point. If y=f(x) is a real-valued function of a real variable, the ratio Δy/Δx=(y₂ − y₁)/(x₂ − x₁) represents the slope of a straight line through the two points P (x₁,y₁) and Q (x₂,y₂) on the graph of the function. If P is taken closer to Q, then x₁ will approach x₂ and Δx will approach zero. In the limit where Δx approaches zero, the ratio becomes the derivative dy/dx=f′(x) and represents the slope of a line that touches the curve at the single point Q, i.e., the tangent line. This property of the derivative yields many applications for the calculus, e.g., in the design of optical mirrors and lenses and the determination of projectile paths.

The Integral Calculus

The second important kind of limit encountered in the calculus is the limit of a sum of elements when the number of such elements increases without bound while the size of the elements diminishes. For example, consider the problem of determining the area under a given curve y=f(x) between two values of x, say a and b. Let the interval between a and b be divided into n subintervals, from a=x₀ through x₁, x₂, x₃, … x_{i − 1}, x_i, … , up to x_n=b. The width of a given subinterval is equal to the difference between the adjacent values of x, or Δx_i=x_i − x_{i − 1}, where i designates the typical, or ith, subinterval. On each Δx_i a rectangle can be formed of width Δx_i, height y_i=f(x_i) (the value of the function corresponding to the value of x on the right-hand side of the subinterval), and area ΔA_i=f(x_i)Δx_i. In some cases, the rectangle may extend above the curve, while in other cases it may fail to include some of the area under the curve; however, if the areas of all these rectangles are added together, the sum will be an approximation of the area under the curve.

This approximation can be improved by increasing n, the number of subintervals, thus decreasing the widths of the Δx's and the amounts by which the ΔA's exceed or fall short of the actual area under the curve. In the limit where n approaches infinity (and the largest Δx approaches zero), the sum is equal to the area under the curve:

The last expression on the right is called the integral of f(x), and f(x) itself is called the integrand. This method of finding the limit of a sum can be used to determine the lengths of curves, the areas bounded by curves, and the volumes of solids bounded by curved surfaces, and to solve other similar problems.

An entirely different consideration of the problem of finding the area under a curve leads to a means of evaluating the integral. It can be shown that if F(x) is a function whose derivative is f(x), then the area under the graph of y=f(x) between a and b is equal to F(b) − F(a). This connection between the integral and the derivative is known as the Fundamental Theorem of the Calculus. Stated in symbols:

The function F(x), which is equal to the integral of f(x), is sometimes called an antiderivative of f(x), while the process of finding F(x) from f(x) is called integration or antidifferentiation. The branch of calculus concerned with both the integral as the limit of a sum and the integral as the antiderivative of a function is known as the integral calculus. The type of integral just discussed, in which the limits of integration, a and b, are specified, is called a definite integral. If no limits are specified, the expression is an indefinite integral. In such a case, the function F(x) resulting from integration is determined only to within the addition of an arbitrary constant C, since in computing the derivative any constant terms having derivatives equal to zero are lost; the expression for the indefinite integral of f(x) is

The value of the constant C must be determined from various boundary conditions surrounding the particular problem in which the integral occurs. The calculus has been developed to treat not only functions of a single variable, e.g., x or t, but also functions of several variables. For example, if z=f(x,y) is a function of two independent variables, x and y, then two different derivatives can be determined, one with respect to each of the independent variables. These are denoted by ∂z/∂x and ∂z/∂y or by D_xz and D_yz. Three different second derivatives are possible, ∂²z/∂x², ∂²z/∂y², and ∂²z/∂x∂y=∂²z/∂y∂x. Such derivatives are called partial derivatives. In any partial differentiation all independent variables other than the one being considered are treated as constants.

Bibliography

See R. Courant and F. John, Introduction to Calculus and Analysis, Vol. I (1965); M. Kline, Calculus: An Intuitive and Physical Approach (2 vol., 1967); G. B. Thomas and R. L. Finney, Calculus and Analytic Geometry (7th ed. 2 vol., 1988).

Calculus

a formal apparatus, based on clearly formulated rules, for operating with symbols of a specified type, which permits an exhaustively exact description of a certain class of problems as well as solution algorithms for certain subclasses of this class. The subclasses in question coincide with the whole class only in the case of the simplest calculi. Examples of calculi are the set of arithmetic rules for operating with numbers (that is, numerical symbols), the literal calculus of elementary algebra, differential calculus, integral calculus, the calculus of variations, and other branches of mathematical analysis and the theory of functions.

Despite its early origin, the term “calculus” was used in mathematics without a rigorous general definition until very recently. With the development of mathematical logic a demand arose for a general theory of calculus as well as for the refinement of the concept of “calculus” itself, which underwent a more systematic formalization. In most cases, however, the following conception of a calculus (originating from D. Hilbert) proves to be sufficient. Consider a certain alphabet (generally speaking, infinite, although also possibly given by means of a finite number of symbols), whose elements, called letters, are used to construct formulas of the calculus under consideration (sometimes called words or expressions) with the help of clearly stated formation rules. Some of these (“well-formed”) formulas are declared to be axioms, and from these, with the help of transformation rules (or rules of deduction), new formulas are “deduced,” which are called theorems of the given calculus. Sometimes the term “calculus” is applied only to the “dictionary” (“expression”) part of the structure described, and it is said that joining the “deductive” part to it (that is, adding both the rules and axioms of formation to the rules of deduction and to the alphabet) produces a formal system. Besides, these terms are often also considered synonymous (and the terms “logistic system,” “formal theory,” “formalism,” and many others are also used as synonyms for this). If such a noninterpreted (“meaningless”) calculus is juxtaposed to a certain interpretation (or, as is said, a purely syntactic treatment is supplemented by semantics), then a formalized language is obtained. The representation of substantive logical (and logico-mathematical) theories in the form of formalized languages is a characteristic feature of mathematical logic.

REFERENCES

Kleene, S. C. Vvedenie v metamatematiku, sections 14–20. Moscow, 1957. (Translated from English.)
Markov, A. A. “Teoriia algorifmov.” Moscow-Leningrad, 1954. (Tr. Matematicheskogo in-ta im. V. A. Steklova, vol. 42.)
Curry, H. B. Osnovaniia matemalicheskoi logiki, chap. 2. Moscow, 1969. (Translated from English.)
Matematicheskaia teoriia logicheskogo vyvoda.Collection of translations. Edited by A. V. Idel’son and G. E. Mints. Moscow, 1967.
Logicheskie i logiko-matematicheskie ischisleniia, 1.Collection of works. Edited by V. P. Orevkov. Leningrad, 1968.

IU. A. GASTEV

calculus

[′kal·kyə·ləs] (anatomy) A small and cuplike structure. (mathematics) The branch of mathematics dealing with differentiation and integration and related topics. (pathology) An abnormal, solid concretion of minerals and salts formed around organic materials and found chiefly in ducts, hollow organs, and cysts.

calculus

1. a branch of mathematics, developed independently by Newton and Leibniz. Both differential calculus and integral calculus are concerned with the effect on a function of an infinitesimal change in the independent variable as it tends to zero. 2. any mathematical system of calculation involving the use of symbols 3. Logic an uninterpreted formal system 4. Pathol a stonelike concretion of minerals and salts found in ducts or hollow organs of the body

calculus

[kal´ku-lus] (pl. cal´culi) (L.) an abnormal concretion, usually composed of mineral salts, occurring within the body, chiefly in hollow organs or their passages. Called also stone. See also kidney stone" >kidney stone and gallstone" >gallstone. adj., adj cal´culous.biliary calculus gallstone.bladder calculus vesical calculus.bronchial calculus broncholith.calcium oxalate calculus oxalate calculus.dental calculus a hard, stonelike concretion, varying in color from creamy yellow to black, that forms on the teeth or dental prostheses through calcification of plaque" >dental plaque; it begins as a yellowish film formed of calcium phosphate and carbonate, food particles, and other organic matter that is deposited on the teeth by the saliva. It should be removed regularly by a dentist or dental hygienist; if neglected, it can cause bacteria to lodge between the gums and the teeth, causing gum infection, dental caries, loosening of the teeth, and other disorders. Called also tartar.gastric calculus gastrolith.intestinal calculus enterolith.lung calculus a hard mass or concretion formed in the bronchi around a small center of inorganic material, or from calcified portions of lung tissue or adjacent lymph nodes. Called also pneumolith.mammary calculus a concretion in one of the lactiferous ducts.nasal calculus rhinolith.oxalate calculus a hard urinary calculus of calcium oxalate" >calcium oxalate; some are covered with minute sharp spines that may abrade the renal pelvic epithelium, and others are smooth. Called also calcium oxalate calculus.phosphate calculus a urinary calculus composed of a phosphate along with calcium oxalate and ammonium urate; it may be hard, soft, or friable, and so large that it may fill the renal pelvis and calices.prostatic calculus a concretion formed in the prostate, chiefly of calcium carbonate and phosphate. Called also prostatolith.renal calculus kidney stone.staghorn calculus a urinary calculus, usually a phosphate calculus, found in the pelvis" >renal pelvis and shaped like the antlers of a stag because it extends into multiple calices.urate calculus uric acid calculus.urethral calculus a urinary calculus in the urethra; symptoms vary according to the patient's sex and the site of lodgment.uric acid calculus a hard, yellow or reddish-yellow urinary calculus formed from uric acid.urinary calculus a calculus in any part of the tract" >urinary tract; it is vesical when lodged in the bladder and renal (see kidney stone" >kidney stone) when in the pelvis" >renal pelvis. Common types named for their primary components are oxalate calculi, phosphate calculi, and uric acid calculi. Called also urolith.uterine calculus any kind of concretion in the uterus, such as a calcified myoma. Called also hysterolith and uterolith.vesical calculus a urinary calculus in the urinary bladder. Called also bladder calculus.

cal·cu·lus

, gen. and pl.

cal·cu·li

(kal'kyū-lŭs, -lī), A concretion formed in any part of the body, most commonly in the passages of the biliary and urinary tracts; usually composed of salts of inorganic or organic acids, or of other material such as cholesterol. Synonym(s): stone (1) [L. a pebble]

calculus

(kăl′kyə-ləs)n. pl. calcu·li (-lī′) or calcu·luses 1. Medicine An abnormal concretion in the body, usually formed of mineral salts and found in the gallbladder, kidney, or urinary bladder, for example.2. Dentistry See tartar.

calculus

Dentistry
Indurated, yellow-brown/black deposits on teeth formed by bacteria in dental plaques from mineralised calcium salts in saliva and subgingival transudates.
Kidneys
A stone in the urinary tract.
Pathology
An abnormal, often calcium-rich mass found in various tissues, seen by light microscopy.

calculus

Plural, calculi Dentistry Tartar Indurated, yellow–brown/black deposits on teeth formed by bacteria in dental plaques from mineralized calcium salts in saliva and subgingival transudates Kidneys A stone in the urinary tract. See Mulberry calculus, Staghorn calculus.

cal·cu·lus

, pl. calculi (kal'kyū-lŭs, -lī) A concretion formed in any part of the body, most commonly in the passages of the biliary and urinary tracts; usually composed of salts of inorganic or organic acids, or of other material such as cholesterol.
See also: dental calculus
Synonym(s): stone (1) . [L. a pebble]

calculus

A stone of any kind formed abnormally in the body, mainly in the urinary system and the gall bladder. Calculi form in fluids in which high concentrations of chemical substances are dissolved. Their formation is encouraged by infection.

calculus (dental)

See DENTAL CALCULUS.

Calculus

Any type of hard concretion (stone) in the body, but usually found in the gallbladder, pancreas and kidneys. They are formed by the accumulation of excess mineral salts and other organic material such as blood or mucous. Calculi (pl.) can cause problems by lodging in and obstructing the proper flow of fluids, such as bile to the intestines or urine to the bladder.Mentioned in: Abdominal Ultrasound, Oral Hygiene

cal·cu·lus

, pl. calculi (kal'kyū-lŭs, -lī) 1. Synonym(s): dental calculus. 2. Concretion formed in any part of the body, most commonly in passages of biliary and urinary tracts; usually composed of salts of inorganic or organic acids, or of other material such as cholesterol.
Synonym(s): stone (1) . [L. a pebble]

Patient discussion about calculus

Q. Why do i get kidney stones? I am 38 and have had three stones pass so far. Is it the coffee, the meat, the stress, or the damned DNA?! My uncle is in his 50s and has passed over 30 stones!A. Kidney stones are very common and even without the genetic or familial background people tend to get them. Of course, the more family predisposition you have, the higher are your chances of developing them, which is probably why you did. Also, a diet rich with dairy and calcium can cause your body to store excess calcium, that tends to calcify and create stones. Not drinking enough fluid is also one of the reasons.

Q. how do i cure tonsil stones (tonsiloth)? A. There are very little literature about this subject, but I heard about treatment in which the crypts (deep and narrow grooves on the tongue in which the stones form) are burned with laser.
As far as I know these stones don't cause damage by themselves so it's not such a common treatment.
You may read more here:
http://en.wikipedia.org/wiki/Tonsillolith

Q. Would kidney stones affect a PSA reading? Would drinking lots of grapefruit juice affect a PSA reading? My husband's PSA reading jumped from a 4.2 to a 17 in @ 2 years' time. How can that be? This man takes all sorts of supplements and really watches his diet. He also takes good care of his body, and does NOT look or act 68.A. You should get your parathyroid gland checked out. Your calcium level might be causing the kidney stones.

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The Differential Calculus

The Integral Calculus

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Calculus

REFERENCES

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Synonyms for calculus

noun a hard lump produced by the concretion of mineral salts

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noun an incrustation that forms on the teeth and gums

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noun the branch of mathematics that is concerned with limits and with the differentiation and integration of functions

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