If conceptually the definite integral for function of one variable represents the area of the region between the graph and the x-axis, that for functions of two variables (double integral) consists of the measure of the space between the graph and the plane which contains its domain, so they describe not an area but a volume of a particular solid called cylindroid; you obtain the same value if you consider the triple integrals (functions of three variables) calculated with the constant f(x, y, z) = 1 ...

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Multiple integral, Multiple integral - Multiple integrals are not the same as iterated integrals, Multiple integral - Multiple integrals, Multiple integral - Some practical applications, Multiple integral - Mathematical definition, Multiple integral - Theorems, Multiple integral - Double integral, Multiple integral - Triple integral, Multiple integral - Methods of integration, Multiple integral - Direct examination, Multiple integral - Formulas of reduction, Multiple integral - Change of variables, Multiple integral - Example of mathematical applications - Calculations of volume, Multiple integral - Multiple improper integral, Multiple integral - Bibliography

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Multiple integral, Multiple integral - Multiple integrals are not the same as iterated integrals, Multiple integral - Multiple integrals, Multiple integral - Some practical applications, Multiple integral - Mathematical definition, Multiple integral - Theorems, Multiple integral - Double integral, Multiple integral - Triple integral, Multiple integral - Methods of integration, Multiple integral - Direct examination, Multiple integral - Formulas of reduction, Multiple integral - Change of variables, Multiple integral - Example of mathematical applications - Calculations of volume, Multiple integral - Multiple improper integral, Multiple integral - Bibliography

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The resolution of problems with multiple integrals consists in most of cases in finding the way to reduce operations in a series of integral of one variable, the only directly solvable. Multiple integral - Direct examination. Sometimes is possible to avoid direct calculation and obtain the result of the integration. In case of constant functions the result is immediate; one need only multiply the measure of the domain for the value of the constant c. If n = 1, on R2 ...

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Multiple integral, Multiple integral - Multiple integrals are not the same as iterated integrals, Multiple integral - Multiple integrals, Multiple integral - Some practical applications, Multiple integral - Mathematical definition, Multiple integral - Theorems, Multiple integral - Double integral, Multiple integral - Triple integral, Multiple integral - Methods of integration, Multiple integral - Direct examination, Multiple integral - Formulas of reduction, Multiple integral - Change of variables, Multiple integral - Example of mathematical applications - Calculations of volume, Multiple integral - Multiple improper integral, Multiple integral - Bibliography

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The substitution rule can be used to determine antiderivatives. One chooses a relation between x and t, determines the corresponding relation between dx and dt by differentiating, and performs the substitutions. An antiderivative for the substituted function can hopefully be determined; the original substitution between x and t is then undone. Similar to our first example above, we can determine the following antiderivative with this method: ...

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Integration by substitution, Integration by substitution - Examples, Integration by substitution - Antiderivatives, Integration by substitution - Substitution rule for multiple variables

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The logarithmic derivative idea is closely connected to the integrating factor method, for first order differential equations. In operator terms, write D = d/dx and let M denote the operator of multiplication by some given function G(x). Then M−1DM can be written (by the product rule) as D + M* where M* now denotes the multiplication operator by the logarithmic derivative G′/G. In practice we are given an opera ...

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Logarithmic derivative, Logarithmic derivative - Formulae, Logarithmic derivative - Integrating factors, Logarithmic derivative - Complex analysis, Logarithmic derivative - The multiplicative group

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AIM is a three-letter abbreviation with multiple meanings, as described below: AOL Instant Messenger A I M Management Group Inc. (AIM Investments) Abductory Inductive Mechanism Abrams Integrated Management Abridged Index Medicus Absorption Isotherm Measurement Accunet Information Manager (AT&T) Accuracy in Media (news media watchdog) Acquisition Information Management Action Item Master Active Inert Missile Ada Interacti ...

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A residue, broadly, is anything left behind by a reaction or event. In complex analysis, the residue is a complex number which describes the behavior of path integrals of a meromorphic function around a singularity. See residue (complex analysis). In modular arithmetic, the residue of an integer n to base b is the remainder r after the largest multiple mb of b no greater than n has been subtracted from n (if n < 0, one adds multiples of b just sufficient to make the result non-negative). The resid ...

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Miles. See mile - unit of measurement (distance) Miles Aircraft Ltd - UK manufacturer of light and military aircraft Miles "Tails" Prower - a fictional fox in the Sonic the Hedgehog series Miles Davis was an American jazz composer and trumpeter and was one of the most influential and innovative musicians of the 20th century. Robert Miles is a Swiss DJ and musician M.I.L.E.S.: Multiple Integrated Laser Engagement System - a method of simula

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WebSphere refers to a brand of IBM software products, although the term also popularly refers to one specific product: WebSphere Application Server (WAS). WebSphere helped define the middleware software category and is designed to set up, operate and integrate e-business applications across multiple computing platforms using Web technologies. It includes both the run-time components (like WAS) and the too ...

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WebSphere - WebSphere products partial WebSphere - Other IBM software brands

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The formula as given can be applied more widely; for example if f(z) is a meromorphic function, it makes sense at all complex values of z at which f has neither a zero nor a pole. Further, at a zero or a pole the logarithmic derivative behaves in a way that is easily analysed in terms of the particular case zn with n an integer, n≠0. The logarithmi ...

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Logarithmic derivative, Logarithmic derivative - Formulae, Logarithmic derivative - Integrating factors, Logarithmic derivative - Complex analysis, Logarithmic derivative - The multiplicative group

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Carcinogenesis (meaning literally, the creation of cancer) is the process by which normal cells are transformed into cancer cells. Carcinogenesis - Introduction. Cell division (proliferation) is a physiological process that occurs in almost all tissues and under many circumstances. Normally homeostasis, the balance between proliferation and programmed cell death, usually in the form of apoptosis, is maintained by tightly regulating both processes to ensure the integrity of organs and tissues. Mutatio ...

Including:

Carcinogenesis - Introduction Carcinogenesis - Properties of malignant cells Carcinogenesis - Mechanisms of carcinogenesis
Carcinogenesis - Proto-oncogenes Carcinogenesis - Tumor suppressor genes Carcinogenesis - Multiple mutations Carcinogenesis - Role of genetic damage Carcinogenesis - Non-mutagenic carcinogens Carcinogenesis - Role of viral infections Carcinogenesis - Etiology
Carcinogenesis - Non-mainstream theories

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A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. The core functionality of a CAS is manipulation of mathematical expressions in symbolic form. Computer algebra system - Types of expressions. The expressions manipulated by the CAS typically include polynomials in multiple variables; standard functions of expressions (sine, exponential, etc.); various special functions (gamma, zeta, erf, Bessel, etc.); arbitrary functions of expressions; derivatives, integral ...

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Computer algebra system - Types of expressions Computer algebra system - Symbolic manipulations Computer algebra system - Other features Computer algebra system - History Computer algebra system - Mathematics used in computer algebra systems

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A virtual globe is a 3D software model of the earth. Depending on its technology, a virtual globe may be as simple as an inexpensive globe sitting in a typical school class room, or as sophisticated as an integrated interface that provides intuitive access to multiple GIS databases. Virtual globe - Types. Most earlier computerized world atlases were not detailed or only had limited area coverages. Today's virtual globes are connected to satellite image servers. These virtual globes are capable of rotation a ...

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Virtual globe - Types
Virtual globe - Offline virtual globes Virtual globe - Online virtual globes Virtual globe - Literary References
Virtual globe - Features Virtual globe - Comparisons

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Power series - Addition and subtraction. When two functions f and g are decomposed into power series around the same center c, the power series of the sum or difference of the functions can be obtained by termwise addition and subtraction. That is, if: then Power series - Multiplication and division. With the same definitions above, for the power series of the product and quotient of the functions can be obtained as follows: ...

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Power series, Power series - Examples, Power series - Radius of convergence, Power series - Operations on power series, Power series - Addition and subtraction, Power series - Multiplication and division, Power series - Differentiation and integration, Power series - Analytic functions, Power series - Formal power series, Power series - Power series in several variables, Power series - Order of a power series

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One can give a further explanation, however from the other direction, based on the special role of functions f(x)g(y). These, in which the roles of the two variables are uncoupled, present no problem in this context; and neither do their linear combinations. Quite generally, given compact spaces X and Y, we can use the Stone-Weierstrass theorem to show that such functions give a subalgebra of C(X×Y) that is dense in the uniform norm: or in other words any continuous function on X×Y can be uniformly approximated b ...

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Double integral, Double integral - Definitions, Double integral - Counterexample, Double integral - Explanation via Lebesgue theory, Double integral - In the positive sense

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A double integral is defined via a 2-dimensional measure in the plane, rather than by integrating twice (see Lebesgue integral). On the other hand, if we define then is an iterated integral, so called because one integrates, and then integrates again. ...

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Double integral, Double integral - Definitions, Double integral - Counterexample, Double integral - Explanation via Lebesgue theory, Double integral - In the positive sense

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Does it matter whether one integrates first with respect to x and then with respect to y or vice-versa? Perhaps surprisingly, in some cases yes, as an example shows: Obviously the sign gets reversed if the order of iterated integration gets reversed, i.e., if "dy dx" replaces "dx dy". But the value of the integral is not zero, and so the values of the two iterated integrals differ from each other. For the details of the evaluation of this integral, see this rearrangemen ...

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Double integral, Double integral - Definitions, Double integral - Counterexample, Double integral - Explanation via Lebesgue theory, Double integral - In the positive sense

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To give the analytic explanation: the double integral exists only if and in that case, the double integral coincides in value with either of the two iterated integrals. Thus, whenever the two iterated integrals differ in value from each other, the double integral of the absolute value of the function must be infinite. See Fubini's theorem. ...

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Double integral, Double integral - Definitions, Double integral - Counterexample, Double integral - Explanation via Lebesgue theory, Double integral - In the positive sense

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The most basic technique for computing integrals of one real variable is based on the fundamental theorem of calculus. It proceeds like this: Choose a function f(x) and an interval [a,b]. Find an antiderivative of f, that is, a function F such that F' = f. By the fundamental theorem of calculus, . Therefore the value of the integral is F(b) − F(a). Note that the integral is not actually the antiderivative (the integral is a number), but the fundamental theorem allows ...

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Integral, Integral - Computing integrals, Integral - Approximation of definite integrals, Integral - Integrals and computerized algebra systems, Integral - Improper integrals, Integral - Definitions of the integral, Integral - Definitions by means of an integral

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The most basic technique for computing integrals of one real variable is based on the fundamental theorem of calculus. It proceeds like this: Choose a function f(x) and an interval [a,b]. Find an antiderivative of f, that is, a function F such that F' = f. By the fundamental theorem of calculus, provided the integrand and integral have no singularities on the path of integration, . Therefore the value ...

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Integral, Integral - Computing integrals, Integral - Approximation of definite integrals, Integral - Integrals and computerized algebra systems, Integral - Improper integrals, Integral - Definitions of the integral, Integral - Definitions by means of an integral

Read more here: » Integral: Encyclopedia II - Integral - Computing integrals