This book is an introduction to the standard methods of proving mathematical theorems. It has been approved by the American Institute of Mathematics' Open Textbook Initiative . Also see the Mathematical Association of America Math DL review , and the Amazon reviews .
In differential geometry , Stokes' theorem (also called the generalized Stokes' theorem ) is a statement about the integration of differential forms on manifolds , which both simplifies and generalizes several theorems from vector calculus . Stokes' theorem says that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω, i.e. This modern form of Stokes' theorem is a vast generalization of a classical result first discovered by Lord Kelvin , who communicated it to George Stokes in a letter dated July 2, 1850. [ 1 ] [ 2 ] [ 3 ] Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to the result bearing his name. [ 4 ] [ 3 ] This classical Kelvin–Stokes theorem relates the surface integral of the curl of a vector field F over a surface Σ in Euclidean three-space to the line integral of the vector field over its boundary ∂Σ:
The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the integral . The first part of the theorem, sometimes called the first fundamental theorem of calculus , shows that an indefinite integration [ 1 ] can be reversed by a differentiation. This part of the theorem is also important because it guarantees the existence of antiderivatives for continuous functions . [ 2 ] The second part, sometimes called the second fundamental theorem of calculus , allows one to compute the definite integral of a function by using any one of its infinitely many antiderivatives . This part of the theorem has invaluable practical applications, because it markedly simplifies the computation of definite integrals .
In vector calculus , the curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field . At every point in the field, the curl of that field is represented by a vector . The attributes of this vector (length and direction) characterize the rotation at that point. The direction of the curl is the axis of rotation, as determined by the right-hand rule , and the magnitude of the curl is the magnitude of rotation. If the vector field represents the flow velocity of a moving fluid , then the curl is the circulation density of the fluid. A vector field whose curl is zero is called irrotational .
In vector calculus , divergence is a vector operator that measures the magnitude of a vector field 's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. For example, consider air as it is heated or cooled.
Grigori Yakovlevich Perelman ( / ˈ p ɛr ɨ l m ən / PERR -il-mən [ dubious ] ; Russian : Григо́рий Я́ковлевич Перельма́н ; born 13 June 1966) is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology . In 1994, Perelman proved the soul conjecture . In 2003, he proved Thurston's geometrization conjecture . This consequently solved in the affirmative the Poincaré conjecture , posed in 1904, which before its solution was viewed as one of the most important and difficult open problems in topology .
An imaginary number is a number that can be written as a real number multiplied by the imaginary unit , which is defined by its property . [ 1 ] An imaginary number has a negative or zero square. For example, is an imaginary number and its square is
Riemann zeta function ζ ( s ) in the complex plane . The color of a point s encodes the value of ζ ( s ): colors close to black denote values close to zero, while hue encodes the value's argument . The white spot at s = 1 is the pole of the zeta function; the black spots on the negative real axis and on the critical line Re( s ) = 1/2 are its zeros. Values with arguments close to zero including positive reals on the real half-line are presented in red.
This article is about Euler's formula in complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic . Euler's formula , named after Leonhard Euler , is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function . Euler's formula states that, for any real number x , where e is the base of the natural logarithm , i is the imaginary unit , and cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians . This complex exponential function is sometimes denoted cis ( x ) (" c osine plus i s ine").
This is a list of symbols found within all branches of mathematics to express a formula or to replace a constant . When reading the list, it is important to recognize that a mathematical concept is independent of the symbol chosen to represent it. For many of the symbols below, the symbol is usually synonymous with the corresponding concept (ultimately an arbitrary choice made as a result of the cumulative history of mathematics), but in some situations a different convention may be used. For example, depending on context, "≡" may represent congruence or a definition. Further, in mathematical logic, numerical equality is sometimes represented by "≡" instead of "=", with the latter representing equality of well-formed formulas . In short, convention dictates the meaning.
The arithmetization of analysis was a research program in the foundations of mathematics carried out in the second half of the 19th century. Kronecker originally introduced the term arithmetization of analysis , by which he meant its constructivization in the context of the natural numbers (see quotation at bottom of page). The meaning of the term later shifted to signify the set-theoretic construction of the real line. Its main proponent was Weierstrass , who argued the geometric foundations of calculus were not solid enough for rigorous work. The highlights of this research program are: the various (but equivalent) constructions of the real numbers by Dedekind and Cantor resulting in the modern axiomatic definition of the real number field; the epsilon-delta definition of limit ; and the naïve set-theoretic definition of function .
In multilinear algebra , a dyadic or dyadic tensor is a second order tensor written in a special notation, formed by juxtaposing pairs of vectors, along with a notation for manipulating such expressions analogous to the rules for matrix algebra . The notation and terminology is relatively obsolete today. Its uses in physics include stress analysis and electromagnetism .
Calculus deals with properties of the real numbers. In order to understand calculus you must first understand what it is about the real numbers that separates them from other kinds of numbers we use from day to day. 1.1 The Counting Numbers These are the first numbers we learn.
Те́нзор (от лат. tensus , «напряженный») — объект линейной алгебры , линейно преобразующий элементы одного линейного пространства в элементы другого. Частными случаями тензоров являются скаляры , векторы , билинейные формы и т. п. Термин «тензор» также часто служит сокращением для термина « тензорное поле », изучением которых занимается тензорное исчисление .