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Fermat's little theorem and some proofs; Gödel's completeness theorem and its original proof; Mathematical induction and a proof; Proof that 0.999... equals 1; Proof that 22/7 exceeds π; Proof that e is irrational; Proof that π is irrational; Proof that the sum of the reciprocals of the primes diverges
Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common antiderivatives.
Download as PDF; Printable version; In other projects ... Advanced. Calculus on Euclidean space ... Proof of Abel's test: ...
Print/export Download as PDF; Printable version; In other projects ... In mathematics, advanced calculus can refer to Multivariable calculus ; Mathematical analysis ...
2 Rules for propositional calculus. ... Download as PDF; Printable version; In other projects ... Proof by rules of inference: ...
Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the x-axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three-dimensional Cartesian plane where z = f(x, y)) and the plane which contains its domain. [1]
Hejhal's proof of a general form of the Selberg trace formula consisted of 2 volumes with a total length of 1322 pages. Arthur–Selberg trace formula. Arthur's proofs of the various versions of this cover several hundred pages spread over many papers. 2000 Almgren's regularity theorem. Almgren's proof was 955 pages long.
This calculus is also known as advanced calculus, especially in the United States. It is similar to multivariable calculus but is somewhat more sophisticated in that it uses linear algebra (or some functional analysis) more extensively and covers some concepts from differential geometry such as differential forms and Stokes' formula in terms of ...
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