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The first edition came out in 1952; Calculus and Analytic Geometry became one of the most famous and widely used texts on the subject. For many of the later editions (from the 5th onwards), Thomas was assisted by co-author Ross L. Finney, which gave rise to the text's metonym Thomas & Finney; such was its ubiquity in calculus teaching.
This 13th-century book contains the earliest complete solution of 19th-century Horner's method of solving high order polynomial equations (up to 10th order). It also contains a complete solution of Chinese remainder theorem, which predates Euler and Gauss by several centuries.
Calculus is also used to find approximate solutions to equations; in practice, it is the standard way to solve differential equations and do root finding in most applications. Examples are methods such as Newton's method , fixed point iteration , and linear approximation .
Spivak acknowledged in the preface of the second edition that the work is arguably an introduction to mathematical analysis rather than a calculus textbook. [13] Another of his well-known textbooks is Calculus on Manifolds, [14] a concise (146 pages) but rigorous and modern treatment of multivariable calculus accessible to advanced undergraduates.
The theory guarantees existence of codimension 1 solutions that are smooth away from a closed set of Hausdorff dimension. In the case of higher codimension Almgren proved existence of solutions with singular set of dimension at most k − 2 {\displaystyle k-2} in his regularity theorem .
In calculus, interchange of the order of integration is a methodology that transforms iterated integrals (or multiple integrals through the use of Fubini's theorem) of functions into other, hopefully simpler, integrals by changing the order in which the integrations are performed. In some cases, the order of integration can be validly ...
Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to ...
In mathematics, secondary calculus is a proposed expansion of classical differential calculus on manifolds, to the "space" of solutions of a (nonlinear) partial differential equation. It is a sophisticated theory at the level of jet spaces and employing algebraic methods.
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