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The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each point in time) with the concept of integrating a function (calculating the area under its graph, or the cumulative effect of small contributions). Roughly speaking, the two operations can be ...
For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus. [1] The names are mostly traditional, so that for example the fundamental theorem of arithmetic is basic to what would now be called number theory. [2]
Implicit function theorem; Increment theorem; Integral of inverse functions; Integration by parts; Integration using Euler's formula; Intermediate value theorem; Inverse function rule; Inverse function theorem
Bolzano's theorem (real analysis, calculus) Bolzano–Weierstrass theorem (real analysis, calculus) Bombieri's theorem (number theory) Bombieri–Friedlander–Iwaniec theorem (number theory) Bondareva–Shapley theorem ; Bondy's theorem (graph theory, combinatorics) Bondy–Chvátal theorem (graph theory) Bonnet theorem (differential geometry)
Elementary Calculus: An Infinitesimal Approach; Nonstandard calculus; Infinitesimal; Archimedes' use of infinitesimals; For further developments: see list of real analysis topics, list of complex analysis topics, list of multivariable calculus topics
The theorem is a generalization of the second fundamental theorem of calculus to any curve in a plane or space (generally n-dimensional) rather than just the real line. If φ : U ⊆ R n → R is a differentiable function and γ a differentiable curve in U which starts at a point p and ends at a point q, then
These rules are given in many books, both on elementary and advanced calculus, in pure and applied mathematics. Those in this article (in addition to the above references) can be found in: Mathematical Handbook of Formulas and Tables (3rd edition) , S. Lipschutz, M.R. Spiegel, J. Liu, Schaum's Outline Series, 2009, ISBN 978-0-07-154855-7 .
In fact, calculus and real analysis textbooks often conflate the two, introducing the definition of the Darboux integral as that of the Riemann integral, due to the slightly easier to apply definition of the former. The fundamental theorem of calculus asserts that integration and differentiation are inverse operations in a certain sense.
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