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From the conjecture and the proof of the fundamental theorem of calculus, calculus as a unified theory of integration and differentiation is started. The first published statement and proof of a rudimentary form of the fundamental theorem, strongly geometric in character, [ 2 ] was by James Gregory (1638–1675).
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 ]
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
In other words, we obtain a simpler and more satisfactory version of the second fundamental theorem of calculus: each differentiable function is, up to a constant, the integral of its derivative: () = ′ ().
Fundamental theorem of calculus II: if the graph is a tree, is a -cochain, and a function (-cochain) is defined on the nodes of the graph by = where a -chain ...
This category has the following 2 subcategories, out of 2 total. ... Pages in category "Theorems in calculus" ... Fubini's theorem; Fundamental theorem of calculus; G.
The fundamental theorem of calculus states that differentiation and integration are inverse operations. [49]: 290 More precisely, it relates the values of antiderivatives to definite integrals. Because it is usually easier to compute an antiderivative than to apply the definition of a definite integral, the fundamental theorem of calculus ...
James Gregory, influenced by Fermat's contributions both to tangency and to quadrature, was then able to prove a restricted version of the second fundamental theorem of calculus, that integrals can be computed using any of a function's antiderivatives. [26] [27] The first full proof of the fundamental theorem of calculus was given by Isaac Barrow.