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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 ]
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 ...
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: () = ′ ().
In single-variable calculus, the fundamental theorem of calculus establishes a link between the derivative and the integral. The link between the derivative and the integral in multivariable calculus is embodied by the integral theorems of vector calculus: [1]: 543ff Gradient theorem; Stokes' theorem; Divergence theorem; Green's theorem.
Fundamental theorem of algebra (complex analysis) Fundamental theorem of arbitrage-free pricing (financial mathematics) Fundamental theorem of arithmetic (number theory) Fundamental theorem of calculus ; Fundamental theorem on homomorphisms (abstract algebra) Fundamental theorems of welfare economics ; Furry's theorem (quantum field theory)
Stronger versions of the theorem only require that the partial derivative exist almost everywhere, and not that it be continuous. [2] This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus.