mathematics, Fubini's theorem on differentiation, named after Guido Fubini, is a result in real analysis concerning the differentiation of series of monotonic...

σ-finite. Combining Fubini's theorem with Tonelli's theorem gives the Fubini–Tonelli theorem. Often just called Fubini's theorem, it states that if X...

Guido Fubini (19 January 1879 – 6 June 1943) was an Italian mathematician, known for Fubini's theorem and the Fubini–Study metric. Born in Venice, he was...

algebras) Froda's theorem (mathematical analysis) Frucht's theorem (graph theory) Fubini's theorem (integration) Fubini's theorem on differentiation (real analysis)...

of order of integration (integration under the integral sign; i.e., Fubini's theorem). A Leibniz integral rule for a two dimensional surface moving in three...

with respect to the supremum norm. Morera's theorem can also be used in conjunction with Fubini's theorem and the Weierstrass M-test to show the analyticity...

In Riemannian geometry, the sphere theorem, also known as the quarter-pinched sphere theorem, strongly restricts the topology of manifolds admitting metrics...

is differentiable almost everywhere, provided that p > n. Calderón's theorem is a relatively direct corollary of the Lebesgue differentiation theorem and...

positive if F is positive. The equality above is a simple case of Fubini's theorem, involving no measure theory. Titchmarsh (1939) proves it in a straightforward...

called indefinite integrals. The fundamental theorem of calculus relates definite integration to differentiation and provides a method to compute the definite...

partial derivatives, differentiation under the integral sign, and Fubini's theorem deal with the interchange of differentiation and integration operators...

integrands under certain conditions. This property is popularly known as Fubini's theorem. In the case of T ⊆ R 2 {\displaystyle T\subseteq \mathbb {R} ^{2}}...

convergence theorem Fatou's lemma Absolutely continuous Uniform absolute continuity Total variation Radon–Nikodym theorem Fubini's theorem Double integral...

follows from Fubini's theorem. The same result holds if f and g are only assumed to be nonnegative measurable functions, by Tonelli's theorem. In the one-variable...

calculate areas and volumes of regions in the plane and in space. Fubini's theorem guarantees that a multiple integral may be evaluated as a repeated...

{\displaystyle f,g,\phi \in {\mathcal {D}}(\mathbb {R} ^{n}),} then by Fubini's theorem ⟨ C f g , ϕ ⟩ = ∫ R n ϕ ( x ) ∫ R n f ( x − y ) g ( y ) d y d x = ⟨...

a choice of orientation forms on M and N defines an orientation of every fiber of f. The analog of Fubini's theorem is as follows. As before, M and...

function theorems) and Riemann integration (including Fubini's theorem) to functions of several variables, the book treats the classical theorems of vector...

region of absolute convergence: this is a consequence of Fubini's theorem and Morera's theorem. Similarly, the set of values for which F(s) converges (conditionally...

basics of the Lebesgue theory, but does not treat material such as Fubini's theorem. Rudin, Walter (1966). Real and complex analysis. New York: McGraw-Hill...

Simon Brendle and Richard Schoen in 2009 to give a proof of the differentiable sphere theorem, which had been a major conjecture in Riemannian geometry since...

transforms iterated integrals (or multiple integrals through the use of Fubini's theorem) of functions into other, hopefully simpler, integrals by changing...

next theorem allows us to compute the integral of a function as the iteration of the integrals of the function in one-variables: Fubini's theorem — If...

example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature and parallel transport. Any smooth surface...

f(x,y)\,dx\,dy.} In general, although these two can be different, Fubini's theorem states that under specific conditions, they are equivalent. The alternative...

Schoen 2009). Their convergence theorem included as a special case the resolution of the differentiable sphere theorem, which at the time had been a long-standing...

f on (a,b) the function Iα f on (a,b) which is also integrable by Fubini's theorem. Thus Iα defines a linear operator on L1(a,b): I α : L 1 ( a , b )...

s_{i}=s_{j}\}} is contained in a hyperplane, hence by an application of Fubini's theorem its measure with respect to the n-fold product of μ is zero. Since...

was to make sheaf cohomology groups vanish, on the other hand, the Grauert–Riemenschneider vanishing theorem is known as a similar result for compact complex...

circumstances the order of integration is significant in this result (contrast Fubini's theorem). As justified using the theory of distributions, the Cauchy equation...