Elementary Calculus: Infinite Sum Theorem

INFINITE SUM THEOREM

Let h(x, y) be continuous on an open region D₀ and let B be a function which assigns a real number B(D) to each region D contained in D₀. Assume that

(i) B has the Addition Property B(D) = B(D₁) + B(D₂).

(ii) B(D) ≥ 0 for every D.

(iii) For every element of area ΔD, B(ΔD) ≈ h(x, y) ΔA (compared to ΔA).

Then

We shall use the notation

ΔB = B(ΔD).

Given (i) and (ii), the theorem shows that if we always have

ΔB ≈ h(x, y) ΔA (compared to ΔA)

then

B(D) ≈ ∑∑_Dh(x, y) ΔA.

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Infinite Sum Theorem INFINITE SUM THEOREM Let h(x, y) be continuous on an open region D₀ and let B be a function which assigns a real number B(D) to each region D contained in D₀. Assume that (i) B has the Addition Property B(D) = B(D₁) + B(D₂). (ii) B(D) ≥ 0 for every D. (iii) For every element of area ΔD, B(ΔD) ≈ h(x, y) ΔA (compared to ΔA). Then We shall use the notation ΔB = B(ΔD). Given (i) and (ii), the theorem shows that if we always have ΔB ≈ h(x, y) ΔA (compared to ΔA) then B(D) ≈ ∑∑_Dh(x, y) ΔA. The proof is simplest in the case that D is a rectangle.
Home Multiple Integrals Infinite Sum Theorem and Volume Infinite Sum Theorem