Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in ${\mathbb{R}}^m$

Depends on

• The lower and upper Darboux integrals over a nondegenerate rectangle in $\mathbb{R}^m$
• Riemann's criterion on a nondegenerate rectangle in $\mathbb{R}^m$: integrability is equivalent to arbitrarily small Darboux gaps
• The multidimensional Darboux and tagged-mesh definitions of the Riemann integral agree
• Refinement raises multidimensional lower sums and lowers upper sums, with a quantitative boundary-slab estimate
• Lower and upper Darboux sums over a grid partition in $\mathbb{R}^m$
• Grid partitions of a rectangle in $\mathbb{R}^m$, their cells, refinements and mesh
• Laws of finite sums and finite products
• Absolute value in an ordered field
• Basic properties of the absolute value
• The reverse triangle inequality

Used by

• A curl-free field has zero circulation around the induced boundary chain of a C² patch Corollary
• A field with vanishing divergence has zero outward flux through the boundary of a glued elementary solid Corollary
• A nonzero-degree map to a connected manifold is surjective Corollary
• Change of variables on bounded open Jordan sets when both integrands are bounded and Riemann integrable Corollary
• Green's second identity on a glued elementary solid region Corollary
• Jordan content is finitely additive when the overlap has content zero Corollary
• The divergence at a point is the limit of outward flux per unit volume Corollary
• The integral of a product function on a product rectangle is the product of the two integrals Corollary
• The normal component of the curl is the limiting circulation per unit area of shrinking discs Corollary
• The volume of a glued elementary solid is a third of the outward flux of the position field Corollary
• De Rham integration cochain Definition
• Improper multiple integrals and absolute convergence on open sets Definition
• A normalized compactly supported top form on Euclidean space Example
• Principal value distribution one over x Example
• Additivity of the integral over finitely many Jordan pieces that fill a Jordan set up to content zero Lemma
• An integrable dominator gives uniform tail control on every compact parameter set Lemma
• Change of variables for a C¹ map injective and regular only on the interior of a compact Jordan set Lemma
• Changing a bounded integrand on a content-zero set does not change its Riemann integral Lemma
• Compactly supported top cohomology propagates across overlapping oriented coordinate balls Lemma
• Finite chart localization gives choice-free integration and compact Stokes Lemma
• Finite Jordan covers bound upper integrals, while interior-disjoint Jordan subfamilies bound lower integrals Lemma
• Finite sums of product tests are dense on product open sets Lemma
• Shared boundary arcs cancel when finitely many elementary regions are glued Lemma
• Stokes theorem for the standard simplex Lemma
• The plane Gaussian integral equals π by polar coordinates Lemma
• The Riemann integral of a compactly supported function is independent of its bounding rectangle Lemma
• The Riemann integral over a Jordan set is independent of the bounding rectangle Lemma
• Zero-integral compactly supported top forms on Euclidean space have compactly supported primitives Lemma
• Linearity and additivity of the form integral Proposition
• Positivity of the oriented integral Proposition
• Absolute convergence makes signed improper multiple integrals independent of exhaustion Theorem
• Change of variables for an injective C¹ map on a compact Jordan set Theorem
• Comparison and absolute comparison tests for improper multiple integrals Theorem
• Differentiation under an improper multiple integral under an integrable derivative bound Theorem
• Every Jordan exhaustion computes a nonnegative improper multiple integral Theorem
• Locally dominated parameter-dependent improper multiple integrals are continuous Theorem
• Orientation-free density integration and its properties Theorem
• Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections Theorem
• The cylindrical-shell formula for a solid of revolution about the y-axis Theorem
• The divergence theorem on an elementary solid region Theorem

…and 1 more result.

Dependency tree · two levels

39 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• J. Lebl, Basic Analysis, Riemann Integral in Several Variables (standard reference, not scraped)
• J. Lebl, Basic Analysis, The Riemann-Lebesgue Criterion (standard reference, not scraped)