Line integral approximation of double integrals

On the Ostrowski Type Integral Inequality for Double Integrals

Line integrals and physical optics . Part II . The conversion ita of the Kirchhoff surface integral to a line integral

A new approach is presented for converting the surface integral, representing the Kirchhoff diffracted field of an aperture on a plane screen, to a line integral. It has the advantages that it is mathematically rigorous and explicit and that it results in a representation that has exactly the same properties as the original Kirchhoff formula, i.e., it admits arbitrary source distributions and i...

Line integrals and physical optics. Part I. The transformation of the solid-angle surface integral to a line integral

The surface integral that measures the solid angle subtended by a planar aperture at a point in space is transformed to a line integral over the boundary of the aperture. The problem is divided into three distinct cases, and in each case a transformation is established. The results have a direct application to the problem of converting the Kirchhoff integral of diffraction by an aperture on a p...

Double Integrals and Iterated Integrals

Corresponding material in the book: Section 15.2, 15.3. Note: We are omitting the question types from the book that require three-dimensional visualization, i.e., those that require sketching figures in three dimensions to compute volumes. What students should definitely get: The procedure for computing double integrals over rectangles using iterated integrals, the procedure for computing doubl...

Statistical Approximation by Double Picard Singular Integral Operators

We first construct a sequence of double smooth Picard singular integral operators which do not have to be positive in general. After giving some useful estimates, we mainly show that it is possible to approximate a function by these operators in statistical sense even though they do not obey the positivity condition of the statistical Korovkin theory.