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...

We show how methods of continuum perturbation theory can be used to simplify perturbative lattice calculations. We use the technique of asymptotic expansions to expand lattice loop integrals around the continuum limit. After the expansion, all nontrivial dependence on momenta and masses is encoded in continuum loop integrals and the only genuine lattice integrals left are tadpole integrals. Usi...

Exact calculations of Feynman’s path integrals (defined on a time lattice) are mainly based on recurrence integral formulas in which the convolution of two functions having a common feature retains the same feature. Therefore, exactly soluble path integrals in quantum mechanics may be classified by their recurrence integral formula used in the calculation. According to this classification, ther...

This paper may be regarded as a new numerical method for the analysis of triangular thin plates using the natural area coordinates. Previous studies on the solution of triangular plates with different boundary conditions are mostly based on the Rayleigh-Ritz principle which is performed in the Cartesian coordinates. Consequently, manipulation of the geometry and numerical calculation of the int...

We develop a class of integrals on a manifold called exponential iterated integrals, an extension of K. T. Chen’s iterated integrals. It is shown that these integrals can be used to compute the matrix entries of certain solvable representations of the fundamental group of the manifold. In particular we find that exponential iterated integrals separate the elements of groups of fibered knots.

We consider coupled weakly birefringent cavities filled-in with nonlinear Kerr material and subject to linearly polarized optical injection. Light propagation in such a system is described by of discrete Lugiato–Lefever-type equations for each linear polarization component the electric field into cavity, cross-phase modulation terms neighboring waveguides overlap integrals. demonstrate that thi...

A half-numeric algorithm for the evaluation of effective core potential integrals over Cartesian Gaussian functions is described. Local and semilocal integrals are separated into two-dimensional angular and one-dimensional radial integrals. The angular integrals are evaluated analytically using a general approach that has no limitation for the l-quantum number. The radial integrals are calculat...