Vector Calculus

We shall point out the source of error, which leads one to define a delta function. We show that a similar argument applies to Stokes' theorem. Also using Stokes' theorem, once more we demonstrate that Maxwell's Equations are source-free. The divergence theorem is given by the following equation, τ (·F)dτ = s F·ds, (1) where τ is the volume bounded by the closed surface s. To simplify the argument, we choose a central field, i.e. F(r) = F (r)ˆ r, (2) wherê r is a unit vector in the direction of r. Substituting (2) in the right-hand side of (1) yields s F·ds = 4πr 2 F (r). (3) Note that the variable r is indefinite, i.e., 0 ≤ r < ∞. Substituting (2) in the left-hand side of (1), and writing the divergence operator in spherical coordinates, yields τ (·F)dτ = r Ω 1 r 2 ∂ ∂r [r 2 F (r)]r 2 drdΩ = r ∂ ∂r [4πr 2 F (r)]dr = 4πr 2 F (r). (4) Note again that the integral with respect to r is indefinite. Comparing (4) to (3) we observe the validity of the divergence theorem for any vector function F(r). We now elaborate on the case of the inverse square field vector, which has zero divergence and it is the subject of confusion. For example, the electric field of a point charge q is, E(r) = qˆr 4πr 2 , (5) which has zero divergence, ·E = 0. (6)

Contemporary Mathematics, 2016

ACM Transactions on Programming Languages and Systems, 2013

Growing interest in graphics processing units has brought renewed attention to the Single Instruction Multiple Data (SIMD) execution model. SIMD machines give application developers tremendous computational power; however, programming them is still challenging. In particular, developers must deal with memory and control-flow divergences. These phenomena stem from a condition that we call data divergence, which occurs whenever two processing elements (PEs) see the same variable name holding different values. This article introduces divergence analysis, a static analysis that discovers data divergences. This analysis, currently deployed in an industrial quality compiler, is useful in several ways: it improves the translation of SIMD code to non-SIMD CPUs, it helps developers to manually improve their SIMD applications, and it also guides the automatic optimization of SIMD programs. We demonstrate this last point by introducing the notion of a divergence-aware register spiller. This sp...