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Producer surplus, or producers' surplus, is the amount that producers benefit by selling at a market price that is higher than the least that they would be willing to sell for; this is roughly equal to profit (since producers are not normally willing to sell at a loss and are normally indifferent to selling at a break-even price).
The integral version of Gauss's equation can thus be rewritten as = Since Ω is arbitrary (e.g. an arbitrary small ball with arbitrary center), this is satisfied if and only if the integrand is zero everywhere. This is the differential equations formulation of Gauss equation up to a trivial rearrangement.
This equation is a special form of the more general weakly singular Volterra integral equation of the first kind, called Abel's integral equation: [7] = Strongly singular: An integral equation is called strongly singular if the integral is defined by a special regularisation, for example, by the Cauchy principal value.
The EFIE describes a radiated field E given a set of sources J, and as such it is the fundamental equation used in antenna analysis and design. It is a very general relationship that can be used to compute the radiated field of any sort of antenna once the current distribution on it is known.
Under steady constant frequency conditions we get from the two curl equations the Maxwell's equations for the Time-Periodic case: = , = + . It must be recognized that the symbols in the equations of this article represent the complex multipliers of e j ω t {\displaystyle e^{j\omega t}} , giving the in-phase and out-of-phase parts with respect ...
Integration by parts is a heuristic rather than a purely mechanical process for solving integrals; given a single function to integrate, the typical strategy is to carefully separate this single function into a product of two functions u(x)v(x) such that the residual integral from the integration by parts formula is easier to evaluate than the ...
Thus we conclude that the operations of integration and differentiation of a Grassmann number are identical. In the path integral formulation of quantum field theory the following Gaussian integral of Grassmann quantities is needed for fermionic anticommuting fields, with A being an N × N matrix:
The integral form of such equations is usually the physically more natural formulation, and the differential equation arises from differentiation. Since the integral equation can also have non-differentiable solutions, the equality of both formulations can break down in some cases, leading to weak solutions and severe numerical difficulties in ...