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A different technique, which goes back to Laplace (1812), [3] is the following. Let = =. Since the limits on s as y → ±∞ depend on the sign of x, it simplifies the calculation to use the fact that e −x 2 is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity.
Since the flux is defined as an integral of the electric field, this expression of Gauss's law is called the integral form. A tiny Gauss's box whose sides are perpendicular to a conductor's surface is used to find the local surface charge once the electric potential and the electric field are calculated by solving Laplace's equation.
Any inverse-square law can instead be written in a Gauss's law-type form (with a differential and integral form, as described above). Two examples are Gauss's law (in electrostatics), which follows from the inverse-square Coulomb's law, and Gauss's law for gravity, which follows from the inverse-square Newton's law of universal gravitation. The ...
In three dimensions, the derivative has a special structure allowing the introduction of a cross product: = + = + from which it is easily seen that Gauss's law is the scalar part, the Ampère–Maxwell law is the vector part, Faraday's law is the pseudovector part, and Gauss's law for magnetism is the pseudoscalar part of the equation.
The integral / for some real constants a, b and c > 0 can be calculated by putting it into the form of a Gaussian integral.First, the constant a can simply be factored out of the integral.
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.
A common integral is a path integral of the form ((, ˙)) where (, ˙) is the classical action and the integral is over all possible paths that a particle may take. In the limit of small ℏ {\displaystyle \hbar } the integral can be evaluated in the stationary phase approximation .
This can be proved by computing the derivative of the right-hand side of the formula, taking into account that the condition on g is here for insuring the continuity of the integral. This gives the following formulas (where a ≠ 0 ), which are valid over any interval where f is continuous (over larger intervals, the constant C must be replaced ...