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The Poincaré lemma states that if B is an open ball in R n, any closed p-form ω defined on B is exact, for any integer p with 1 ≤ p ≤ n. [ 1 ] More generally, the lemma states that on a contractible open subset of a manifold (e.g., R n {\displaystyle \mathbb {R} ^{n}} ), a closed p -form, p > 0, is exact.
The formula may be derived by applying repeated integration by parts to successive intervals [r, r + 1] for r = m, m + 1, …, n − 1. The boundary terms in these integrations lead to the main terms of the formula, and the leftover integrals form the remainder term.
Let U be an open set in R n. A differential 0-form ("zero-form") is defined to be a smooth function f on U – the set of which is denoted C ∞ (U). If v is any vector in R n, then f has a directional derivative ∂ v f, which is another function on U whose value at a point p ∈ U is the rate of change (at p) of f in the v direction:
Name Dim Equation Applications Bateman-Burgers equation: 1+1 + = Fluid mechanics Benjamin–Bona–Mahony: 1+1 + + = Fluid mechanics Benjamin–Ono: 1+1 + + = internal waves in deep water
In other words, a volume form gives rise to a measure with respect to which functions can be integrated by the appropriate Lebesgue integral. The absolute value of a volume form is a volume element, which is also known variously as a twisted volume form or pseudo-volume form. It also defines a measure, but exists on any differentiable manifold ...
The main idea is to express an integral involving an integer parameter (e.g. power) of a function, represented by I n, in terms of an integral that involves a lower value of the parameter (lower power) of that function, for example I n-1 or I n-2. This makes the reduction formula a type of recurrence relation. In other words, the reduction ...
However, the important thing to note is that z 1/2 = e (Log z)/2, so z 1/2 has a branch cut. This affects our choice of the contour C . Normally the logarithm branch cut is defined as the negative real axis, however, this makes the calculation of the integral slightly more complicated, so we define it to be the positive real axis.
The quadratic formula =. is a closed form of the solutions to the general quadratic equation + + =. More generally, in the context of polynomial equations, a closed form of a solution is a solution in radicals; that is, a closed-form expression for which the allowed functions are only n th-roots and field operations (+,,, /).