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For example, for the tangent bundle of a Riemannian manifold, the structure group is O(n) and Ω is a 2-form with values in the Lie algebra of O(n), i.e. the antisymmetric matrices. In this case the form Ω is an alternative description of the curvature tensor , i.e.
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.
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
It is common to denote the partial derivatives of r with respect to u α by r α, α = 1, 2. Regularity of the parametrization means that r 1 and r 2 are linearly independent for any (u 1,u 2) in the domain of r, and hence span the tangent plane to S at each point. Equivalently, the cross product r 1 × r 2 is a nonzero
We solve the van der Pol oscillator only up to order 2. This method can be continued indefinitely in the same way, where the order-n term ϵ n x n {\displaystyle \epsilon ^{n}x_{n}} consists of a harmonic term a n cos ( t ) + b n cos ( t ) {\displaystyle a_{n}\cos(t)+b_{n}\cos(t)} , plus some super-harmonic terms a n , 2 cos ( 2 t ...
A periodic Bernoulli polynomial P n (x) is a Bernoulli polynomial evaluated at the fractional part of the argument x. These functions are used to provide the remainder term in the Euler–Maclaurin formula relating sums to integrals. The first polynomial is a sawtooth function.
The bottom-up development involves noticing that u = ci XOR (x XOR y), where XOR means eXclusive OR [true when either input is true but not when both are true], and that co = ci x + x y + y ci. One such development takes twelve NOR gates in all: six 2-input gates and two 1-input gates to produce u in 5 gate delays, plus three 2-input gates and ...