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[3] Suppose that a vector-valued differential 2-form is regarded as assigning to each p a multilinear map s p: T p M × T p M → E p which is completely anti-symmetric. Then the exterior covariant derivative d ∇ s assigns to each p a multilinear map T p M × T p M × T p M → E p given by the formula
The last formula, where summation starts at i = 3, follows easily from the properties of the exterior product. Namely, dx i ∧ dx i = 0. Example 2. Let σ = u dx + v dy be a 1-form defined over ℝ 2. By applying the above formula to each term (consider x 1 = x and x 2 = y) we have the sum
A large body of results from ordinary differential calculus, such as binomial formula and Taylor expansion, have natural q-analogues that were discovered in the 19th century, but remained relatively obscure for a big part of the 20th century, outside of the theory of special functions.
The term differential is used nonrigorously in calculus to refer to an infinitesimal ("infinitely small") change in some varying quantity. For example, if x is a variable, then a change in the value of x is often denoted Δx (pronounced delta x). The differential dx represents an infinitely small change in the variable x. The idea of an ...
The differential was first introduced via an intuitive or heuristic definition by Isaac Newton and furthered by Gottfried Leibniz, who thought of the differential dy as an infinitely small (or infinitesimal) change in the value y of the function, corresponding to an infinitely small change dx in the function's argument x.
The backward differentiation formula (BDF) is a family of implicit methods for the numerical integration of ordinary differential equations.They are linear multistep methods that, for a given function and time, approximate the derivative of that function using information from already computed time points, thereby increasing the accuracy of the approximation.
The Newmark-beta method is a method of numerical integration used to solve certain differential equations.It is widely used in numerical evaluation of the dynamic response of structures and solids such as in finite element analysis to model dynamic systems.
Importantly, the geometry of the domain on which a ¯-closed differential form is ¯-exact is more restricted than for the Poincaré lemma, since the proof of the Dolbeault–Grothendieck lemma holds on a polydisk (a product of disks in the complex plane, on which the multidimensional Cauchy's integral formula may be applied) and there exist ...