Search results
Results from the WOW.Com Content Network
In mathematics, a rotor in the geometric algebra of a vector space V is the same thing as an element of the spin group Spin(V).We define this group below. Let V be a vector space equipped with a positive definite quadratic form q, and let Cl(V) be the geometric algebra associated to V.
One way to define the curl of a vector field at a point is implicitly through its components along various axes passing through the point: if ^ is any unit vector, the component of the curl of F along the direction ^ may be defined to be the limiting value of a closed line integral in a plane perpendicular to ^ divided by the area enclosed, as ...
Geometric calculus, an extension of GA that incorporates differentiation and integration, can be used to formulate other theories such as complex analysis and differential geometry, e.g. by using the Clifford algebra instead of differential forms.
This meaning is somehow inverse to the meaning in the group theory. Rotations of (affine) spaces of points and of respective vector spaces are not always clearly distinguished. The former are sometimes referred to as affine rotations (although the term is misleading), whereas the latter are vector rotations. See the article below for details.
1. Means "much less than" and "much greater than". Generally, much is not formally defined, but means that the lesser quantity can be neglected with respect to the other. This is generally the case when the lesser quantity is smaller than the other by one or several orders of magnitude. 2.
Given a commutative ring A, an A-extension or an extension of an A-algebra is defined in the same way by replacing "ring" with "algebra over A" and "abelian groups" with "A-modules". An extension is said to be trivial or to split if ϕ {\displaystyle \phi } splits; i.e., ϕ {\displaystyle \phi } admits a section that is a ring homomorphism [ 2 ...
The resulting extension remains linear and bounded, and is thus continuous, which makes it a continuous linear extension. This procedure is known as continuous linear extension . Theorem
An extension of A by B is called split if it is equivalent to the trivial extension 0 → B → A ⊕ B → A → 0. {\displaystyle 0\to B\to A\oplus B\to A\to 0.} There is a one-to-one correspondence between equivalence classes of extensions of A by B and elements of Ext 1