half-angle identities

... 11-5 Half-Angle Identities You can derive the double-angle identities for cosine and tangent in the same way. There are three forms of the identity for cos 2θ, which are derived by using sin2θ + cos2θ = 1. It is common to rewrite expressions as functions of θ only. ...

... 11-5 Half-Angle Identities You can derive the double-angle identities for cosine and tangent in the same way. There are three forms of the identity for cos 2θ, which are derived by using sin2θ + cos2θ = 1. It is common to rewrite expressions as functions of θ only. ...

Hopf algebras

... 1. The category Set whose objects are sets, and where the set of morphisms between two sets is given by all mappings between those sets. 2. Let k be a commutative ring, then Mk denotes the category with as objects all (right) kmodules, and with as morphisms between two k-modules all k-linear mapping ...

... 1. The category Set whose objects are sets, and where the set of morphisms between two sets is given by all mappings between those sets. 2. Let k be a commutative ring, then Mk denotes the category with as objects all (right) kmodules, and with as morphisms between two k-modules all k-linear mapping ...

The Product Topology

... Product Topology in terms of Basis I The basis B that as we used to define the product topology is relatively large since we obtain it by pairing up every open set U in X with every open set V in Y . ...

... Product Topology in terms of Basis I The basis B that as we used to define the product topology is relatively large since we obtain it by pairing up every open set U in X with every open set V in Y . ...

The Classification of Three-dimensional Lie Algebras

... details on constructing a special bilinear form on a simple Lie algebra (the importance of which will become apparent), and section 7 completes the classification of threedimensional Lie algebras by classifying them over fields of characteristic two. The final section, section 8, merely compiles the ...

... details on constructing a special bilinear form on a simple Lie algebra (the importance of which will become apparent), and section 7 completes the classification of threedimensional Lie algebras by classifying them over fields of characteristic two. The final section, section 8, merely compiles the ...

School Plan

... - Using Units of Measurement - Using Units of Measurement o ACMMG137 – Solve problems using length & area o ACMMG139 – Interpret & use timetables Measurement & Geometry Measurement & Geometry ...

... - Using Units of Measurement - Using Units of Measurement o ACMMG137 – Solve problems using length & area o ACMMG139 – Interpret & use timetables Measurement & Geometry Measurement & Geometry ...

The Fourier Algebra and homomorphisms

... So why is A(G) an algebra? I want to build a bit of theory here. Define a map ∆ : C[G] → C[G] ⊗ C[G] = C[G × G] by ∆(s) = s ⊗ s, and extend by linearity. Then ∆ is a homomorphism, and also (∆ ⊗ ι)∆ = (ι ⊗ ∆)∆, so ∆ is co-associative. Actually ∆ gives a isometry Cr∗ (G) → Cr∗ (G × G). (This is automa ...

... So why is A(G) an algebra? I want to build a bit of theory here. Define a map ∆ : C[G] → C[G] ⊗ C[G] = C[G × G] by ∆(s) = s ⊗ s, and extend by linearity. Then ∆ is a homomorphism, and also (∆ ⊗ ι)∆ = (ι ⊗ ∆)∆, so ∆ is co-associative. Actually ∆ gives a isometry Cr∗ (G) → Cr∗ (G × G). (This is automa ...

Applying Universal Algebra to Lambda Calculus

... Barendregt’s book [4]). At the beginning researchers have focused their interest on a limited number of equational extensions of lambda calculus, called λ-theories. They arise by syntactical or semantic considerations. Indeed, a λ-theory may correspond to a possible operational semantics of lambda c ...

... Barendregt’s book [4]). At the beginning researchers have focused their interest on a limited number of equational extensions of lambda calculus, called λ-theories. They arise by syntactical or semantic considerations. Indeed, a λ-theory may correspond to a possible operational semantics of lambda c ...

# Geometric algebra

A geometric algebra (GA) is a Clifford algebra of a vector space over the field of real numbers endowed with a quadratic form. The term is also sometimes used as a collective term for the approach to classical, computational and relativistic geometry that applies these algebras. The Clifford multiplication that defines the GA as a unital ring is called the geometric product. Taking the geometric product among vectors can yield bivectors, trivectors, or general n-vectors. The addition operation combines these into general multivectors, which are the elements of the ring. This includes, among other possibilities, a well-defined formal sum of a scalar and a vector.Geometric algebra is distinguished from Clifford algebra in general by its restriction to real numbers and its emphasis on its geometric interpretation and physical applications. Specific examples of geometric algebras applied in physics include the algebra of physical space, the spacetime algebra, and the conformal geometric algebra. Geometric calculus, an extension of GA that incorporates differentiation and integration can be used to formulate other theories such as complex analysis, differential geometry, e.g. by using the Clifford algebra instead of differential forms. Geometric algebra has been advocated, most notably by David Hestenes and Chris Doran, as the preferred mathematical framework for physics. Proponents claim that it provides compact and intuitive descriptions in many areas including classical and quantum mechanics, electromagnetic theory and relativity. GA has also found use as a computational tool in computer graphics and robotics.The geometric product was first briefly mentioned by Hermann Grassmann, who was chiefly interested in developing the closely related exterior algebra, which is the geometric algebra of the trivial quadratic form. In 1878, William Kingdon Clifford greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chose to call them ""geometric algebras""). For several decades, geometric algebras went somewhat ignored, greatly eclipsed by the vector calculus then newly developed to describe electromagnetism. The term ""geometric algebra"" was repopularized by Hestenes in the 1960s, who recognized its importance to relativistic physics.