Modular Lie Algebras

... usually write in the standard shorthand notation a ∗ b or even ab instead of the long notation µ(a, b). To an element a ∈ A one can associate two linear operators La , Ra : A → A where La (b) = ab and Ra (b) = ba. The names of this operators are intuitive: La is the left multiplication operator and ...

... usually write in the standard shorthand notation a ∗ b or even ab instead of the long notation µ(a, b). To an element a ∈ A one can associate two linear operators La , Ra : A → A where La (b) = ab and Ra (b) = ba. The names of this operators are intuitive: La is the left multiplication operator and ...

Introduction to Tensor Calculus

... Before we start with the main topic of this booklet, tensors, we will first introduce a new notation for vectors and matrices, and their algebraic manipulations: the index notation. It will prove to be much more powerful than the standard vector notation. To clarify this we will translate all well-k ...

... Before we start with the main topic of this booklet, tensors, we will first introduce a new notation for vectors and matrices, and their algebraic manipulations: the index notation. It will prove to be much more powerful than the standard vector notation. To clarify this we will translate all well-k ...

Section 13.1 Vectors in the Plane

... Chapter Preview We now make a significant departure from previous chapters by stepping out of the x y-plane R2 into three-dimensional space R3 . The fundamental concept of a vector—a quantity with magnitude and direction—is introduced in two and three dimensions. We then put vectors in motion by int ...

... Chapter Preview We now make a significant departure from previous chapters by stepping out of the x y-plane R2 into three-dimensional space R3 . The fundamental concept of a vector—a quantity with magnitude and direction—is introduced in two and three dimensions. We then put vectors in motion by int ...

Vector Spaces and Operators

... In quantum mechanics the state of a physical system is a vector in a complex vector space. Observables are linear operators, in fact, Hermitian operators acting on this complex vector space. The purpose of this chapter is to learn the basics of vector spaces, the structures that can be built on thos ...

... In quantum mechanics the state of a physical system is a vector in a complex vector space. Observables are linear operators, in fact, Hermitian operators acting on this complex vector space. The purpose of this chapter is to learn the basics of vector spaces, the structures that can be built on thos ...

+ v

... Vector spaces in which the scalars are complex numbers are called complex vector spaces, and those in which the scalars must be real are called real vector spaces. ...

... Vector spaces in which the scalars are complex numbers are called complex vector spaces, and those in which the scalars must be real are called real vector spaces. ...

Vector Spaces

... These are essentially the same properties enjoyed by geometric vectors and algebraic or coordinate vectors. Actually, functions have more properties: you can multiply them, diﬀerentiate them, and so on. But many properties of functions just rely on addition and scalar multiplication. Polynomials beh ...

... These are essentially the same properties enjoyed by geometric vectors and algebraic or coordinate vectors. Actually, functions have more properties: you can multiply them, diﬀerentiate them, and so on. But many properties of functions just rely on addition and scalar multiplication. Polynomials beh ...

Representing Rotations and Orientations in Geometric Computing

... that a fixed axis v̂ and an angle θ can always be found such that the rotation of the white bunny about axis v̂ by angle θ will bring it to the orientation of the blue bunny. a negative angle denotes a clockwise rotation. A scalar value larger than 2π or smaller than −2π denotes multiple spinning of ...

... that a fixed axis v̂ and an angle θ can always be found such that the rotation of the white bunny about axis v̂ by angle θ will bring it to the orientation of the blue bunny. a negative angle denotes a clockwise rotation. A scalar value larger than 2π or smaller than −2π denotes multiple spinning of ...

Review of Matrix Algebra

... 5) Physical meaning? Projection of one directional quantity (vector, second order tensor, or higher tensor) on to another directional quantity (vector, second order tensor, or higher tensor) ...

... 5) Physical meaning? Projection of one directional quantity (vector, second order tensor, or higher tensor) on to another directional quantity (vector, second order tensor, or higher tensor) ...

# Bivector

In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered an order zero quantity, and a vector is an order one quantity, then a bivector can be thought of as being of order two. Bivectors have applications in many areas of mathematics and physics. They are related to complex numbers in two dimensions and to both pseudovectors and quaternions in three dimensions. They can be used to generate rotations in any dimension, and are a useful tool for classifying such rotations. They also are used in physics, tying together a number of otherwise unrelated quantities.Bivectors are generated by the exterior product on vectors: given two vectors a and b, their exterior product a ∧ b is a bivector, as is the sum of any bivectors. Not all bivectors can be generated as a single exterior product. More precisely, a bivector that can be expressed as an exterior product is called simple; in up to three dimensions all bivectors are simple, but in higher dimensions this is not the case. The exterior product is antisymmetric, so b ∧ a is the negation of the bivector a ∧ b, producing the opposite orientation, and a ∧ a is the zero bivector.Geometrically, a simple bivector can be interpreted as an oriented plane segment, much as vectors can be thought of as directed line segments. The bivector a ∧ b has a magnitude equal to the area of the parallelogram with edges a and b, has the attitude of the plane spanned by a and b, and has orientation being the sense of the rotation that would align a with b.