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Maxwell's equations are used in physics to describe the relationship between electric and magnetic fields. Normally given as four differential equations they have a particularly compact form when the fields are expressed as a spacetime bivector from ⋀ 2 R 3,1. If the electric and magnetic fields in R 3 are E and B then the electromagnetic ...
The derivatives that appear in Maxwell's equations are vectors and electromagnetic fields are represented by the Faraday bivector F. This formulation is as general as that of differential forms for manifolds with a metric tensor, as then these are naturally identified with r -forms and there are corresponding operations.
the bivector has two distinct principal null directions; in this case, the bivector is called non-null. Furthermore, for any non-null bivector, the two eigenvalues associated with the two distinct principal null directions have the same magnitude but opposite sign, λ = ±ν, so we have three subclasses of non-null bivectors: spacelike: ν = 0
Continuous charge distribution. The volume charge density ρ is the amount of charge per unit volume (cube), surface charge density σ is amount per unit surface area (circle) with outward unit normal n̂, d is the dipole moment between two point charges, the volume density of these is the polarization density P.
conserved, bivector Angular velocity: ω: The angle incremented in a plane by a segment connecting an object and a reference point per unit time rad/s T −1: bivector Area: A: Extent of a surface m 2: L 2: extensive, bivector or scalar Centrifugal force: F c: Inertial force that appears to act on all objects when viewed in a rotating frame of ...
Examples of geometric algebras applied in physics include the spacetime algebra (and the less common algebra of physical space). 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 ...
A two-vector or bivector [1] is a tensor of type () and it is the dual of a two-form, meaning that it is a linear functional which maps two-forms to the real numbers (or more generally, to scalars). The tensor product of a pair of vectors is a two-vector. Then, any two-form can be expressed as a linear combination of tensor products of pairs of ...
Using the Maxwell equations, one can see that the electromagnetic stress–energy tensor (defined above) satisfies the following differential equation, relating it to the electromagnetic tensor and the current four-vector , + = or , + =, which expresses the conservation of linear momentum and energy by electromagnetic interactions.