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The number of simple bivectors needed to form a general bivector rises with the dimension, so for n odd it is (n − 1) / 2, for n even it is n / 2. So for four and five dimensions only two simple bivectors are needed but three are required for six and seven dimensions.
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.
Scofield Reference Bible, page 1115. This page includes Scofield's note on John 1:17. The Scofield Bible had several innovative features. Most important, it printed what amounted to a commentary on the biblical text alongside the Bible instead of in a separate volume, the first to do so in English since the Geneva Bible (1560). [2]
The term "Bible" can refer to the Hebrew Bible or the Christian Bible, which contains both the Old and New Testaments. [2]The English word Bible is derived from Koinē Greek: τὰ βιβλία, romanized: ta biblia, meaning "the books" (singular βιβλίον, biblion). [3]
A bivector is an element of the antisymmetric tensor product of a tangent space with itself. In geometric algebra, also, a bivector is a grade 2 element (a 2-vector) resulting from the wedge product of two vectors, and so it is geometrically an oriented area, in the same way a vector is an oriented line segment.
A Dictionary of the Bible (1863), edited by William Smith, title page for the third volume. A Bible dictionary is a reference work containing encyclopedic entries related to the Bible, typically concerning people, places, customs, doctrine and Biblical criticism. Bible dictionaries can be scholarly or popular in tone.
The torque or curl is then a normal vector field in this 3rd dimension. By contrast, geometric algebra in 2 dimensions defines these as a pseudoscalar field (a bivector), without requiring a 3rd dimension. Similarly, the scalar triple product is ad hoc, and can instead be expressed uniformly using the exterior product and the geometric product.