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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 ...
As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge. The divergence of a tensor field T {\displaystyle \mathbf {T} } of non-zero order k is written as div ( T ) = ∇ ⋅ T {\displaystyle \operatorname {div} (\mathbf {T} )=\nabla \cdot \mathbf {T} } , a contraction of a tensor field ...
The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
The cross product of two vectors u and v would be represented as: By some conventions (e.g. in France and in some areas of higher mathematics), this is also denoted by a wedge, [ 13 ] which avoids confusion with the wedge product since the two are functionally equivalent in three dimensions: u ∧ v {\displaystyle \mathbf {u} \wedge \mathbf {v} }
The tensor product of V and its dual space is isomorphic to the space of linear maps from V to V: a dyadic tensor vf is simply the linear map sending any w in V to f(w)v. When V is Euclidean n-space, we can use the inner product to identify the dual space with V itself, making a dyadic tensor an elementary tensor product of two vectors in ...
If imagined as a parallelogram, with the origin for the vectors at 0, then signed area is the determinant of the vectors' Cartesian coordinates (a x b y − b x a y). [21] The cross product a × b is orthogonal to the bivector a ∧ b. In three dimensions all bivectors can be generated by the exterior product of two vectors.
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, . [1] The term vector calculus is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration.
It forms a loop in the first quadrant with a double point at the origin and asymptote + + =. It is symmetrical about the line y = x {\displaystyle y=x} . As such, the two intersect at the origin and at the point ( 3 a / 2 , 3 a / 2 ) {\displaystyle (3a/2,3a/2)} .