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In algebra, a multilinear polynomial [1] is a multivariate polynomial that is linear (meaning affine) in each of its variables separately, but not necessarily simultaneously. It is a polynomial in which no variable occurs to a power of 2 {\displaystyle 2} or higher; that is, each monomial is a constant times a product of distinct variables.
Here, properties of tensors, including high-rank moments as well, are considered to repeat basically features of solid spherical functions but having their own specifics. Using of invariant polynomial tensors in Cartesian coordinates , as shown in a number of recent studies, is preferable and simplifies the fundamental scheme of calculations ...
The Cartesian labels are replaced by tensor indices in the basis vectors e x ↦ e 1, e y ↦ e 2, e z ↦ e 3 and coordinates a x ↦ a 1, a y ↦ a 2, a z ↦ a 3. In general, the notation e 1, e 2, e 3 refers to any basis, and a 1, a 2, a 3 refers to the corresponding coordinate system; although here they are restricted to the Cartesian ...
The tensor product of two vector spaces is a vector space that is defined up to an isomorphism.There are several equivalent ways to define it. Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.
In mathematics, the tensor algebra of a vector space V, denoted T(V) or T • (V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product.It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing V, in the sense of the corresponding universal property ...
A general 3d rotation of a vector a, about an axis in the direction of a unit vector ω and anticlockwise through angle θ, can be performed using Rodrigues' rotation formula in the dyadic form a r o t = R ⋅ a , {\displaystyle \mathbf {a} _{\mathrm {rot} }=\mathbf {R} \cdot \mathbf {a} \,,}
where the Cartesian coordinates x, y, z are measured in m. If the units of length are changed into cm , then the numerical values of the coordinate functions must be rescaled by a factor of 100: x ′ = 100 x , y ′ = 100 y , z ′ = 100 z . {\displaystyle x'=100x,\quad y'=100y,\quad z'=100z.}
The real part of the other side is a polynomial in cos x and sin x, in which all powers of sin x are even and thus replaceable through the identity cos 2 x + sin 2 x = 1. By the same reasoning, sin nx is the imaginary part of the polynomial, in which all powers of sin x are odd and thus, if one factor of sin x is factored out, the remaining ...