Information: - In mathematics and vector algebra, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space (R) and is denoted by the symbol . Given two linearly independent vectors a and b, the cross product, a b, is a vector that is perpendicular to both a and b and therefore normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with dot product (projection product). - In mathematics, a block matrix or a partitioned matrix is a matrix that is "interpreted" as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices. Any matrix may be interpreted as a block matrix in one or more ways, with each interpretation defined by how its rows and columns are partitioned. - Tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors. Elementary examples of such relations include the dot product, the cross product, and linear maps. Geometric vectors, often used in physics and engineering applications, and scalars themselves are also tensors. A more sophisticated example is the Cauchy stress tensor T, which takes a direction v as input and produces the stress T on the surface normal to this vector for output, thus expressing a relationship between these two vectors, shown in the figure (right). - In continuum mechanics, the Cauchy stress tensor formula_1, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components formula_2 that completely define the state of stress at a point inside a material in the deformed state, placement, or configuration. The tensor relates a unit-length direction vector n to the stress vector T across an imaginary surface perpendicular to n: - For '' outer product '' in geometric algebra , see Exterior algebra . In linear algebra , the term outer product typically refers to the tensor product of two vectors . The result of applying the outer product to a pair of coordinate vectors is a matrix . The name contrasts with the inner product , which takes as input a pair of vectors and produces a scalar . The outer product of vectors can be also regarded as a special case of the Kronecker product of matrices . Some authors use the expression '' outer product of tensors '' as a synonym of '' tensor product '' . The outer product is also a higher - order function in some computer programming languages such as R , APL , and Mathematica . - In mathematics, the dot product or scalar product (sometimes inner product in the context of Euclidean space, or rarely projection product for emphasizing the geometric significance), is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation can be defined either algebraically or geometrically. Algebraically, it is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. The name "dot product" is derived from the centered dot "" that is often used to designate this operation; the alternative name "scalar product" emphasizes that the result is a scalar (rather than a vector). - In Set theory (and, usually, in other parts of mathematics), a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets. That is, for sets "A" and "B", the Cartesian product is the set of all ordered pairs where and . Products can be specified using set-builder notation, e.g. - In mathematics, the Kronecker product, denoted by , is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to matrices, and gives the matrix of the tensor product with respect to a standard choice of basis. The Kronecker product should not be confused with the usual matrix multiplication, which is an entirely different operation. - Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces. - In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication. - Mathematics (from Greek "mathma", knowledge, study, learning) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. - In mathematics, the tensor product formula_1 of two vector spaces "V" and "W" is itself a vector space, together with an operation of bilinear composition, denoted by formula_2, from ordered pairs in the Cartesian product formula_3 into formula_1, in a way that generalizes the outer product. The tensor product of "V" and "W" is the vector space generated by the symbols formula_5, with formula_6 and formula_7, in which the relations of bilinearity are imposed for the product operation formula_2, "and no other relations" are assumed to hold. The tensor product space is thus the "freest" (or most general) such vector space, in the sense of having the least constraints. - In mathematics, matrix multiplication or the matrix product is a binary operation that produces a matrix from two matrices. The definition is motivated by linear equations and linear transformations on vectors, which have numerous applications in applied mathematics, physics, and engineering. In more detail, if is an matrix and is an matrix, their matrix product is an matrix, in which the entries across a row of are multiplied with the entries down a columns of and summed to produce an entry of . When two linear transformations are represented by matrices, then the matrix product represents the composition of the two transformations. After reading the paragraphs above, we are interested in knowing the entity with which 'outer product' exhibits the relationship of 'part of'. Find the answer from the choices below. Choices: - algebra - applied mathematics - computer - continuum - linear algebra - mathematics - matrix - name - order - physics - set theory - theory - v
The answer to this question is:
mathematics