First-order tensor

Field variables identified by their magnitude and two associated directions are called second-order tensors (by analogy a scalar is said to be a zero-order tensor and a vector is a first-order tensor). An important example of a second-order tensor is the physical function stress which is a surface force identified by magnitude, direction and orientation of the surface upon which it is acting. Using a mathematical approach a second-order Cartesian tensor is defined as an entity having nine components T/j, i, j = 1, 2, 3, in the Cartesian coordinate system of ol23 which on rotation of the system to ol 2 3 become... 

Matrix and tensor notation is useful when dealing with systems of equations. Matrix theory is a straightforward set of operations for linear algebra and is covered in Section A.I. Tensor notation, treated in Section A.2, is a classification scheme in which the complexity ranges upward from scalars (zero-order tensors) and vectors (first-order tensors) through second-order tensors and beyond. 

We have considered scalar, vector, and matrix molecular properties. A scalar is a zero-dimensional array a vector is a one-dimensional array a matrix is a two-dimensional array. In general, an 5-dimensional array is called a tensor of rank (or order) s a tensor of order s has ns components, where n is the number of dimensions of the coordinate system (usually 3). Thus the dipole moment is a first-order tensor with 31 = 3 components the polarizability is a second-order tensor with 32 = 9 components. The molecular first hyperpolarizability (which we will not define) is a third-order tensor. 

One of the outstanding characteristic features of the modem analytical instrumentation is the ability of providing analytical signals in the form of different order tensors, i.e. vectors (the first order tensors), matrices (the second order tensors) and even higher order tensors. The multivariate data supplied by such instruments embody more information than the traditional univariate signal and are more suitable for the qualitative and quantitative analysis. 

The algebra of multi-way arrays is described in a field of mathematics called tensor analysis, which is an extension and generalization of matrix algebra. A zero-order tensor is a scalar a first-order tensor is a vector a second-order tensor is a matrix a third-order tensor is a three-way array a fourth-order tensor is a four-way array and so on. The notions of addition, subtraction and multiplication of matrices can be generalized to multi-way arrays. This is shown in the following sections [Borisenko Tarapov 1968, Budiansky 1974],... 

The cross-product can be expressed with the help of the first-order tensor product... 

The other components of the first-order tensor product can be obtained by a similar procedure hence... 

Ordered sets of numbers may collectively be called tensors, with a matrix being a second-order tensor, and a vector a first-order tensor. Since tensors of order higher than two are relatively rare, the terms vector and matrix are more commonly used. Sometimes it is also convenient to consider a vector asalxAioriVxl matrix, and a scalar as a 1 x 1 matrix. 

Vectors. Velocity, force, momentum, and acceleration are considered vectors since they have magnitude and direction. They are regarded as first-order tensors and are written in boldface letters in this text, such as v for velocity. The addition of the two vectors B -i- C by parallelogram construction and the subtraction of two vectors B — C is shown in Fig. 3.6-1. The vector B is represented by its three projectionsB, By, and Bj on the x, y, and z axes and... 

Note 2 According to the definition of pth order tensor, scalars are zero-order tensors and vectors are first-order tensors. 

There are different ways to write down tensors and their components, whose usefulness depends on the context. If we talk about the tensor itself, independent of a coordinate system, we use the symbolic notation. Different typographical styles can be found in the literature. In this book, a first-order tensor is underscored once a, b,. ..), a, second-order tensor twice (and usually denoted with a capital letter. A, B,. ..). Higher-order tensors get a tilde... 

The essence of this transformation is that the components of force transform linearly with the turn of the axes, that is, they can be represented by the sums of the initial components, with the coefficients given by the direction cosines to the first power. This establishes the characteristic of the force as a physical variable. We have also defined a vector as a first-order tensor. The first order of the tensor is defined in this case by the first power of the cosines in the transformation equation. A vector can be multidimensional while still remaining a first-order tensor. A scalar can be defined as a tensor of zeroth order. 

It is worth recalling here that each tensor has an order (I, II, III, IV, etc.). Tensor order reflects the physical properties of a tensor and is determined by the power of the direction cosines product, that is, the power of the product of linear transformation coefficients. The tensor order physically reflects the possibility of visualizing the various properties of a field or a body from different viewpoints. Tensor order is also an indicator of the different ways in which spatial anisotropy is revealed. Scalar quantities, that is, temperature, mass, and amount of heat, are zeroth-order tensors the vectors of velocity or force are the first-order tensors mechanical stresses and strains are second-order tensors, while the elasticity modulus is a fourth-order tensor, as will be shown in the following text. 

At the same time, when a physical property is represented by a tensor of a given order, it can be characterized by a particular number of components in a given space. For example, the three spatial components and time form a 4D field vector, which is a first-order tensor. Another noteworthy example is that of the fonrth-order elasticity tensor, which in an isotropic medium is degenerated into two scalar quantities the Young s modulus and the Poisson ratio. 

An important advantage of multivariate calibration over univariate calibration is that, because many measurements are obtained from the same solution, the signal from the analytes and that from the interferences can be separated mathematically, so concentrations can be determined without the need for highly selective measurements for the analyte. This advantage has been termed the first-order advantage, and eqn (4.3) is also called the first-order calibration model. The term first-order means that the response from a test specimen is a vector (a first-order tensor). This nomenclature and the advantages of first-order calibration have been well described in the theory of analytical chemistry. To use this advantage, however, there is one major requirement the multivariate measurements of the calibrators must contain... 

Free index non-repeated subscripts are called free subscripts since they are free to take on any value in 3D space. The count of the free indices on a variable indicates the order of the tensor, e.g. Fj is a vector (first order tensor), Oy is a second order tensor. 

W.S. Tong, C.K. Tang, and G. Medioni (2001) First order tensor voting, and application to 3-D scale analysis. Proc. of Computer Vision and Pattern Recognition, 175-182. 

---