What is a Tensor and how can one formulate a Tensor for application to physical systems and quantities? Explain it in detail along with mathematical expressions and equations.

Answer 1

In physics and mathematics, a tensor is a mathematical object that describes a geometric entity in a vector space, including physical quantities. Specifically, tensors are multi-dimensional arrays of numbers that transform in a specific way under coordinate transformations. They provide a concise mathematical representation of physical systems and quantities, allowing us to analyze their behavior under various transformations and operations.

Formally, a tensor of rank (or order) n on a vector space V can be defined as a multilinear map that takes n vectors from V as inputs and produces a scalar (or number) as output. However, it is often more practical to describe tensors as multi-dimensional arrays, where each element of the array corresponds to a specific component of the tensor.

Let's consider an example of a second-order tensor, also known as a matrix. A second-order tensor describes a linear transformation between two vector spaces, often represented by rows and columns of numbers. For instance, let's denote a second-order tensor as T, and its components as T_{ij}, where i and j range from 1 to n (the dimensionality of the vector space).

A common representation of a tensor is using the Einstein summation convention, which states that if an index appears as both a subscript and a superscript within a single term, it implies summation over that index. Using this convention, we can express the components of a second-order tensor as follows:

T = T_{ij}

Where i and j range from 1 to n, indicating n^2 components in total. These components represent the transformation properties of the tensor under coordinate transformations.

To understand how tensors transform under coordinate transformations, we need to introduce the concept of basis vectors and dual vectors. Basis vectors define the directions in which a tensor can operate, while dual vectors represent the components of a tensor when acted upon by a basis vector.

For example, let's consider a second-order tensor in three-dimensional Euclidean space. The components with respect to the Cartesian coordinate system can be denoted as T_{ij}. Now, if we change to a new coordinate system with basis vectors denoted by e'_i, the components of the tensor in the new coordinate system would be T'_{ij}.

The transformation of the components of the tensor can be expressed using the transformation rule:

T'_{ij} = A_i^k A_j^l T_{kl}

Where A_i^k represents the transformation matrix for the contravariant tensor components and A_j^l represents the transformation matrix for the covariant tensor components. The summation conventions apply to the indices k and l.

This transformation rule ensures that the tensor maintains its physical meaning and behaves correctly under coordinate transformations. By studying these transformation rules, we can derive the properties and behavior of tensors under various operations and transformations.

In summary, tensors are mathematical objects that provide a concise and abstract representation of physical systems and quantities. They are multi-dimensional arrays of numbers with specific transformation properties under coordinate transformations. By formulating tensors using appropriate mathematical expressions and equations, we can analyze the behavior of physical systems and quantities and derive meaningful insights from them.