This article is about tensors on a single vector space. A vector is represented as a 1-dimensional array in a basis, and is a 1st-order tensor. Tensor analysis pdf free download are single numbers and are thus 0th-order tensors. Because they express a relationship between vectors, tensors themselves must be independent of a particular choice of basis.
The total order of a tensor is the sum of these two numbers. In some areas, tensor fields are so ubiquitous that they are simply called “tensors”. Although seemingly different, the various approaches to defining tensors describe the same geometric concept using different languages and at different levels of abstraction. Whether an index is displayed as a superscript or subscript depends on the transformation properties of the tensor, described below. The components of a more general tensor transform by some combination of covariant and contravariant transformations, with one transformation law for each index.
Combinations of covariant and contraviant components with the same index allow us to express geometric invariants. Similarly, a linear operator, viewed as a geometric object, does not actually depend on a basis: it is just a linear map that accepts a vector as an argument and produces another vector. The transformation law for the how the matrix of components of a linear operator changes with the basis is consistent with the transformation law for a contravariant vector, so that the action of a linear operator on a contravariant vector is represented in coordinates as the matrix product of their respective coordinate representations. Here the primed indices denote components in the new coordinates, and the unprimed indices denote the components in the old coordinates. The terms “order”, “type”, “rank”, “valence”, and “degree” are all sometimes used for the same concept.
The definition of a tensor as a multidimensional array satisfying a transformation law traces back to the work of Ricci. A downside to the definition of a tensor using the multidimensional array approach is that it is not apparent from the definition that the defined object is indeed basis independent, as is expected from an intrinsically geometric object. Although it is possible to show that transformation laws indeed ensure independence from the basis, sometimes a more intrinsic definition is preferred. A different choice of basis will yield different components. This motivates viewing multilinear maps as the intrinsic objects underlying tensors. For some mathematical applications, a more abstract approach is sometimes useful.
In principle, one could define a “tensor” simply to be an element of any tensor product. In many applications, especially in differential geometry and physics, it is natural to consider a tensor with components that are functions of the point in a space. This was the setting of Ricci’s original work. This table shows important examples of tensors on vector spaces and tensor fields on manifolds.
Symmetrically, lowering an index corresponds to moving diagonally up and to the right on the table. Orientation defined by an ordered set of vectors. Reversed orientation corresponds to negating the exterior product. There are several notational systems that are used to describe tensors and perform calculations involving them. Several distinct pairs of indices may be summed this way. It is independent of basis elements, and requires no symbols for the indices. This notation captures the expressiveness of indices and the basis-independence of index-free notation.
There are several operations on tensors that again produce a tensor. On components, these operations are simply performed component-wise. When described as multilinear maps, the tensor product simply multiplies the two tensors, i. On components, the effect is to multiply the components of the two input tensors pairwise, i. It thereby reduces the total order of a tensor by two. The operation is achieved by summing components for which one specified contravariant index is the same as one specified covariant index to produce a new component.