Calculus Definitions > What is a Tensor?
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One way to think about tensors is that they are containers that describe data or physical entities in n-dimensions. They can be represented by grids of numbers, called N-way arrays (Pan, 2014).
The rank of a tensor is the number of indices. The first three ranks (also called orders) for tensors (0, 1, 2) are scalar, vector, and matrix. Although these three are technically simple tensors, a mathematical object isn’t usually called a “tensor” unless the rank is 3 or above. There are exceptions. For example, rank 2 tensors (which can be represented by a matrix) hold special importance in many areas of engineering and physics, including electromagnetism, mechanics and quantum theory, because of their practicality; Therefore, instead of calling them matrices, you might hear them referred to as rank 2 tensors instead.
- Rank 0 Tensor: The familiar scalar is the simplest tensor and is a rank 0 tensor. Scalars are just single real numbers like ½, 99 or -1002 that are used to measure magnitude (size). Scalars can technically be written as a one-unit array: [½],  or [-1002], but it’s not usual practice to do so.
- Rank 1 Tensor: Vectors are rank 1 tensors. There are many ways to write vectors, including as an array:
The one-dimensional array for vectors always extends in a downward direction. Rank 1 tensors are usually represented by lowercase bold letters, e.g. u, v, w.
- Rank 2 Tensor: The next level up is a Rank 2 tensor, which can be represented by a matrix. Matrices are rectangular arrays of numbers arranged into columns and rows (similar to a spreadsheet). They have a rank of 2 because of the two-dimensional array:
Rank 2 tensors are usually represented by uppercase bold letters, e.g. U, V, W. More formally, a rank 2 tensor is a mathematical operator that acts on one vector and generates another (Kelly, 2015). For example:
- Rank 3 and Above: When “tensor” is mentioned in texts, it’s usually referring to rank 3 and above. If you view tensors as containers, a rank 3 tensor is one that packs in an additional layer, much in the same way a matrix packs in an additional layer compared to the vector, and the vector packs in an extra layer compared to a scalar. Each step up in ranking means that you’re dealing with an extra layer of information in the container. In geometrical terms, a 3D-Tensor is a cube of numbers:
Rank 3 tensors and higher differ from matrices in one very important aspect. Let’s say you had a tensor located within a container with other mathematical objects. If you transform the entities in the structure, then the tensor obeys the transformation law. Any set of 9 numbers that follow this equation form the components of a tensor (Schnack, 2007):
A′ijkl…. = aip ajq akrais….Apqrs….
An index (plural indices) is a way to organize quantities of numbers, equations, functions and similar objects.
Indices can be written as a superscript (called a raised index) or as a subscript (called a lowered index). For example:
- Raised index: qi,
- Lowered index: qi.
The letter “i” in these examples is the index, and while i is probably the most common letter used as a placeholder, you might see a variety of other letters, including various letters from the Latin or Greek alphabet.
Note: although indices look like exponents, they aren’t. This can be confusing, but you can usually figure out if it’s an exponent by the context. For example, if you see x2 in an equation, like x2 + 2, then it’s an exponent. But if you’re working with tensors and you see that same notation, then it’s an index.
Index placement—the choice of raised or lowered indices—is usually a matter of convention:
- Superscript indices: coordinates, vectors.
- Subscript indices: covariant vectors.
Matrices have two indices, which can be raised or lowered. Tensors have two or more indices, which can also be raised or lowered. For example, the following matrix can be written with index notation as Aij:
There are many sub-conventions here, which are extremely important to follow. For example, the Kronecker delta
forms the components of a tensor. But switch the placement around, like
and you no longer have a tensor.
How Indices Work
The index usually represents a series of positive integers. For example, let’s say you had a series of equations, each with a different variable from 1 to 9. You could list all of the different equations:
But a much simpler way would be to represent that list with an index number,
which tells you that “i” can take on any value (with the understanding here that the variables are from 1 to 9).
Tensor calculus is, at its most basic, the set of rules and methods for manipulating and calculating with tensors.
Tensors are mathematical objects which have an arbitrary (but defined) number of indices. For example, a nth-rank tensor in m-dimensional space will have n indices, and it will have mn components.
Scalars have no indices, vectors have one, and matrices have two, so you can think of tensors as a generalization of vectors (or of matrices, for that matter). Tensor calculus, then, is a generalization of linear algebra.
Example of a Tensor
A tensor with 3 indices may be written
An ordered set of numbers that are labeled with three indices. If you want to visualize this, think of a matrix, but in 3 dimensions rather than the flat 2 dimensional matrices you are used to. Dullemond & Peeters visualize it like this:
In applications of tensors, each index has a meaning assigned to it.
Importance of Tensor Calculus
Although it may seem an abstract field of mathematics, tensors actually make up a very good framework for formulating and solving many physics problems; in areas like fluid mechanics, electromagnetism, quantum field theory and elasticity. Albert Einstein used it to work out his theory of general relativity, and since then it has been included in mathematical physics curriculum.
One special property of tensor calculus is that when physics problems are framed in it they are independent of coordinate systems on the manifold, which makes for much neater problem solving than infinitesimal calculus.
What is a Tensor? References
Notes on Index Notation PHYS 471. Retrieved January 5, 2019 from: https://jmureika.lmu.build/PHYS471/InClass/IndexNotation.pdf
Abraham, R.; Marsden, J. E.; and Ratiu, T. S. Manifolds, Tensor Analysis, and Applications, 2nd ed. New York: Springer-Verlag, 1991.
Arfken, G. “Tensor Analysis.” Ch. 3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 118-167, 1985
Clarke, David. A Primer on Tensor Calc. Retrieved from http://ap.smu.ca/~dclarke/home/documents/byDAC/tprimer.pdf on March 17, 2019
Dullemond & Peeters. Introduction to Tensor Calculus. Retrieved from http://www.ita.uni-heidelberg.de/~dullemond/lectures/tensor/tensor.pdf on March 17, 2019
Feng, J. The Poor Man’s Introduction to Tensors. Retrieved February 5, 2020 from: https://web2.ph.utexas.edu/~jcfeng/notes/Tensors_Poor_Man.pdf
Fleisch, D. A Student’s Guide to Vectors and Tensors. New York: Cambridge University Press, 2012.
Kelly (2015). 1.8. Tensors. Retrieved February 2, 2020 from: http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/Chapter_1_Vectors_Tensors/Vectors_Tensors_08_Tensors.pdf
Lawden, D. F. An Introduction to Tensor Calculus, Relativity, and Cosmology, 3rd ed. Chichester, England: Wiley, 1982.
Pan, R. (2014). Tensor Transpose and Its Properties from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.763.3988&rep=rep1&type=pdf
Rowland, Todd and Weisstein, Eric W. “Tensor.” From MathWorld–A Wolfram Web Resource. Retrieved from http://mathworld.wolfram.com/Tensor.html on March 17, 2019
Schnack, D. “Scalars, Vectors, Tensors, and Dyads.” 2007. http://www.physics.wisc.edu/grads/courses/726-f07/files/Section_2_Vectors_06.pdf. Retrieved January 5, 2020 from: http://www.physics.wisc.edu/grads/courses/726-f07/files/Section_2_Vectors_06.pdf
TimothyRias, Tensor image: Components_stress_tensor_cartesian.svg: Sanpazderivative work: TimothyRias [CC BY-SA (https://creativecommons.org/licenses/by-sa/3.0)]