Tensors

Type of physical science: Tensors, Algebra, Analytic geometry, Mathematical methods

Field of study: General topics in mathematics

Tensors are mathematical objects that are generalizations of vectors and scalars into higher dimensions. Tensors and the related branch of mathematics called "tensor analysis" allow scientists to construct mathematical and physical laws that remain valid when transforming between different coordinate systems. Tensors are used in many branches of physics, including classical and quantum mechanics, solid-state physics, and special and general relativity theory.

Overview

Tensors and tensor analysis make up a useful and powerful part of mathematical physics and applied mathematics. Examples of tensors include ordinary scalars and vectors--familiar objects in physics--as well as more complicated objects. The word "tensor" means "to stretch." As such, it is connected to the concepts of stress and strain in the theory of elasticity and the deformation of materials.

As with many mathematical objects, tensors are defined by precise rules. A tensor is described by two parameters, its dimensionality D and its rank R. The lowest rank R a tensor may have is zero. The number of components contained in a tensor of rank zero is always one, no matter what the dimension D. Tensors of rank zero are commonly called "scalars." Physically, scalars are quantities represented only by a number. The number of pages in this article is an example of a scalar. The temperature outside is another example of a scalar.

The next class of tensors has rank R equal to one. The number of components of a second-rank tensor is D¹, where D is the dimension of the tensor. In a two-dimensional space, a first-rank tensor has 2¹ = 2 components, while in a three-dimensional space, it has 3¹ = 3 components. First-rank tensors are commonly called "vectors" by mathematicians and physicists. The following form gives the usual notation for a first-rank tensor, labeled v in this example, in two dimensions: v = (v¹ , v²)

The subscript for the components of the tensor serve to distinguish the different elements. Vectors are quantities that have both magnitude and direction. Examples include the length and angle of a ray that goes from the lower left-hand corner of this page to the upper right-hand corner, the velocity of a moving object, the force acting on an object, and the electric field in a region of space. It is important to distinguish between speed and velocity. The speed of an object is a scalar quantity and gives, for example, the number of meters it moves in a certain time interval, regardless of the direction in which it moves. Velocity, on the other hand, is a vector quantity and is specified by giving both the speed of the object and the direction it moves. Thus, vectors contain more information (the number of components) than scalars.

A tensor of rank two is commonly called a "matrix," since it can be represented by a rectangular array of numbers. In a two-dimensional space, a second-rank tensor has 2² = 4 components, and in a three-dimensional space, 3² = 9 components. The following form gives the usual notation for a second-rank tensor, labeled m in this example, in two dimensions:

The subscripts for the components of the tensor give the row index and column index, respectively.

While the rules for the number of components for tensors with higher ranks can be generalized, the geometric structure of the components is much harder to visualize. If the number of components in the tensor is given by Z, then Z is related to the dimension D and the rank R by the relation Z = DR. A tensor of rank three can be envisioned as a stack of the rectangular arrays of components for a second-rank tensor into a three-dimensional grid of components, similar to the way a sandwich is made by stacking various layers of food. Above rank three, it is nearly impossible to visualize the structure of the components of a tensor. While this visualization problem poses some difficulty when working with higher-rank tensors, it does not forbid their use in the mathematical sense as long as the rules that define them and allow for their manipulation are sound. One of the chief powers of mathematics is the generalization of simple objects, rules, and procedures into more abstract forms.

Tensors can be manipulated using the rules of algebra developed for matrices. Addition, subtraction, multiplication, and division of tensors with identical rank and dimension can be performed easily. For example, two second-rank tensors of dimensions two, a and b, can be added by simply adding the corresponding components: In order to move beyond a description of tensors to some of their mathematical properties, the idea of a transformation must be explored. In the simplest sense, a transformation is an action that changes one number into another. For example, the function f(x) = x² changes a number to its square. Thus, for the case x = 2, f = 2² = 4; the action changes the number two to the number four. Transformations can also be applied to more abstract mathematical objects identified by labels or variables. In these situations, the actions are often called "operators." For example, the same action described above could be applied to a function g(x) = (1 + x) in the following way. The action f acting on the function g(x) gives: f[g(x)]=[g(x)]²=[1 + x]²=1 + 2x + x² Certain transformations have the property that they leave the object being acted upon changed. Transformations of this type are called "invariant." Particularly simple examples of invariant transformations are the additive and multiplicative identities. The additive identity is an action that adds the number zero to any other number. Thus, 4 + 0 = 4. This transformation is invariant, since the result is identical to the original number. The multiplicative identity also displays invariance: 4 × 1 = 4.

A more sophisticated type of invariance can be seen in two dimensions. Consider two different Cartesian coordinate systems, the only difference being that one is rotated by some angle q with respect to the other. The coordinates of a point in space are denoted by (x, y) in one system and by (x¬, y¬) in the other. These coordinates are related by the following trigonometric equations: x¬=x cos q-y sin q y¬=x cos q+y sin q Using the Pythagorean theorem, the square of the distance from the origin to the point in space is given by x² + y² and x¬² + y¬² in each coordinate system, respectively. This quantity is an example of an invariant for this particular transformation, since its value is the same in each coordinate system.

Next, consider this transformation idea applied to tensors. Imagine again two different coordinate systems in three dimensions, not necessarily Cartesian. Label the coordinates in the first system by the symbols x, y, and z and the coordinates in the second system by x¬, y¬, and z¬. A scalar or tensor of rank zero has the same value in both coordinate systems; it is an invariant. If S and S¬ are its values in each coordinate system respectively, then S = S¬. A physical example of this relationship would be the temperature at a point in an object; it has the same value no matter what coordinate system is chosen.

For a vector or tensor of rank one, the transformation relationship is more complicated. If the vector in the first coordinate system, v, has components vx'vy' and vx and vector y¬ in the second system has components vx¬,vy¬, and vz¬, then the transformation for the x-component of vector v is: vx¬ =(¶x¬/¶x)vx+(¶x¬/¶y)vy+(¶x¬/¶z)vz with similar relationships for the other two components of vector v¬. The terms (¶x¬/¶x), (¶x¬/¶y), and (¶x¬/¶z) are called "partial derivatives." These partial derivatives describe how the measures of length in the two coordinate systems relate. As a concrete example, imagine two three-dimensional Cartesian coordinate systems that are identical, except that in the second, the grid points are spaced twice as far apart as in the first system. Then (¶x¬/¶x) = 2, (¶x¬/¶y) = 0, and (¶x¬/¶z) = 0. This means that the x-component of the vector v¬ in the second coordinate system is twice as long as its x-component in the first system. While all this formalism may seem unnecessarily complicated for this example, it becomes necessary when the transformations are more complicated and the number of dimensions and the rank of the tensors increase.

The chief utility of the tensor formalism and methods of tensor analysis is that they allow for a systematic way of constructing mathematical relations and physical laws displaying invariance under coordinate transformations. In the physical sciences, physical laws are often constructed by direct observation of natural phenomena via controlled experimentation. While this experimentation is usually performed in a particular coordinate system, it would be inappropriate to call a physical law constructed from this experimentation a "law of nature" if its predictions are different in different coordinate systems. Tensors help to ensure that physical laws display this necessary invariance.

Applications

One of the most important areas of application of tensors in physics is in the area of classical mechanics. Isaac Newton's second law of motion states that the acceleration of an object is directly proportional to the net applied force acting on it and inversely proportional to its mass. The term "mass," in this context, is often referred to as "inertia," since it describes how effective a force is in causing an acceleration of an object. Newton's second law of motion can also be applied to extended objects. An extended object is one that comprises a rigid arrangement of many particles. A bicycle wheel is an example of an extended object. Since an extended object can have various rotations in addition to simple translations (motions along lines or curves), other inertial parameters are necessary. The moment of inertia for an extended object is the constant of proportionality that describes how effective external forces are at causing specific rotations along particular axes. For a three-dimensional extended object, there are at most nine (3² = 9) different moment-of-inertia parameters. These nine components can be grouped into a second-rank tensor and then manipulated mathematically as a single object. Since the physical laws involving the moment of inertia for an object--Newton's second law being one example--are required to be invariant under coordinate transformations, it is often important to find the specific coordinate system in which the moment-of-inertia tensor can be expressed with the fewest components. In a three-dimensional space, a special coordinate system can be derived in which the moment-of-inertia tensor has only three independent components. The mathematical procedure for deriving this coordinate system is called the "principal-axis transformation" and is a basic technique of tensor analysis.

Another important application of tensors in physics is in the field of elasticity theory. When material objects are subjected to external forces, commonly called "stresses," they often change shape. This process of changing shape is sometimes called "deformation." A physical measure of the deformation of a material is called the "strain." In many materials, the stress is directly proportional to the strain, with the relevant constant of proportionality called the "spring constant" or the "elastic constant." This relationship was proposed by Robert Hooke (1635-1703) and is now called "Hooke's law." While many materials obey Hooke's law, especially if their geometry is such that one dimension is much larger than the other two--a long, thin metal wire, for example--some do not. Imagine, for example, an apparatus devised to stretch a rectangular bar of copper metal along its long axis. If we measure the three dimensions of the bar before and after the experiment, we would find that while the length of the bar increased, its cross-sectional dimensions decreased. This phenomenon is called "necking." The observation that a stress applied in a particular direction causes a strain in another direction is an indication that the material is anisotropic. Anisotropic materials are ones that have physical properties that vary with direction. Careful analysis shows that three-dimensional objects can have at most eighty-one different elastic constants, of which twenty-one are independent. These eighty-one elastic constants can be represented by a fourth-rank tensor in three-dimensions, since 3⁴ = 81. These eighty-one components can be manipulated mathematically as a single object using the tensor formalism, simplifying the situation considerably.

A final example of the application of tensors to the physical sciences can be found from the field of electromagnetism. The flow of electric current in conducting materials has a particularly simple representation using the tensor formalism. Ohm's law, named in honor of Georg Ohm (1787-1854), is one of the fundamental relationships describing the flow of electric current in these materials. As in the case of elasticity, many materials have physical properties that are anisotropic. The resistance of a material and its related quantity, the conductivity, is such a property. In the microscopic formulation of Ohm's law, the conductivity is the constant of proportionality that relates the electric current in the material to the electric field in the material. For a three-dimensional object, the conductivity has nine components and can be expressed as a second-rank tensor.

Context

Carl Friedrich Gauss (1777-1855), a German mathematician, was one of the first to study tensors and their properties. Gauss's work was based on his studies of intrinsic coordinates, a method of geometrical analysis. Fellow Germans Bernhard Riemann (1826-1866) and Elwin Bruno Christoffel (1829-1900) extended Gauss's work into the areas of the geometry of n-dimensional manifolds and the calculus of quadratic forms, respectively. Another German, Hermann Grassmann (1809-1877), applied Gauss's methods to number theory. In the 1880's, Italian geometer Gregorico Ricci-Curbastro (1853-1925) and his student Tulio Levi-Civita (1873-1941) formalized these previous studies into the field of mathematics then called "absolute differential calculus." This branch of mathematics is now called "tensor analysis."

Until about 1910, further studies in tensor analysis were rather limited outside the domain of mathematics. However, with the development of relativity theory by the German physicist Albert Einstein (1879-1955), the pace of research, especially in the physical sciences, increased. While Einstein's special theory of relativity, developed in the first decade of the twentieth century, did not employ the methods of tensor analysis, his general theory of relativity proposed in 1916 did. Einstein's special theory of relativity defines the kinematics, or description of motion, for objects that move at speeds comparable to the speed of light. The influence of objects by the gravitational force is not included in the special theory. In the general theory, Einstein tackled the problem of the gravitational force head-on. As a way to eliminate any explicit reference to the gravitational force acting on an object, Einstein chose to account for its effect by altering the geometry of the space-time in which it moves. This geometry is defined in such a way that the motion of the object is exactly the same as if gravity were assumed.

In order to solve this geometrical problem, Einstein and Swiss mathematician Marcel Grossman (1878-1936) used the prior work in tensor analysis of Riemann and Levi-Civita. They proposed that the geometry of space-time for the universe was not flat but in fact was warped or curved by the distribution of matter within it. The motion of objects in this curved space-time occurs along curves called "geodesics." A geodesic is the shortest (or longest) path between two nearby points. A geodesic for the surface of the earth is a great circle, for example.

In addition to this proposal about the geometry of space-time, Einstein's general theory demands that the laws of nature remain the same for all observers in relative motion. Mathematically, this is a statement of invariance, one of the fundamental properties of tensors. The perplexing phenomena of length contraction and time dilation are a direct result of this invariance. The work of Einstein and others on general relativity was a clear demonstration of the utility of tensors and tensor analysis in the physical sciences.

Principal terms

CARTESIAN COORDINATE SYSTEM: A coordinate system based on three mutually perpendicular lines passing through the origin, with equal grid spacing along each axis

COORDINATE SYSTEM: A grid of points in space located at prescribed places on axes defined by a precise mathematical relationship; the intersection of the axes is called the "origin"; coordinate systems are distinguished by the angles at which the axes intersect and the curvature of the axes

DIMENSION: The number of independent axes in a coordinate system

INVARIANT: An object or mathematical relationship that does not change under the action of a transformation; for example, any number is invariant to multiplication by one, while a sphere is invariant to rotation about any axis through its center, since it looks the same from any orientation

MATRIX: A mathematical object or a physical quantity described by a two-dimensional rectangular array of scalars; matrices are useful because they enable many numbers to be represented as a single object

RANK: A tensor in a space of D dimensions has rank R if there are DR components; in a three-dimensional space, a scalar has one component and rank zero; a vector has three components and rank one; and a matrix has nine components and thus has rank two

SCALAR: A mathematical object or a physical quantity described by a single number; mass, temperature, and the number of pages in this volume are all examples of scalars

TRANSFORMATION: One or more mathematical equations used to convert a mathematical object into another; a function such as y = f(x) is an example of a transformation, since it changes the number x to another number y

VECTOR: A mathematical object or a physical quantity described by more than one number; examples of vectors include velocity, acceleration, force, angular momentum, and magnetic field

Bibliography

Arfken, George. Mathematical Methods for Physicists. 2d ed. New York: Academic Press, 1970. A college-level textbook on a wide range of mathematical methods. Chapter 3, on tensor analysis, gives a sophisticated introduction to the topic, emphasizing the formal mathematical structure of tensors. Applications in the fields of solid-state physics and electromagnetism are discussed. Includes chapters on scalars and vectors as well as a reference list.

Ashcroft, Neil W., and N. David Mermin. Solid State Physics. Philadelphia: Saunders College Publishing, 1976. A college-level text on experimental and theoretical solid-state physics. The use of tensors in describing the elastic properties of materials is treated extensively.

Boriesenko, A. I., and I. E. Tarapov. Vector and Tensor Analysis with Applications. Englewood Cliffs, N.J.: Prentice-Hall, 1968. An applied treatment of the subject for students of the physical sciences. The book makes extensive use of differential calculus in the presentation. Applications in the fields of fluids dynamics and electromagnetic theory are included.

Craig, Homer V. Vector and Tensor Analysis. New York: McGraw-Hill, 1943. A college-level text for undergraduates on tensor analysis from a calculus point of view. An extensive review of the necessary calculus background is given. The final three chapters of the book present applications of tensors to the fields of classical dynamics, special relativity, and general relativity.

Goldstein, Herbert. Classical Mechanics. 2d ed. Reading, Mass.: Addison-Wesley, 1980. The most widely used text for advanced students in the physical sciences on the topic. The book includes succinct treatments of the use of tensors in coordinate transformations, in the calculation of moments of inertia for extended objects, and in special relativity.

Hoffmann, Banesh. About Vectors. Englewood Cliffs, N.J.: Prentice-Hall, 1966. An introductory book on vector analysis intended for lower-level undergraduate students. This book is not a textbook but rather is written to provoke thought and discussion. Includes a thorough discussion of scalars and vectors, emphasizing their formal mathematical and geometric structures. Also includes a chapter on tensors, aiming to introduce them based upon their transformation properties.

Sokolnikoff, I. S. Tensor Analysis: Theory and Applications to Geometry and Mechanics of Continuous Media. New York: John Wiley & Sons, 1964. This graduate-level text for students in the physical sciences and applied mathematics is one of the best advanced references available. The book includes a thorough introductory chapter on linear transformations, emphasizing the geometry and physics of the situations. Applications in the fields of geometry, analytical mechanics, relativistic mechanics, and the mechanics of deformable media are treated in depth.

Wrede, Robert. Introduction to Vector and Tensor Analysis. New York: Dover, 1972. A broad introduction to vector and tensor analysis for the advanced undergraduate or graduate student in the physical sciences and mathematics. The text stresses the interrelationships between the algebra and geometry of both vectors and tensors. Applications in the field of general relativity are given.