Tensor Calculus

a mathematical theory that studies the properties of, and operations on, mathematical objects called tensors. Tensor calculus is an extension and generalization of vector calculus and matrix theory. It has many uses in such disciplines as differential geometry, the theory of Riemannian manifolds, relativity, mechanics, and electrodynamics.

Coordinate systems are a common means of describing many physical and geometric facts. In this approach objects are labeled by single numbers or by sets of numbers, and relations among objects are described by equations connecting the respective numbers or sets of numbers. Scalars, for example, mass and temperature, are described by single numbers that do not change upon transition from one coordinate system to another. (We are treating physical phenomena here from the standpoint of classical physics.) Vectors, for example, force and velocity, are given by three numbers, which are called the components of the vector; these three numbers change according to a definite law upon transition from one coordinate system to another. Scalars and vectors are the simplest members of a class of objects called tensors that appear in many physical and geometric problems. The more complex tensors are described in each coordinate system by a set of numbers—the components of the tensor—that change upon transition from one coordinate system to another according to a law more complex than the corresponding law for vectors. When we use a coordinate system, then, in addition to the numbers that describe an object or physical phenomenon, there appear numbers describing the relation of the object or phenomenon to that system.

For example, consider the numbers J_ij, i, j = 1, 2, 3, where J_ii is the axial moment of inertia of a rigid body with respect to the x_i,-axis, and J_ii. i ≠ j, are the negatives of the products of inertia. A change of coordinates induces a change in the position of the x_i-axis and thus a change in the value of the axial moment of inertia J_ii. It follows that J_ii is not a physical magnitude having significance independent of the coordinate system. One manifestation of this fact is that knowledge of the value of J_ii in one coordinate system is not sufficient to determine it in another. On the other hand, the totality of numbers J_ii has significance that is independent of the choice of a coordinate system: if we know the J_ii in one rectangular coordinate system, then we can find their values in any other rectangular coordinate system by means of the formula . where and are certain numbers. The omission of summation signs is a standard convention. If an index appears twice, once as upper and once as lower index, then we sum over that index and assign to it the relevant integral values—in our example the values 1, 2, 3.

Like vector calculus, tensor calculus is a mathematical apparatus that eliminates the effect of the choice of a particular coordinate system in the sense that the components of a tensor in one coordinate system determine its components in any other coordinate system. In the tensor calculus we develop methods for obtaining invariant relations and invariants—that is, relations between tensors, and functions of the tensor components, that are not affected by coordinate changes.

It should be clear by now that one of the fundamental problems of the tensor calculus is the formulation of the laws of mechanics, geometry, and physics in a manner independent of the choice of a coordinate system.

Cartesian tensors. A Cartesian tensor is determined in every rectangular coordinate system in three-dimensional space by 3^k numbers

pi₁ ... i_k i_r= 1, 2, 3

which change to the numbers

when the coordinate system (x₁, x₂, x₃) is changed to the coordinate system . Here, . The number k is called the rank of the tensor, and the

pi₁ ... i_k

are called its components, or coordinates. There are similar definitions of tensors in n-dimensional Euclidean space.

Let us consider some examples of tensors. The components a_i, i = 1, 2, 3, of a vector a form a tensor of rank one. The vectors a = {a_i} and b = {b_i} determine a tensor with components p_ij = a_i b_i known as a dyad. If a(x₁, x₂, x₃) is a vector field, then each of its points determines a tensor with components p_ij = ∂a_i/∂X_j known as the derivative da/dr of the vector r = {a_j} with respect to the vector r{x₁,x₂, x₃. The numbers J_ij mentioned earlier form a tensor of rank two—the inertia tensor.

Tensors of rank two. In addition to tensors of rank one (vectors), the tensors most frequently encountered in applications to mechanics are tensors of rank two.

A tensor is called symmetric if p_ij = p_ij and skew symmetric if p_ij = p_ji. A symmetric tensor has six essential components. A skew symmetric tensor has three essential components: ω₁, = p₃₂, = –p₂₃, ω²= p_l3 = –p₃₁ and ω³ = p₂₁ = –p₁₂ (p₁₁ = p₂₂ = p₃₃ = 0). The components ω₁, ω₂, ω₂ and ω₃ transform like the components of a pseudovector. In general, pseudovectors, such as angular velocity and the cross product of two vectors, may be regarded as skew symmetric tensors of rank two. The tensor whose components in every coordinate system are p₁₁ = p₂₂ = p₃₃ = 1 and p_ij = 0 for i ≠ j is called the unit tensor. Its components are denoted by the Kronecker symbol δ_ij. The inertia, stress, and unit tensors are symmetric. Every tensor can be uniquely represented as a sum of a symmetric and a skew symmetric tensor. If a(r) is the displacement vector associated with a small deformation, then the symmetric part of da/dr is called the deformation tensor. Its skew symmetric part is (½) curl a. The tensor da/dr is symmetric only if the field a (r) is irrotational. To the decomposition of the tensor da/dr into a symmetric and a skew symmetric part there corresponds the decomposition of the relative displacement da into a pure deformation and a rotation of the body as a whole.

We recall that invariants of a tensor are functions of its components that do not depend on the choice of a coordinate system. An example of an invariant is the trace p₁₁ + p₂₂ + p₃₃ of a tensor. For the inertia tensor, the trace is twice the polar moment of inertia with respect to the origin; for the tensor da/dr. the trace is the divergence of the vector field a(r).

Affine tensors. In many problems it is necessary to consider tensors in affine coordinate systems. In such coordinate systems, the axes need not be orthogonal, and the units of length on different axes need not be equal. The connection between two affine coordinate systems is determined by the components of the new basis vectors relative to the old basis vectors e_i and by the components of the basis vectors e_i, relative to the basis vectors . Accordingly, we may distinguish covariant, contravariant, and mixed tensors: the transformation laws of covariant tensors involve the numbers . the transformation laws of contravariant tensors involve the numbers . and the transformation laws of mixed tensors involve the and the . More precisely, a mixed tensor of rank r + s that is s times covariant and r times contravariant [briefly, a tensor of type (s, r)] is specified in each coordinate system by 3^r+s numbers

that transform as a result of a change of coordinates according to the rule

In the case of the rectangular coordinate system, . For Cartesian tensors, consequently, we need not distinguish between covariant (lower) and contravariant (upper) indices.

The coefficients of the equation of a quadric surface form a covariant tensor of rank two, and the elements of the matrix of a linear transformation form a once covariant and once contravariant tensor. A system of three numbers x¹, x², x³ that transform in the manner of the components of a vector x = xⁱe_i forms a once contravariant tensor, and a system of numbers that transform in the manner of a scalar product x_i = xe_i forms a once covariant tensor. Relative to affine coordinate transformations, the Kronecker symbol is a mixed tensor—hence the upper and lower indices, in contrast to the situation discussed above in the section “Tensors of rank two.” The numbers g_ij = e_ie_j, where the e_i form a basis, are the covariant metric tensor. The length of a vector x = xⁱe_i is

and the scalar product of two vectors x and y is g_ijxⁱy^j. The numbers g^ij with form the contravariant metric tensor.

There are analogous definitions of tensors in n-dimensional space. The components of p-vectors are important examples of tensors in n-dimensional space.

The order of the indices is an essential part of the definition of a tensor. In general, a permutation of the indices of a tensor results in a change of its components. A tensor is said to be symmetric with respect to a set of indices on the same level if the interchange of any two of these indices leaves unaltered all of its components. It is skew symmetric with respect to these indices if such an interchange changes the algebraic sign of its components. More generally, any invariant linear dependence between the components of a tensor is called a symmetry condition of the tensor.

Operations on tensors. The four basic operations on tensors are addition, multiplication, contraction with respect to two or more indices, and transposition of indices. Since a tensor is given by its components in different coordinate systems, operations on tensors are defined by formulas that give in each coordinate system the components of the result of the operation in terms of the components of the tensors involved in the operation. Of course, the result must be a tensor.

ADDITION OF TENSORS. The sum of two tensors and of the same type—that is, with the same number of upper indices and the same number of lower indices, is the tensor .

MULTIPLICATION OF TENSORSS. The product of two tensors and (possibly of different types) is the tensor with components . In general, the product of a tensor of rank zero, that is, a scalar λ, by a tensor is the tensor .

CONTRACTION OF A TENSOR. The result of the contraction of a tensor with respect to the upper index a and the lower index d is the tensor with components . For example, contraction of a matrix with respect to the indices i and j yields the trace of the matrix, and the biscalar product of the tensors and is the result of contraction of their product with respect to all indices. Contraction of a tensor with respect to all indices yields an invariant.

TRANSPOSITION OF INDICES. If and are tensors such that then we say that obtained from by transposing the indices c and e. The transposed indices must be on the same level.

Tensor analysis. In applications we usually encounter tensor fields rather than tensors. In the study of elastic deformations, for example, the deformation and stress tensors at all points of a body are considered. Given a rectangular coordinate system in space, we can regard a tensor field T(P) as the totality of functions t_i1 ... i_k(x^l, x₂, x³), defined at the points P(x¹, x₂, x₃) of some region, that transform as a result of a change of coordinates according to formulas of type (1). Then each partial derivative δt_i1 ... _ik/δX_i forms a tensor of rank one higher than the original tensor. For example, differentiation of a scalar field yields the gradient field, and differentiation of the gradient field yields the field of the symmetric tensor ∂²f/∂xⁱ∂xⁱ.

Tensor analysis makes use not only of Cartesian and affine coordinates but also of arbitrary, sufficiently smooth curvilinear coordinates xⁱ. In the neighborhood of each point such coordinates can be replaced by affine coordinates using as basis vectors the partial derivatives ∂r/∂xⁱ of the radius vector r at P. The scalar products e_ie_j are then equal to the components of the metric tensor g_ij at P, and the length of an infinitesimal vector , P(xⁱ), Q(xⁱ + dx_i). is given by the equation ds² = g_ijdxⁱdx^j. The metric forms for curvilinear and affine coordinates, consequently, are equal up to order two. Thus, at any point there is a local affine coordinate system, and the components of a tensor field relate to these systems. If the curvilinear coordinates (x¹, ..., xⁿ) change to (y¹, . . ., yⁿ). then the local coordinate systems change also, and the local basis vectors transform in accordance with the formula

In other words, the coefficients and vary from point to point, and their respective values are ∂xⁱ/∂yⁱ and ∂y_i/∂xⁱ. For this reason, we define a tensor field relative to a curvilinear coordinate system as a system of functions

of the curvilinear coordinates. These functions are defined in some region and transform as a result of a transition from one curvilinear coordinate system to another according to equation (2), where and . In this case, the partial derivatives of the field components with respect to the coordinates xⁱ do not form a tensor field. The reason for this fact is that upon transition from point to point there is a change not only in the tensor components but also in the local coordinate system to which the tensor relates.

It follows that in order to determine the increment of a tensor we must consider not only the increments of its components due to the transition from the point P(xⁱ) to the infinitesimally close point P(xⁱ + dxⁱ) but also the change in the local coordinate system. In other words, we must not identify the components of the increment of a tensor with the increments of its components. For example, in the case of vector fields u(P), where u has contravariant components uⁱ, the increment of the vector field is given, apart from higher-order terms, by the equation . Here, is the Christoffel symbol, which is linked to the metric tensor g_ij by means of the relation

It should be noted that the Christoffel symbols are not tensors. The summand duⁱ takes into account the dependence of the components of the increment of the tensor on the increments of its components, and the summand takes into account the dependence of the components of the increment of the tensor on the change of coordinate system associated with the transition from point to point.

The vector Duⁱ is called the covariant, or absolute, differential of the vector field u(P), and

is the covariant, or absolute, derivative of the field. Similarly, the covariant derivative of a covariant vector field is equal to

The covariant derivative of a tensor field is defined by the equation

The covariant derivative of a tensor field is a tensor field whose covariant rank is one greater than the covariant rank of the original field. If the curvilinear coordinates are rectangular, then co-variant differentiation reduces to ordinary differentiation—that is, to the operation of forming the field

In this case, the Christoffel symbols vanish.

For sums and products of tensors, the rules of covariant differentiation coincide with those of ordinary differentiation. Covariant differentiation commutes with contraction. Also, covariant differentiation is commutative in the sense that Δ_r, Δ_mT = Δ_mΔ_rT. It should be noted that the covariant derivative of the metric tensor is zero.

Historical survey. The development of the theory of algebraic forms and the theory of quadratic differential forms in the 19th century prepared the ground for the tensor calculus. The work in the theory of quadratic differential forms was directly related to differential geometry, in particular, the geometry of surfaces (K. Gauss) and the geometry of n-dimensional metric spaces (G. Riemann).

The modern form of the tensor calculus is due to the Italian mathematician C. Ricci and is sometimes called the Ricci calculus. Ricci’s ideas attracted little attention until the appearance of A. Einstein’s general theory of relativity in 1916. The mathematics of the theory of relativity is the mathematics of the tensor calculus.