Mathematical Handbook for Scientists and Engineers

Tensor algebra was developed by successive generalizations of the theory of vector spaces (Secs. 12.4-1 and 14.2-1 to 14.2-7), linear algebras (Secs. 12.4-2 and 14.3-5), and their representations (Sec. 14.1-2). Tensor analysis is particularly concerned with tensors in their aspect as point functions and applies especially to the description of curved spaces (Chap. 17) and of continuous fields in physics. Tensor methods frequently link complicated numerical data measured in different frames of reference to relatively simple abstract models.

16.1-2. Coordinate Systems and Admissible Transformations. Consider a class (space, region of a space) of objects (points) labeled with corresponding ordered sets of n < ∞ continuously variable real numbers (coordinates) x¹, x²,. . . , xⁿ. A coordinate transformation (“alias” point of view, see also Sec. 14.1-3) “admissible” in the sense of the following sections is a relabeling of each point with n new coordinates related to the original coordinates x¹, x², . . . , xⁿ by n transformation equations

such that, throughout the region of points under consideration, (1) each function is single-valued and continuously differentiable and (2) the Jacobian (Sec. 4.5-5) det is different from zero.

The class of “admissible” transformations constitutes a group (Sec. 12.2-1) with respect to the operation of forming their “product,” i.e., of applying two transformations successively (see also Sec. 12.2-8). The Jacobian of the product of two transformations is the product of the individual Jacobians. Each admissible transformation (1) has a unique inverse whose Jacobian is the reciprocal of the original Jacobian.

16.1-3. Description (Representation) of Abstract Functions by Components. Dummy (Umbral) Index Notation. Tensor analysis deals with abstract objects associated with the points (x¹, x², . . . , xⁿ) of an n-dimensional space (n < ∞, Sec. 14.1-2) as point functions (see also Sec. 5.4-1) defined on a region of the space. Each point function Q(x¹, x², . . . , xⁿ), say, will be described or represented by an ordered set of n^R < ∞; numerical functions (components of Q, see also Sec. 14.1-2) Q(j¹, j² . . . , j^R; x¹, x², . . . , xⁿ) of the coordinates x¹, x², . . . , xⁿ where each index j₁, j₂, . . . , j_R runs from 1 to n.

Depending on the type of object (Secs. 16.2-1 and 16.2-2), certain of the indices labeling each component are written as superscripts, and others as subscripts. Thus an object Q may be described by n^r+s components where all indices run from 1 to n. With this notation, it is possible to abbreviate sets of sums like

and also sums involving vectors, like

through the use of the following conventions.

1. Summation Convention. A summation from 1 to n is performed over every dummy (umbral) index appearing once as a superscript and once as a subscript.

Any dummy index may be changed at will, since dummy indices are “canceled” by the summation (EXAMPLE: A^ikB^k = A^ikB_i = A^ihB_h). Different summation indices should be used in the case of multiple summations.

2. Range Convention. All free indices appearing only as superscripts or only as subscripts are understood to run from 1 to n; so that an equation involving R free indices stands for n^R equations.

The superscripts and subscripts on the two sides of any equation must match.

3. In derivatives like ∂aⁱ/∂x^k, k is considered as a subscript.

The dummy-index notation defined by the above conventions is used throughout the remainder of this chapter. Thus expressions like A^ikB_k are understood to be sums, unless the contrary is explicitly indicated, as in (A^ikB_k)_{no sum}.

In many applications, the dummy-index notation is considerably more powerful than the matrix notation employed in Chap. 13 (see Table 14.7-1 for a comparison of notations).

16.1-4. Schemes of Measurements and Induced Transformations. Invariants (see also Secs. 14.1-5 and 16.2-1). A scheme of measurements for a class (or a set of classes) of abstract point functions is a reference system (or a set of reference systems, one for each class of functions) for the (biunique) representation of each point function by a corresponding set of numerical components.

Each function will be represented by the same number of components in all schemes of measurements considered. In physics, each component value usually corresponds to the result of a physical measurement.

It is useful to associate a definite scheme of measurements (x scheme of measurements) with each system of space coordinates x¹, x²,. . . , xⁿ. The components of a point function Q(x¹, x²,. . . , xⁿ) in the x scheme of measurements and the components of Q in the scheme of measurements are then related by an induced transformation

associated with (induced by) each admissible coordinate transformation (1).

A class C of abstract point functions Q(x¹, x², . . . , xⁿ) described in an x scheme of measurements by numerical components constitutes a class of invariants or geometrical objects (sometimes called tensors in the most general sense) if their mathematical properties can be defined in terms of (abstract) operations independent of the scheme of measurements (see also Secs. 12.1-1 and 16.4-1). Then the induced transformations (2) are related to the defining operations of C as follows:

1. The correspondence between any admissible coordinate transformation (1) and the corresponding induced transformation (2) is an isomorphism (Sec. 12.1-6) preserving the operation of forming the product of two transformations.

2. Every induced transformation is an isomorphism preserving the defining operations of C (see also Sec. 16.4-1). Defining operations involving two or more classes C₁, C₂, . . . of invariants (e.g., scalars and vectors) are also preserved by the induced transformations of C₁, C₂,. . . .

A class C of point functions may also be invariants only with respect to a subgroup of the group of admissible coordinate transformations (Sec. 12.2-8).

16.2. ABSOLUTE AND RELATIVE TENSORS

16.2-1. Definition of Absolute and Relative Tensors in Terms of Their Induced Transformation Laws (see also Table 16.2-1 and Sec. 6.3-3). Throughout this handbook (and in practically all applications), tensors are understood to be real absolute and relative tensors represented by real components. Each type of absolute or relative tensor is defined by a linear and homogeneous induced transformation law (Sec. 16.1-4) relating the tensor components in different schemes of measurements. Given a space of points (x¹, x², . . . ,xⁿ) and a group of admissible coordinate transformations (16.1-2),

1. An (absolute) scalar (scalar invariant; absolute tensor of rank 0) α is an object represented in an x scheme of measurements by a function α(x^l, x², . . . , xⁿ) and in an scheme of measurements by a function related to α(x^l, x² . . . ,xⁿ) at each point by the induced transformation

2. An (absolute) contravariant vector (absolute contravariant tensor of rank 1) a is an object represented in an x scheme of measurements by an ordered set of n functions (components)

aⁱ(x¹ x² . . . , xⁿ) and in an scheme of measurements by an ordered set of n components related to the aⁱ(x¹, x², . . . , xⁿ) at each point by the induced transformation

3. An (absolute) covariant vector (absolute covariant tensor of rank 1) a is an object represented in an x scheme of measurements by an ordered set of n functions (components) a_i(x¹, x², . . . , xⁿ) and in scheme of measurements by an ordered set of n components related to the a(x¹, x², . . . , xⁿ) at each point by the induced transformation

4. An (absolute) tensor A of rank r + s, contravariant of rank r and covariant of rank s, is an object represented in an x scheme of measurements by an ordered set of n^r+s functions (components) and in an scheme of measurements by an ordered set of n^r+8 components related to the at each point by the induced transformation

5. A relative tensor (pseudotensor) A of weight W and of rank r + s, contravariant of rank r and covariant of rank s, is an object represented in an x scheme of measurements by n^r+8 functions (components) and in an scheme of measurements by an ordered set of n^r+8 componentsrelated to the (x¹, x², . . . , xⁿ) at each point by the induced transformation

where W is a real integer. Relative tensors are called densities for W = 1 and capacities for W = — 1 (EXAMPLES: Volume and surface elements are scalar and vector capacities; see also Secs. 6.2-3b and l7.3-3c).

The defining transformation (5) includes Eqs. (1) to (4) as special cases (W= 0 for absolute tensors). A tensor represented by is a mixed tensor if and only if neither r nor s equals zero. The induced transformation (5) characterizing every absolute or relative tensor quantity is linear and homogeneous in the tensor components. The corresponding inverse transformation is

NOTE: Relative ordering of superscripts and subscripts is frequently used to conserve symbols. Thus Aⁱ_k and A_i^k denote different sets of components (see also Secs.16.7-2 and 16.9-1).

16.2-2. Infinitesimal Displacement. Gradient of an Absolute Scalar (see also Secs. 5.5-2, 5.7-1, 6.2-3, and 16.10-7). The coordinate differentials dxⁱ represent an absolute contravariant vector, the infinitesimal displacement dr.

Given a suitably differentiable absolute scalar α, the components ∂α/∂xⁱ represent an absolute covariant vector called the gradient ∇α of α. A given absolute covariant vector described by ai is the gradient of an absolute scalar if and only if for all i, k.

16.3. TENSOR ALGEBRA: DEFINITION OF BASIC OPERATIONS

16.3-1. Equality of Tensors. Two tensors A and B of the same type, rank, and weight are equal (A = B) at the point (x¹, x², . . . , xⁿ) if and only if their corresponding components in any one scheme of measurements are equal at this point:

Corresponding components of A and B are then equal in every scheme of measurements (see also Sec. 16.4-1). Tensor equality is symmetric, reflexive, and transitive (Sec. 12.1-3).

NOTE: Relations between the “values” of tensor point functions at different points are not defined in ordinary tensor algebra. Some such relations are discussed, for the special case of tensors defined on Riemann spaces, in Sec. 16.10-9.

16.3-2. Null Tensor. The null tensor 0 of any given type, rank, and weight is the tensor whose components in any one scheme of measurements are all equal to zero. Thus A = 0 at the point (x^l, x², . . . , xⁿ) if and only if

All components of A are then equal to zero in every scheme of measurements.

16.3-3. Tensor Addition. Given a suitable class of tensors all of the same type, rank, and weight, the sum C = A + B of two tensors A and B is the tensor described in any one scheme of measurements (and hence in all schemes of measurements) by the sums of corresponding components of A and B:

A + B is of the same rank, type, and weight as A and B. Tensor addition is commutative and associative.

16.3-4. Multiplication of a Tensor by an Absolute Scalar. The product B = αA of a tensor A and an absolute scalar a is the tensor represented in every scheme of measurements by the products of the components of A and the scalar α:

αA is of the same rank, type, and weight as A. Multiplication by scalars is commutative, associative, and distributive with respect to both tensor and scalar addition.

In particular, (— 1)A ≡ —A is the negative (additive inverse) of A, with A — A = 0.

16.3-5. Contraction of a Mixed Tensor. One may contract a mixed tensor A described by by equating a superscript to a subscript and summing over the pair. The resulting n^r+8–2 sums describe a tensor of the same weight as A, contravariant of rank r – 1 and covariant of rank s — 1. In general, a mixed tensor can be contracted in more than one way and/or more than once.

EXAMPLE: An absolute or relative mixed tensor A of rank 2 described by A_kⁱ may be contracted to form the absolute or relative scalar (trace of A)

16.3-6. (Outer) Product of Two Tensors (see also Sec. 12.7-3). The (outer) product C = AB of two tensors A and B, respectively, of weight W and W′ and represented by and is the tensor described by

AB is contravariant of rank r + p and covariant of rank s + q and is of weight W + W′. Outer multiplication is associative; it is distributive with respect to tensor addition. It is not in general commutative, since the relative order of the indices in Eq. (5) must be observed.

EXAMPLES: aⁱb^k = A^ik; aⁱb_k = A_kⁱ; a_ib_k = A_ik The product (4) is a special case of an outer product.

16.3-7. Inner Products. If the outer product of two tensors A and B, described by Eq. (5), can be contracted (Sec. 16.3-5) so that one or more superscripts of are summed against one or more subscripts of and/or conversely, the resulting sums represent an inner product of the tensors A and B. In general, several such inner products can be formed.

Every inner product of two tensors A and B is a tensor of the same weight as AB. The rank of the inner product is equal to that of AB diminished by twice the number of index pairs summed. Inner multiplication is distributive with respect to tensor addition.

16.3-8. Indirect Tests for Tensor Character. Let Q be an object described in an x scheme of measurements by n^R components Q(j₁, j₂, . . . ,j_R; x¹,x² . . . , xⁿ), and let X be a tensor described by The outer product QX is defined, as in Sec. 16.3-6, by the n^R+r+8 components Inner products of Q and X are represented by sums formed through contraction (Sec. 16.3-5) of QX, so that one or more indices of Q(j₁, j₂, . . . , j_R) are summed against one or more superscripts and/or subscripts of

The object Q is a tensor if and only if the outer product QX, or a given type of inner product of Q and X, is a tensor Y of fixed rank, type, and weight for any arbitrary tensor X of fixed rank, type, and weight.

An analogous theorem results if X is, instead, the outer product of R distinct arbitrary vectors of fixed types and weights. In either case one infers the rank and type of Q by matching superscripts and/or subscripts. The weight of Q must be the difference of the respective weights of Y and X.

EXAMPLES: (1) Q is an absolute tensor contravariant of rank r and covariant of rank s if, for every absolute vector a represented by aⁱ,

where the components on the right describe an absolute tensor depending on a.

(2) Q is an absolute tensor contravariant of rank r and covariant of rank s if, for every absolute tensor A represented by

where α represents an absolute scalar depending on A.

NOTE: An object Q described by n² components Q_ik is an absolute covariant tensor of rank 2 if and only if

represents a scalar invariant for every absolute contravariant vector a described by aⁱ and Q_ik = Q_ki.

16.4. TENSOR ALGEBRA: INVARIANCE OF TENSOR EQUATIONS

16.4-1. Invariance of Tensor Equations. For each admissible coordinate transformation (16.1-1), the induced transformation laws used to define tensor components preserve the results of tensor addition, contraction, and outer (and hence also inner) multiplication, as well as tensor equality. Every relation between tensors expressible in terms of combinations of such operations (including convergent limiting processes) is invariant with respect to the group of admissible coordinate transformations. If the relation applies to tensor components in any one scheme of measurements, it holds in all schemes of measurements (see also Secs. 12.1-5 and 16.1-4).

EXAMPLE: implies and conversely;this relation may be symbolized by the abstract equation A + B = C.

One may, then, speak of tensors and tensor operations without reference to a specific scheme of measurements. Each suitable class of tensor point functions is a class of invariants and constitutes an abstract model definable (to within an isomorphism, Sec. 12.1-6) in terms of mathematical operations without reference to components (see also Secs. 12.1-1 and 16.1-4).

Thus suitable classes of absolute tensors of rank 0, 1, and 2, respectively, constitute scalar fields (Sec. 12.3-lc), vector spaces (Sec. 12.4-1), and linear algebras (Sec. 12.4-2; see also Sec. 16.9-2). Classes of tensors of rank 2, 3, . . . may be built up as direct products of vector spaces (Sec. 12.7-3; see also Sec. 16.6-lc).

16.5. SYMMETRIC AND SKEW-SYMMETRIC TENSORS

16.5-1. Symmetry and Skew-symmetry. An object Q described by n^R components Q(j₁, j₂, . . . , j_R) each labeled with an ordered set of R indices j₁, j₂, . . . , j_R is

1. Symmetric with respect to any pair of indices, say j₁ and j₂, if and only if

2. Skew-symmetric (antisymmetric) with respect to any pair of indices, say j₁and j₂, if and only if

for all sets of values of i, k, j₃, . . . , j_R, each running from 1 to n. Q is (completely) symmetric or (completely) skew-symmetric with respect to a set of indices if and only if Q is, respectively, symmetric or skew-symmetric with respect to every pair of indices of the set.

The symmetry or skew-symmetry of an absolute or relative tensor with respect to any pair of superscripts or any pair of subscripts is invariant with respect to the group of admissible coordinate transformations.

16.5-2. Kronecker Deltas. The (generalized) Kronecker delta of rank 2r is the absolute tensor represented by n^2_r components defined as follows:

1. = + 1 or — 1 if all superscripts i₁, i₂, . . . , i_r are different, and the ordered set of subscripts k₁, k₂ , . . . , k_r is obtained, respectively, by an even or odd permutation (even or odd number of transpositions) of the ordered set i₁, i₂, . . . , i_r

2. = 0 for all other combinations of superscripts and subscripts

Of particular interest is the Kronecker delta of rank 2 described by

Contraction of any mixed tensor A by summation over a superscript i and a subscript i′(Sec. 16.3-5) is equivalent to inner multiplication (Sec. 16.3-7) of A by δ_i^i′.

Each Kronecker delta is completely skew-symmetric (Sec. 16.5-1) with respect to both the set of superscripts and the set of subscripts. Kronecker deltas of rank 2r > 2n are zero.

If is symmetric with respect to any pair of superscripts, then

If is symmetric with respect to any pair of subscripts, then

Note also

16.5-3. Permutation Symbols (see also Sec. 16.7-2c). The permutation symbols

represent completely skew-symmetric relative tensors of rank n and of weight +1 and —1, respectively. Note that

1. 0 if two or more of the indices i₁, i₂, . . . i_n are equal.

2. 1 if the ordered set i₁, i₂, . . . , i_n is obtained by an even permutation of the set 1, 2, . . . , n.

3. -1 if the ordered set i₁, i₂. . . , i_n is obtained by an odd permutation of the set 1, 2, . . . ,n.

And note

16.5-4. Alternating Product of Two Vectors (see also Secs. 16.8-4 and 16.10-6). The alternating product (sometimes called bivector) of two contravariant vectors and the alternating product of two covariant vectors are skew-symmetric tensors of rank two respectively represented by

The weight of the alternating product is the sum of the weights of the two factors.

16.6. LOCAL SYSTEMS OF BASE VECTORS

16.6-1. Representation of Vectors and Tensors in Terms of Local Base Vectors. (a) Given an x scheme of measurements, each (absolute or relative) contravariant vector described by aⁱ(x¹, x², . . . , xⁿ) may be represented as an invariant linear form

(see Sec. 16.1-3 for umbral-index notation) in the n absolute contravariant local base vectors (Sec. 14.2-3) e₁(x¹, x², . . . , xⁿ), e₂(x¹, x², . . . , xⁿ), . . . , e_n(x¹, x² . . . , xⁿ) associated with the x scheme of measurements. The i^th base vector e_i has the components δ_i¹, δ_i², . . . , δ_iⁿ.

(b) Similarly, each (absolute or relative) covariant vector b described by b_i(x¹, x², . . . , xⁿ) may be represented in the form

in terms of the n absolute covariant local base vectors e¹(x¹, x², . . . , xⁿ), e²(x¹, x², . . . , xⁿ), . . . , eⁿ(x¹, x², . . . , xⁿ) associated with the x scheme of measurements. The i^th base vector eⁱ has the components δ₁ⁱ, δ₂ⁱ, . . . , δ_nⁱ. Note that the vectors (1) and the vectors (2) will, in general, belong to different vector spaces.

in the local base vectors e_i and eⁱ.

16.6-2. Relations between Local Base Vectors Associated with Different Schemes of Measurements. The 2n local base vectors e_i(x^l, x², . . . , xⁿ) and eⁱ(x¹, x², . . , xⁿ) may be thought of as defining the x scheme of measurements (Sec. 16.1-4) in invariant language. New local base vectors and associated with an scheme of measurements have the components δ_kⁱ and δ_i^k, respectively, in the scheme of measurements and are related to their respective counterparts associated with the x scheme of measurements as follows:

Note that e_i(x¹, x², . . . , xⁿ) and are, in general, different vectors (of the same vector space), not different descriptions of the same vector. The base vectors e_i transform formally like (cogrediently with) absolute covariant vector components (Sec. 16.2-1), whereas the eⁱ transform like absolute contravariant vector components. Absolute contravariant and covariant vector components aⁱ and b_i (and hence contravariant and covariant base vectors e_i and eⁱ) transform contra-grediently, so that inner products like aⁱb_i are invariant (see also Sec. 16.4-1).

16.7. TENSORS DEFINED ON RIEMANN SPACES.ASSOCIATED TENSORS

16.7-1. Riemann Space and Fundamental Tensors. Riemann spaces permit the definition of scalar products of vectors in such a manner that the resulting definitions of distances and angles (Sec. 16.8-1; see also Sec. 14.2-7) lead to useful generalizations of Euclidean geometry (see also Secs. 17.4-1 to 17.4-7). A finite-dimensional space of points labeled by ordered sets of real* coordinates x¹, x² . . . xⁿ is a Riemann space if it is possible to define an absolute covariant tensor of rank 2 (Sec. 16.2-1) described (in an x scheme of measurements) by components g_ik (x^l, x², . . . , xⁿ) having the following (invariant) properties throughout the region under consideration:

* The theory presented in Secs. 16.7-1 to 16.10-11 applies to vectors and tensors with real components defined on Riemann spaces described by real coordinates. The theory also applies to the Riemann spaces considered in relativity theory, where the introduction of an imaginary coordinate (Sec. 17.4-6) is essentially a notational convenience

The matrix [g_ik(x¹, x², . . . , xⁿ)] is frequently, but not necessarily, positive definite (Sec. 13.5-2); the indefinite case is of interest in relativity theory (see also Secs. 16.8-1 and 17.4-6).

The metric tensor (see also Sec. 17.4-2) described by the g_ik(x^l, x², . . . , xⁿ) and the absolute symmetric tensor of rank 2 (conjugate or associated metric tensor) whose components g^ik(x^l, x², . . . , xⁿ) are defined by

where G^ik (= G^ki) is the cofactor (Sec. 1.5-2) of g_ik in the determinant det [g_ik] are the fundamental tensors of the Riemann space.

The components of either fundamental tensor define the element of distance ds and the entire intrinsic differential geometry of the Riemann space (Secs. 17.4-1 to 17.4-7). A system of coordinates x¹, x², . . . , xⁿ is, respectively, right-handed or left-handed if the scalar density is positive or negative; an arbitrary choice of sign for any one coordinate system defines every admissible coordinate system as either right-handed or left-handed, since

(see also Secs. 6.2-3b and 6.4-3c).

16.7-2. Associated Tensors.* Raising and Lowering of Indices. An (absolute or relative) contravariant vector represented by aⁱ and a covariant vector represented by a_k, defined on a Riemann space and related at every point (x¹, x², . . . , xⁿ) by

are called associated vectors. More generally, an associated tensor of a given tensor described by is obtained by raising a subscript k through inner multiplication by g^ki, or by lowering a superscript i through inner multiplication by g_ji; or by any combination of such operations. A tensor of rank greater than one has more than one associated tensor. Since it is desirable to denote the components of all

* See also footnote to Sec. 13.3-1.

tensors associated with a given tensor A by the same symbol A, it is necessary to order superscripts and subscripts with respect to each other (see also Sec. 16.2-1). Thus the result of raising the subscript k₂ in is denoted by

Raising of previously lowered indices and/or lowering of previously raised indices restores the original tensor components.

NOTE: The contravariant and covariant permutation symbols (Sec. 16.5-3) are not associated relative tensors but are related by

16.7-3. Equivalence of Associated Tensors. The correspondence between associated tensors defined on a Riemann space is an equivalence relation partitioning the class of all tensors (Sec. 12.1-3b). The components of any associated tensor of a tensor A defined on a Riemann space are, then, considered as a different description (representation) of the same tensor A (see also Sec. 16.9-1).

In particular, vector components a^k and a_i related by Eq. (2) are interpreted as the contravariant and covariant representations of the same vector a in the x scheme of measurements used. In the notation of Sec. 16.6-1,

so that the base vectors e¹, e², . . . , eⁿ and e¹, e², . . . , eⁿ defined on a Riemann space may be regarded as (reciprocal) bases of the same vector space (see also Sec. 16.8-2). They are related formally like associated vector components:

Substitution of the appropriate expression (6) for some e^k or e_i in the expansion (16.6-3) of any tensor A corresponds, respectively, to raising or lowering the index in question (see also Sec. 16.9-1).

16.7-4. Operations with Tensors Defined on Riemann Spaces. If the tensors in question are defined on a Riemann space

1. Any two tensors having the same rank and weight may be added in the manner of Sec. 16.3-3, after their components have been reduced to the same configuration of superscripts and subscripts through raising and/or lowering of indices.

2. A tensor may be contracted over any pair of indices in the manner of Sec. 16.3-5, after one of the indices has been appropriately raised or lowered. Contraction over two superscripts i, k corresponds, then, to inner multiplication by g_ik; contraction over two subscripts i, k corresponds to inner multiplication by g^ik.

3. Inner products of two tensors A and B are defined as contractions of their outer product AB over an index (or indices) of A and a corresponding index (or indices) of B as in (2) above.

16.8. SCALAR PRODUCTS AND RELATED TOPICS

16.8-1. Scalar Product (Inner Product) of Two Vectors Defined on a Riemann Space. In accordance with Sec. 16.7-4, it is possible to define the scalar product (inner product, see also Secs. 5.2-6, 6.4-2a, and 14.2-6) a • b of any two absolute or relative vectors a and b represented by (real) components aⁱ or a_k and bⁱ or b_k:

The magnitude (norm, absolute value; Sec. 14.2-5) |a| of an absolute or relative vector a described by (real) components aⁱ or a_k is the absolute or relative scalar invariant

A unit vector is an absolute vector of magnitude one. The cosine of the angle γ between two absolute or relative vectors a and b is the absolute scalar invariant

Equations (2) and (3) imply the elementary definition (5.2-5) of the scalar product.

NOTE: If the quadratic form g_ikaⁱa^k is indefinite (indefinite metric, see also Sec-17.4-4) at a point (x¹, x², . . . , xⁿ), the square a • a of an absolute or relative vector a represented by components aⁱ at that point is positive, negative, or zero depending on the sign of g_ikaⁱa^k, and |a| = 0 does not necessarily imply a = 0.

16.8-2. Scalar Products of Local Base Vectors. Orthogonal Coordinate Systems (see also Secs. 6.3-3, 6.4-1, and 17.4-7a). The magnitudes of, and angles between, the local base vectors e¹, e², . . . , eⁿ and e¹, e², . . . , eⁿ at each point (x¹, x², . . . , xⁿ) (Sec. 16.6-1) are given by

The vectors e_i are directed along the corresponding coordinate lines, and the e^k are directed along the normals to the coordinate hypersurfaces. Each e^k is perpendicular to all eⁱ except e^k. A system of coordinates x¹, x², . . . , xⁿ is an orthogonal coordinate system if and only if g_ik(x^l, x², . . . , xⁿ) ≡ 0 for i ≠ k; in this case the e_i (and thus also the e^k) are mutually orthogonal. Not every Riemann space admits orthogonal coordinates.

NOTE: Two sets of base vectors eⁱ and e_k satisfying the relation eⁱ • e_k = δⁱ_k constitute reciprocal bases of the vector space in question (see also Sec. 14.7-6).

16.8-3. Physical Components of a Tensor (see also Sec. 6.3-4). The local unit vectors u_i in the coordinate directions corresponding to the subscript are related to the e_i and e^k by

The physical components â_i and Â _j₁ _j₂ . . . _{j_R} of vectors and tensors are defined by

where the Â _j₁ _j₂ . . . _{j_R} are obtained by comparison of Eqs. (7) and (16.6-3). The physical component of a vector a in the direction of another vector b is defined as a • b/|b|.

16.8-4. Vector Product and Scalar Triple Product (see also Secs. 5.2-7, 5.2-8, 6.3-3, and 6.4-2). The vector product a X b of two absolute or relative vectors a and b defined on a three-dimensional Riemann space (n = 3) is the vector represented by the components (see also Sec. 16.5-3)

where the scalar triple product [abc] is defined, as in Sec. 5.2-8, by

The formulas of Table 5.2-2 and Sec. 5.2-9 hold.

NOTE: The definition (8) of the vector product implies the elementary relation (5.2-6) and defines the vector product of two absolute vectors as an absolute vector. Some authors replace by 1 in the definition (8), so that the vector product of two absolute vectors becomes an “axial” vector (as contrasted to “polar” or absolute vectors) described either as a relative contravariant vector of weight +1, or as a relative covariant vector of weight –1.

16.9. TENSORS OF RANK TWO (DYADICS) DEFINED ON RIEMANN SPACES

16.9-1. Dyadics. Absolute or relative tensors of rank 2 (dyadics) A, B, . . . defined on Riemann spaces and described, for instance, in terms of their respective mixed components Aⁱ_k, Bⁱ_k, . . . (see also Sec. 16.7-2 for ordering of indices) are of interest in many applications. They are sometimes thought to warrant the special notation outlined in the following sections.

Every dyadic A may be represented as a sum of n dyads (outer products of two vectors):

Either the antecedents p_j, or the consequents q^j may be arbitrarily assigned as long as they are linearly independent (Sec. 14.2-3). In particular

In terms of physical components Â_ik (Sec. 16.8-3)

In the case of orthogonal coordinates (Sec. 16.8-2)

16.9-2. Inner-product Notation (see also Sec. 16.8-1). The following notation is sometimes useful for the description of inner products involving (real) dyadics and vectors defined on Riemann spaces:

where

With these definitions, the algebra of dyadics is precisely the algebra of linear operators described in Secs. 14.3-1 to 14.3-6; a dyadic associates a linear transformation with each point (x¹, x², . . . , xⁿ). Table 14.7-1 relates the tensor notation to the “classical” notation of Chap. 14 and to the matrix notation for dyadics and vectors.

Symmetry and skew-symmetry of a dyadic are defined in the manner of Sec. 16.5-1. Thus the dyadic A is symmetric if and only if A^ki = A^ik, or if and only if A_ki = A_ik; but this does not necessarily imply that A^k_i = Aⁱ_k, nor does the last relation imply the symmetry of A (see also Sec. 14.7-5).

Tr(A) = A^j_j = p_i • qⁱ is called the scalar of the dyadic (1). The double-dot product of A and B is the scalar

For n = 3, it is possible to define the cross products

The vector v_A = p_j X q^j is called the vector of the dyadic (1). v_A =0 if and only if A is symmetric. If A is skew-symmetric, then for every vector a

so that vector multiplication is equivalent to inner multiplication by a skew-symmetric dyadic.

16.9-3. Eigenvalue Problems (see also Sec. 14.8-3). Eigenvalues and eigenvectors of dyadics are defined at each point (x¹, x², . . . , xⁿ) in the manner of Sec. 14.8-3. The coefficients appearing in the characteristic equation (Sec. 14.8-5) corresponding to a dyadic are absolute or relative scalars. Given any symmetric dyadic A in Euclidean space with continuously differentiable components and a region V where det [Aⁱ_k] ≠ 0, there exists an orthogonal coordinate system (Sec. 16.8-2) such that the matrix [Aⁱ_k] is diagonal throughout V (normal coordinates, see also Sec. 17.4-7).

A symmetric dyadic A defined on a three-dimensional Euclidean space may be represented geometrically by a quadric surface (3.5-1), with a_ik = Aⁱ_k for i, k == 1, 2, 3 (see also Sec. 3.5-1).

16.10. THE ABSOLUTE DIFFERENTIAL CALCULUS. COVARIANT DIFFERENTIATION

16.10-1. Absolute Differentials. (a) A small change or differential of a tensor quantity cannot be defined directly as the difference between “value” of the tensor function resulting from changes dx¹, dx², . . . , dxⁿ in the coordinates x¹, x², . . . , xⁿ, since the tensor algebra of Secs. 16.3-1 to 16.3-7 does not define relations between tensor “values” at different points. The absolute differentials dα, da, db, dA, dB, . . . of absolute scalars, vectors, and tensors α, a, b, A, B, . . . defined on a Riemann space in terms of suitably differentiable components are, instead, defined by the following postulates:

1. The absolute differentials dα, da, and dA are absolute tensor quantities of the same respective ranks and types as α, a, and A.

2. The absolute differential dα of an absolute scalar α is represented in the x scheme of measurements by

3. The following differentiation rules hold:

In particular, Eq. (2) implies

so that

The postulates listed above result in a self-consistent and invariant (Secs. 16.4-1 and 16.10-7) generalization of the vector calculus described in Chap. 5; the postulates are satisfied if one chooses

where the functions are the Christoffel

three-index symbols of the second kind defined in Sec. 16.10-3.

NOTE: Equations (3) and (4) express each component of the absolute differential da as the sum of a “relative differential” daⁱ or da and a term due to the point-to-point changes in the base vectors (i.e., in the metric). One may define vector derivatives ∂a/∂xⁱ by

(b)The postulates of Sec. 16.10-lα imply that the absolute differential of any absolute tensor A defined on a Riemann space in terms of suitably differentiable components is an absolute tensor of the same type and rank, with components D given by

with *

16.10-2. Absolute Differential of a Relative Tensor. Absolute differentials of relative tensor quantities (Sec. 16.2-1) are defined in the manner of Sec. 16.10-1. In particular, the absolute differential dα of a relative scalar a of weight W is represented in the x scheme of measurements by

Equation (10) reduces to Eq. (1) for W = 0. The absolute differential dA of any (suitably differentiable) relative tensor of weight W takes the form (8) with a term

added in the expression (9) for so that

dA is a relative tensor of the same rank, type, and weight as A.

16.10-3. Christoffel Three-index Symbols. (a) The Christoffel three-index symbols of the first kind and the Christoffel three-index symbols of the second kind associated with an x scheme of measurements in a Riemann space are functions of the coordinates x¹, x², . . . , xⁿ, viz.,

where g_ik and g^ik are the fundamental-tensor components of the Riemann space in the x scheme of measurements. The Christoffel three-index symbols are not in general tensor components but transform according to the Christoffel transformation equations

where the functions are the Christoffel three-index symbols associated with an scheme of measurements related to the x scheme of measurements by a suitably differentiable coordinate transformation.

(b) The Christoffel three-index symbols satisfy the following relations:

(c) In the important special case of orthogonal coordinates x¹, x², . . . , xⁿ (Secs. 6.4-1 and 16.8-2), one has g_ik = g_iiδⁱ_k, so that

16.10-4. Covariant Differentiation. Because of the analogy between Eq. (4) or (9) and ordinary partial differentiation, the operation of obtaining the functions from the tensor components is called covariant differentiation (with respect to the metric described by the g_ik). If the components represent an absolute tensor A, then the components describe an absolute tensor

contravariant of rank r and covariant of rank s + 1. ∇A is commonly called the covariant derivative of A.

Note that, in general, neither the functions nor the “relative” differentials d are tensor components.

16.10-5. Rules for Covariant Differentiation (see also Sec. 16.10-7). Computations are frequently simplified by the fact that the ordinary rules for differentiation of sums and products (Table 4.5-2) apply formally to covariant differentiation. Note

The last two rules apply to the covariant differentiation of inner products (Secs. 16.3-7 and 16.8-1). Note also

Equation (24) shows that the fundamental tensors behave like constants with respect to covariant differentiation. The covariant derivative of every associated tensor of A is an associated tensor of ∇A (see also Secs. 16.7-2 and 16.7-3).

The covariant derivative of every Kronecker delta and permutation symbol (Secs. 16.5-2 and 16.5-3) is zero.

16.10-6. Higher-order Covariant Derivatives. Successive covariant differentiation* of tensor components yield higher-order covariant derivatives described by components like _j₁_j₂. . ._{j_m}. These tensors are not in general symmetric with respect to pairs of subscripts j; the order of covariant differentiation may be inverted if and only if the Riemann space in question is flat (Sec. 17.4-6c). For any vector described by a_i

where R^r_ijk are the components of the mixed curvature tensor (Sec. 17.4-5)

Table 16.10-1. Differential Invariants Denned on Riemann Spaces

of the Riemann space.

* It is understood that the components g_ik of the metric tensor as well as the tensor components to be differentiated are repeatedly differentiable.

16.10-7. Differential Operators and Differential Invariants (see also Secs. 5.5-2 and 5.5-5). (a) If one defines

the covariant derivative (22) of a tensor A may be written as an “outer product” (Sec. 16.3-6) of A and the (invariant) vector differential operator

(del or nabla) whose “component” D/∂x transform like covariant vector components (Sec. 16.2-1). For every admissible transformation (16.1-1) of coordinates describing the same Riemann space

(b)Tensor relations involving covariant differentiation as well as tensor addition and multiplication are invariant with respect to the group of admissible coordinate transformations (see also Secs. 16.4-1 and 16.10-1). Tensor quantities obtained through outer and/or inner multiplication of other tensor quantities by the invariant operator (26) are called differential invariants. Table 16.10-1 lists the most useful differential invariants.

16.10-8. Absolute (Intrinsic) and Directional Derivatives (see also Sec. 5.5-3). Given a regular arc in Riemann space,

the components dxⁱ/dt represent a contravariant vector dr/dt “directed” along the tangent to the given curve (Sec. 17.4-2). The absolute (intrinsic) derivative dA/dt of a suitably differentiable absolute or relative tensor A with respect to the parameter t along the given curve is a tensor of the same rank, type, and weight as A:

If the components of A depend explicitly as well as implicitly on t,

The directional derivative dA/ds of A in the direction of the given curve (ds ≠ 0, Sec. 17.4-2) is the absolute derivative of A with respect to the arc length s along the curve.

16.10-9. Tensors Constant along a Curve. Equations of Parallelism. A tensor A is defined as constant along a regular curve arc (28) (i.e., its “values” at neighboring points on the curve are “equal”) if and only if its absolute derivative (30) (and thus also its absolute differential ) along the curve is zero. Sums and products of such tensors are also constant along the curve in question, so that, for example, the absolute values of, and the angles between, constant vectors are constant. Every vector a whose components aⁱ(x^l, x², . . . , xⁿ) or a_i(x¹, x², . . . , xⁿ) satisfy the differential equations

as the coordinates x¹, x², . . . , xⁿ vary along a curve (28) undergoes a “parallel displacement” along the curve.

NOTE: A vector obtained by “parallel displacement” of a given vector along a closed curve is not in general equal to the original vector when the starting point is reached (see also Sec. 17.4-6).

16.10-10. Integration of Tensor Quantities. Volume Element. Integrals of tensor quantities over curves in Riemann space may be defined in terms of scalar integrals over a suitable parameter as in Secs. 5.4-5 and 6.2-3a. The volume element dV is the scalar capacity (Sec. 16.2-1) defined by (see also Secs. 6.2-3b and 6.4-3c)

Volume integrals over scalar invariants are scalar invariants, but volume integrals over tensors of rank R > 0 are not, in general, tensors.

Volume elements in suitable subspaces take the place of surface elements in three-dimensional space. Generalization of the integral theorems of Secs. 5.6-1 and 5.6-2 exist (Sec. 16.10-11 and Ref. 16.6, Chap. 7).

16.10-11. Differential Invariants and Integral Theorems for Dyadics (see also Sec. 16.9-1). The divergence of a suitably differentiable dyadic A defined on a Riemann space is the vector ∇ • A, where the operator ∇ (Sec. 16.10-7) acts like a covariant vector. Note

where Ã is the transposed dyadic of . The dyadic ∇a is the gradient of a (Table 16.10-1).

The following integral theorems analogous to those of Secs. 5.6-1 and 5.6-2 hold for suitable functions, surfaces, and curves:

Vector analysis Chap. 5

Abstract algebra Chap. 12

Linear transformations Chap. 14

Rotation of a vector Chap. 14

Differential geometry Chap. 17

Partial differentiation Chap. 4

Special coordinate systems Chap. 6

16.11-2. References and Bibliography.

16.1. Brillouin, L.: Les Tenseurs en mécanique et en elasticitǩ, Masson, Paris, 1949.

16.2. Eisenhart, L. P.: An Introduction to Differential Geometry, Princeton University Press, Princeton, N.J., 1947.

16.3. Lagally, M.: Vorlesungen über Vektor-Rechnung, Edwards, Ann Arbor, Mich., 1947.

16.4. Lichnerowicz, A.: Elements of Tensor Calculus, Wiley, New York, 1962.

16.5. Phillips, H. B.: Vector Analysis, Wiley, New York, 1933.

16.6. Rainich, G. Y.: Mathematics of Relativity, Wiley, New York, 1950.

16.7. Sokolnikov, I. S.: Tensor Analysis, 2nd Ed., Wiley, New York, 1964.

16.8. Synge, J. L., and A. Schild: Tensor Calculus, University of Toronto Press, 1949.