Foundations of Classical and Quantum Electrodynamics

1

The Mathematical Methods of Electrodynamics

1.1 Vector and Tensor Algebra

1.1.1 The Definition of a Tensor and Tensor Operations

In three-dimensional space, select a rectangular and rectilinear (Cartesian¹)) system of coordinates x1, x2, x3. Regard the space as Euclidean. This means that all axioms of Euclidean geometry²) and their consequences considered in school courses on mathematics are valid in it. For instance, the square of the distance between two close points is given by the following expression:

Along with the original system of coordinates, consider some other systems of common origin yet rotated with respect to the original one (Figure 1.1).

Figure 1.1 The rotation of the Cartesian system of coordinates.

A scalar or invariant is a quantity that does not change when the system of coordinates is rotated, that is, it is the same in either the original or the rotated system of coordinates

(1.1) 2.1.gif

For instance, dl² = dl′² = inv.

In three-dimensional space, a vector is the titality of three quantities Vα(α = 1, 2, 3) defined in all coordinate systems and transformed according to the following rule:

(1.2) 2.2.gif

(summing of elements over the repeated symbol β, from 1 to 3 is assumed). Here Vβ are the projections of the vector on an axis of the original system of coordinates, V′α are the projections of the vector on an axis of the rotated system, and aαβ are the coefficients of the transformation, which are the cosines of the angles between the β axis of the original system and the α axis of the rotated system. They may be written through the single vectors (orts) of the coordinate axes:

(1.3) 2.3.gif

In three-dimensional space, a tensor of rank 2 is a nine-component quantity Tαβ (each index varies independently assuming three values: 1, 2, 3) which is defined in every system of coordinates and, when a coordinate system is rotated, is transformed as the products of the components of the two vectors Aα Vβ, in the following way:

(1.4) 2.4.gif

In three-dimensional space, a tensor of rank s is a 3s-component quantity Tλ…v that is transformed as the product of s components of vectors:

(1.5) 2.5.gif

Scalars and vectors may be regarded as tensors of rank 0 and 1, respectively.

Rotation matrix sr_ac.gif has the following properties:

1. Orthogonality

(1.6) 2.6.gif

where

(1.7)

is Kronecker symbol³);

2. The determinant of a rotation matrix equals 1:

(1.8) 3.1.gif

3. The product of two rotation matrices

(1.9) 3.2.gif

describes the evolution of a system resulting from two consecutive rotations, first with matrix sr_gc.gif and then with matrix sr_ac.gif .⁴) In the general case, rotation matrices are noncommutative, that is,

(1.10) 3.3.gif

As follows from property 1, a reverse matrix defined by the relation

(1.11)

results from the original matrix when the latter is transposed, that is, its columns are substituted for lines and vice versa:

(1.12) 3.5.gif

The reverse transformation (1.2) looks like this:

(1.13) 3.6.gif

All vectors are transformed identically according to rule (1.2) when a coordinate system is rotated. But they may behave in one of two ways if a system of coordinates is inverted, that is,

(1.14) 3.7.gif

Here the transformation matrix is aαβ = –δαβ. Vectors whose components, just like coordinates xa, change their signs during inversions are called polar (regular, real) vectors. Vectors whose components do not change sign as the result of inversions of coordinate systems are called axial vectors or pseudovectors (an angular velocity, a cross-product of two polar vectors A × B, etc.) This definition also includes tensors of arbitrary rank s: when the inversion of coordinates occurs, the components of polar (regular) tensors acquire a factor of (–1)s and the components of pseudotensors acquire a factor of (–1)s+1.

The sum of two tensors of the same rank produces a third tensor of the same rank with components

(1.15) 3.8.gif

The direct products of the components of two tensors (without summing) constitute a tensor whose rank equals the sum of the ranks of the factors, for instance,

(1.16) 4.1.gif

where Qαβγ is a tensor of rank 3.

The contraction of a tensor means the formation of a new tensor whose components are produced by the selection of components with two identical symbols and, further, their summing. For instance, Qαβγ = Aα is a vector and Qαβγ = Bβ is another vector. Contraction decreases the rank of the tensor by 2, for instance,

(1.17) 4.2.gif

is a scalar.

When an equality between tensors is written, the rule of the same tensor dimensionality must be observed: only tensors of the same rank may be equated. This means that the number of free symbols (over which no summation is done) must be the same in the first and second members of an equality. The number of pairs of mute symbols (those over which summing is done) may be any on the right and on the left.

Tensors may be symmetric (antisymmetric) with respect to a pair of indices α and β if their components satisfy the conditions

(1.18) 4.3.gif

Tensor components may be either real or complex numbers. In the latter case, the concepts of Hermitian⁵) and anti-Hermitian tensors play an important role. The definition of a Hermitian tensor is as follows:

(1.19) 4.4.gif

where the asterisk indicates complex conjugation. The definition of an anti-Hermitian tensor is as follows:

(1.20) 4.5.gif

In applications, invariant unit tensors δαβ and eαβγ are very important. The former is a symmetric polar tensor whose components coincide with the Kronecker symbol (1.7), whereas the latter is antisymmetric over any pair of indices, and its components are determined by the following conditions:

(1.21)

It is called the Levi-Civita tensor.⁶) Both tensors, transforming during rotations according to rule (1.7), are peculiar in that their components have the same values in all coordinate systems:

(1.22) 5.1.gif

Problems

1.1. Prove equality (1.8). What is the determinant of the transformation matrix if rotation is accompanied by the inversion of the coordinate axes?

1.2. Prove the equalities δ′αβ = δαβ and e′αμv = eαμv for an arbitrary rotation of a coordinate system.

1.3. Write down the rule of transformation for the components of a pseudotensor of rank s that would be valid not just for the rotation but also for the mirror reflections of the coordinate axes.

1.4. Represent an arbitrary tensor of rank 2 Tαβ as the sum of a symmetric tensor (Sαβ = Sβα) and an antisymmetric tensor (Aαβ = –Aβα). Make sure that this representation is unique.

1.5. Represent an arbitrary complex tensor of rank 2 Tαβ as the sum of a Hermitian tensor sr1h.gif and an anti-Hermitian tensor sr2h.gif . Make sure that this representation is unique.

1.6. Show that

1. the contraction of a symmetric tensor and an antisymmetric tensor equals zero: SαβAαβ = 0

2. the contraction of two Hermitian tensors or two anti-Hermitian tensors of rank 2 is a real number.

3. the contraction of a Hermitian tensor and an anti-Hermitian tensor of rank 2 is a purely imaginary number.

1.7. Show that the symmetry of a tensor is a property that is invariant with respect to rotations, that is, a tensor that is symmetric (antisymmetric) over a pair of indices in a certain system of reference remains symmetric (antisymmetric) over these indices in every system rotated with respect to the original one.

1.8. Using rules (1.2)–(1.6) of tensor transformation, show that

1. Aα is a vector (pseudovector) if AαBα = inv and Bα is a vector (pseudovector).

2. Aα isavectorif Aα = TαβBβ in any system of coordinates and Tαβ is a tensor of rank 2, and Bβ is a vector;

3. Tαα = inv, where Tαβ is a tensor of rank 2.

4. εαβ is a tensor of rank 2 if Aα and Bα are vectors and Aα = εαβBβ in all systems of coordinates. What is εαβ if Aα is a vector and Bα is a pseudovector? What is εαβ if Aα and Bα are both pseudovectors?

5. AαβλBαβ is a vector if Aαβλ and Bαβ are tensors of ranks 3 and 2, respectively.

6. TαβPαβ is a pseudoscalar if Tαβ and Pαβ are a tensor and a pseudotensor of rank 2, respectively.

1.9. Show the rule of the transformation of an aggregate of volumetric integrals Tαβ = ∫ xα xβdV in the cases of rotation and mirror reflection (xα, xα are Cartesian coordinates).

1.10. Show that the components of an antisymmetric tensor of rank 2 Aαβ = – Aβα (either polar or axial) may be identified by the components of a certain vector Cα (either axial or polar) because they are transformed in the same way in the case of rotation or reflection. In this case, Cα is called the vector dual to tensor Aαβ.

1.11. Prove the following equalities:

(1.23)

How are the vector, the dual vector, and the mixed products transformed in the cases of rotation and reflection if all three vectors are polar?

1.12. Show that if the respective components of two vectors are proportional in a certain system of coordinates, then they are also proportional in any other system of coordinates. Vectors such as these are called parallel vectors.

1.13. The area of an elementary parallelogram constructed on the small vectors dr and dr′ is represented by vector dS directed along a normal to the plane of the parallelogram and, by the absolute value, is equal to its area. Write down dSα in tensor notation.

1.14. Write down, in tensor notation, the volume dV of the elementary parallelepiped constructed on the small vectors dr, dr′, dr″. How is it transformed in the cases of rotation and reflection?

1.15. Prove the identities

(1.24)

1.16. In a spherical system of coordinates, the two directions n and n′ are determined by the angles sr_v.gif , α and sr_v.gif ′, α′. Find the cosine of the angle θ between them.

1.17. In certain cases, it may be more convenient to consider the complex cyclic components

(1.25) 7.1.gif

of the vector A instead of its Cartesian components. Express the scalar and vector products of two vectors through their cyclic components. Also, express the cyclic components of the radius vector through spherical functions.⁷)

1.18. Write down the matrix sr_gc.gif of the transformation of the components of a vector in the case of the rotation of the Cartesian system of coordinates around the Ox3 axis by angle α.

1.19. Form the matrices of the transformation of basic orts when changing from Cartesian to spherical coordinates and back and from Cartesian to cylindrical coordinates and back.

1.20. Find the matrix sr_gc.gif of the transformation of the components of a vector in the case of the rotation of the coordinate axes determined by the Euler angles⁸) α1, θ, and α2 (Figure 1.2) by mutually multiplying matrices corresponding to rotation around the Ox3 axis by angle α1, around the line of nodes ON by angle θ, and around the Ox′3 axis by angle α2.

Figure 1.2 The specification of the rotation of Cartesian axes by Euler angles α1, θ, α2.

1.21. Find the matrix sr_dc.gif (α1θ α2) used for transforming the cyclic components of vector (1.25) when rotating the system of coordinates. The rotation is determined by the Euler angles α1, θ, and α2 (Figure 1.2).

1.22. Show that the matrix of an infinitesimal rotation of a coordinate system may be written as sr_ac.gif = 1 + sr_3c.gif , where sr_3c.gif is an antisymmetric matrix (εαβ = –εβα). Find the geometric meaning of εαβ.

1.23. Show that the representation of a small rotation by vector δφ used in the solution of the previous problem is only possible in relation to quantities of the first order of smallness. In the next order, the vector of the resulting rotation is not equal to the sum of the vectors of individual rotations and the relevant matrices do not commute.

1.1.2 The Principal Values and Invariants of a Symmetric Tensor of Rank 2

The selection of a system of coordinates wherein a certain tensor has the simplest structure is of great practical importance. Consider the selection of such a system for a symmetric tensor of rank 2.

If vector n satisfies the condition

(1.26) 8.1.gif

where S is a certain scalar, then the direction that is determined by vector n is called the principal direction of the tensor, vector n is called the proper vector of the tensor, and S is called its principal value.

Example 1.1

Reducing a real sr3h.gif symmetric (Sαβ = Sβα) tensor of rank 2 to diagonal form means finding such a system of axes wherein only the diagonal components of the tensor are not equal to zero. Specify a way of calculation of the principal values and the principal directions of such tensor.

Solution. Use the system of algebraic equations (1.26) to find the proper vectors and principal values of the tensor in question. Normalize the proper vectors to 1: sr4h.gif . The equations (1.26) and the properties of the tensor Sαβ show us that the proper values of S are real scalars: sr5h.gif . They follow from the condition of equality to zero of the determinant of the system (1.26):

(1.27) 8.2.gif

This is a cubic algebraic equation whose solution, in relation to S, includes three real roots: S(1), S(2), S(3). In the general case, they are different from each other, although multiple roots (S(1) = S(2) ≠ S(3) or S(1) = S(2) = S(3)) are possible. Here, the bracketed indices are not tensor symbols!

In the case of different roots, inserting the values found for S, one by one, in the system in (1.26) results in two projections of each of the proper vectors s6h.gif through the third one, which is determined by the condition of normalization. All the proper vectors are real because the coefficients of (1.26) are real. They are mutually perpendicular, which follows from the same system of equations: (S(1) – S(2))(n(1) · n(2)) = 0. The same goes for the other two pairs. Regarding the proper vectors as the orts of the system of coordinates (they determine the principal axes of the tensor), use (1.26) to find the form of the tensor in this system of axes:

(1.28) 9.1.gif

In the case of two repeated roots, S(1) = S(2), the proper vectors n(1) and n(2) are determined ambiguously, that is, any pair of mutually perpendicular directions may be selected in the plane perpendicular to n(3). If all three roots are the same, then any three mutually perpendicular directions may be regarded as the principal axes. box.gif

Problems

1.24. Is it possible to reduce an arbitrary real tensor of rank 2(Tαβ ≠ Tβα) to the diagonal form by rotating its system of coordinates in physical three-dimensional space? What about a Hermitian tensor of rank sr7h.gif ?

1.25. Write down a real symmetric tensor of rank 2 Sαβ in an arbitrary system of coordinates through its principal values S(1), S(2), S(3) and the orts sr8h.gif of the principal axes.

1.26. Using the characteristic (1.27), compile the invariants relative to rotation from the components of an arbitrary tensor of rank 2 Tαβ.

1.27. Using the theorem for the expansion of the determinant in the elements of a row or a column, find the components of the inverse tensor sr9h.gif . Its definition coincides with that of (1.11) for the inverse matrix. Indicate the condition of the existence of an inverse tensor.

1.28. Prove the identities

Using the third identity, prove the formula of vector algebra

1.29. Write down the following in the invariant vector form:

1. sr10h.gif

2. sr11h.gif

1.30. Prove the identity

where Tαβ is an arbitrary tensor of rank 2, A and B are vectors, and C is the vector of the dual antisymmetric part of the tensor Tαβ.

1.31. Present the product (A · (B × C))(A′ · (B′ × C′)) as the sum of members that contain only the scalar products of the vectors.

Hint: Apply the theorem for the multiplication of determinants or use the pseudotensor eαβγ.

1.32. Show that the only vector whose components are the same in all systems of coordinates is a null vector, that any tensor of rank 3 whose components are the same in all systems of coordinates is proportional to eαβγ, and that any tensor of rank 4 whose components are the same in all systems of coordinates is proportional to sr12h.gif

1.33. Regard n as a unit vector whose directions in space are equiprobable. Find the mean values of its components and their products – nα, nα nβ, nα nβ nγ, nα nβ nγ nv – using the transformational properties of the quantities sought.

1.34. Find the average values for all directions of the expressions (a · n)², (a · n)(b · n), (a · n)n, (a × n)², (a × n) · (b × n), (a · n)(b · n)(c · n)(d · n), if n is a unt vector whose all directions are equiprobable and a, b, c, d are constant vectors.

Hint: Use the results obtained in the previous problem.

1.35. Write down all possible invariants of polar vectors n, and n′ and pseudovector l.

1.36. What independent pseudoscalars may be made of two polar vectors n and n′ and one pseudovector l? What independent pseudoscalars may be made of three polar vectors n1, n2, and n3?

1.1.3 Covariant and Contravariant Components

In physics, many problems require nonorthogonal and curvilinear systems of coordinates be used so that the relations between the old and new coordinates are nonlinear and different from (1.2). The transition to new coordinates may not come down to just the simple and obvious rotation of axes. One of the most important areas where such a mathematical apparatus needs to be used is special and, especially, general relativity.

Closing this section, we will come up with the definition of tensors with respect to overall transformations of coordinates and consider their basic properties in three-dimensional Euclidean space. This is appropriate because in three-dimensional space the meaning of many concepts and relation is more obvious and transparent than in four-dimensional space–time of the relativistic theory. We will begin by immersing ourselves in these issues by considering a case that is half way between Cartesian rectangular coordinates and common coordinates when the coordinate axes of the reference frame are still rectilinear but become nonorthogonal (oblique or affine coordinates).

Example 1.2

Three noncoplanar and nonorthogonal unit vectors e1, e2, and e3 are selected as the basic vectors in a three-dimensional Euclidean space. Three systems of rectilinear lines passing through every point of the space and parallel to the basic vectors are the coordinate lines. Build a mutual basis e1, e2, e3 which, by definition, is connected to the original basis by the following relations:

(1.29) 11.1.gif

Will the vectors of the mutual basis be unit vectors?

Expand an arbitrary vector A (including also the radius vector r) in vectors eα and eβ of the original and mutual bases. Show the geometric meaning of its components in both cases (in the first case, they are called contravariant and are labeled with upper indices, A¹, A², A³. In the second case, they are covariant, and are labeled with lower indices, A1, A2, A3).

Solution. In accordance with (1.29), e¹ must be perpendicular to e2 and e3. Look for it in the form of e¹ = ke2 × e3 and, from the condition of normalization e¹ · e1 = 1, find

where k–1 = sr13h.gif is the volume of the parallelepiped built on the vectors of the original basis. sr13h.gif > 0 if the right-hand system of coordinates is selected. Therefore,

(1.30) 12.1.gif

where α, β, and γ form a cyclic permutation. Radius vector r and any other vectors are expanded in basic vectors in the usual way:

(1.31)

Multiplying the first equality, in a scalar way, by eα, we find

(1.32) 12.3.gif

Therefore, the geometric meaning of the covariant components is revealed by projecting the radius vector, in the usual way, by lowering perpendiculars from the end of the vector onto the coordinate axes. When this has been done, the directions of the contravariant basic vectors, by which the covariant components of the vector are multiplied, do not coincide with the directions of the coordinate axes (Figure 1.3) and have no unit lengths. For instance, if vector e3 is orthogonal to e1 and e2 and the angle between the latter is ϕ, then |e¹| = |e²| = 1/sin ϕ and the length of the hypotenuse OB = |x1 e¹| = x1/sin ϕ > x1. However, the length of the leg OC = x1. As follows from (1.31) and Figure 1.3, the contravariant components result from projecting the vector onto the coordinate axes with segments parallel to the axes. For them, a representation identical to (1.32) is valid:

(1.33) 12.4.gif

Figure 1.3 The clarification of the geometric meaning of the covariant and contravariant components of a vector.

Example 1.3

Determine the nine-component quantities:

(1.34) 13.1.gif

where eα and eβ are the basic vectors of the original and mutual nonorthogonal bases, introduced in Example 1.2. The values gαβ and gαβ are called the covariant and contravariant components of a metric tensor.

Prove the following relations that connect the covariant and contravariant components of an arbitrary vector (the rules of rasing and lowering indices):

(1.35)

Here, sr14h.gif is a Kronecker symbol.

Find the determinants of a covariant and a contravariant metric tensor and express them through the volumes sr13h.gif and sr15h.gif of parallelepipeds built on the vectors of the original and mutual bases.

Solution. The expression below follows from expansion (1.31):

Multiplying it, in a scalar way, by eα and using the definitions of mutual basis (1.30) and metric tensor (1.34), we get the first expression in (1.35); multiplying this expansion, in a scalar way, by eα, we get the second expression in (1.35); and inserting the second expression in (1.35) in the first expression in (1.35), we get the third expression in (1.35).

If we label g = |gαβ| and use definition (1.34) and the formula from the first task in Problem 1.29, we find the following:

In the same way, we get |gαβ| = sr15h.gif ². As follows from (1.35), |gαβ|g = 1, therefore, sr16h.gif and sr15h.gif = sr13h.gif –1. box.gif

Problems

1.37. When we transition from one oblique rectilinear system of coordinates to another, the basic vectors eα determining the directions of the coordinate axes are transformed in accordance with the following law:

(1.36) 14.1.gif

where aαβ is the transformation matrix.⁹)

1. Express its elements through the scalar products of the basic vectors of the original and transformed systems.

2. Build the reverse transformation matrix.

3. Show that the same matrices define the transformations of the vectors of the mutual basis.

4. Find the rules of the transformation of the covariant and contravariant components of an arbitrary vector.

5. Find the rules of the transformation of the covariant and contravariant components of a metric tensor.

1.38. Show the laws of the transformation of the vectors of the original and mutual bases in the case of the mirror reflection of the system of coordinates.

1.39. Express the scalar product of two vectors in three different forms: through the covariant and contravariant components and through both of them. Prove its invariance with respect to the transformations (1.36) of the coordinate system. Express, in various forms, the square of the distance dl² between two close points.

1.40. Write down the vector product of two vectors C = A × B in terms the covariant and contravariant components of the factors.

1.41. Write down the cosine of the angle between vectors A and B in terms of their covariant and contravariant components.

1.1.4 Tensors in Curvilinear and Nonorthogonal Systems of Coordinates

We will now consider arbitrary transformations in the case of a transition from a Cartesian to a certain curvilinear and, generally speaking, nonorthogonal system of coordinates or between curvilinear and nonorthogonal systems of coordinates (Borisenko and Tarapov, 1966, Section 2.8). The connection between the coordinates xα and x′β (α, β = 1, 2, 3) of two coordinate systems described by certain general form relations is

(1.37) 14.2.gif

(we will now indicate coordinate numbers with upper indices). The linear homogeneous function fα(x′¹, x′², x′³) with constant coefficients corresponds to the affine transformation (1.36). The rotation of the orthogonal rectilinear coordinate system is determined by the orthogonal matrix of coefficients with a unit determinant.

So that (1.37) can be solved with respect to x′β and the reverse transformation x′β = φβ(x¹, x², x³) can be found, the functional determinant J must be different from zero,

(1.38) 15.1.gif

which hereafter will be presumed. The differentials of the coordinates are transformed in accordance with

(1.39) 15.2.gif

where the coefficients of the transformation sr17h.gif , in the general case, become the functions of the coordinates. The connection between the differentials remains linear, as in the case of affine transformations, which, generally speaking, is not the case for the connection between the coordinates themselves. Although (1.37) describes the transition from the orthogonal Cartesian system of coordinates xα to an arbitrary system qβ (to make things clearer, we hereafter will label curvilinear coordinates as q), we will write the square of the distance between close points with the use of (1.39) as

(1.40) 15.3.gif

where the values

(1.41) 15.4.gif

are called the covariant components of the metric tensor, and its contravariant components gμv = gvμ are determined by the conditions

(1.42) 15.5.gif

which means that the tensors gμv and gμv are mutually inverse. Because the coefficients of transformation (1.39) satisfy the relation

(1.43) 15.6.gif

the contravariant components of the metric tensor may be written as¹⁰)

(1.44) 15.7.gif

The latter relations, just like (1.41), may be regarded as the rule of the transformation of the metric tensor from Cartesian coordinates (δσk) to arbitrary curvilinear coordinates qα. It is easy to see that the same rule applies to the transformation of the metric tensor from a curvilinear system qα to another curvilinear system q′β:

(1.45)

where gαβ is defined in accordance with (1.44).

One can easily make sure that the relations written above mostly repeat the formulas obtained when considering the oblique-angled (affine) system of coordinates, being their generalizations, in a certain way. For instance, multiplying both parts of (1.39) by the Cartesian orts sr19h.gif and relabeling x′β as qβ, we get the increase of the radius vector

This means that the basic vectors eβ of the curvilinear system (not unit in the general case) may be written as

(1.46) 16.3.gif

The right-hand side of the latter equality includes Cartesian orthogonal unit vectors. As follows from (1.46), the connection between the basic vectors of the curvilinear systems of coordinates q′μ and qβ looks the same way as (1.46):

(1.47) 16.4.gif

Further on, we will define the vectors of the mutual basis eβ of the curvilinear system. As follows from (1.46) and the conditions in (1.29),

(1.48) 16.5.gif

which means that

(1.49) 16.6.gif

(we use the equality of the lower and upper symbols for Cartesian vectors). Finally, considering (1.41) and (1.44), we see that the relations in (1.34) remain valid for curvilinear coordinates,

(1.50)

as do the rules of raising and lowering indices (1.35).

We now will give a definition of tensor, as it relates to the general transformations of coordinates.

A tensor of rank 2 in the three-dimensional space is a nine-component quantity whose contravariant components are transformed as products of the differentials of coordinates, that is, in accordance with the following:

(1.51)

This definition is directly generalized to include tensors of any rank. For instance, scalar S is not transformed, whereas the covariant components of a tensor of rank 1 (vector) are transformed in accordance with

(1.52) 17.2.gif

The fundamental difference between the above definition of a tensor and the previous ones (for the cases of rotation and affine transformation) is that now the transformation coefficients depend on the locations. This means that the definition of a tensor is of a local nature. For instance, the products of the components of vectors located at different points qα ≠ pα, that is, Aα(q)Bβ(p), do not form a tensor.

Unlike Cartesian coordinates, the totality of arbitrary curvilinear coordinates qα, α = 1, 2, 3, does not form a vector because the coordinates do not comply with rule of transformation (1.51). Most significantly, these peculiarities manifest themselves in differentiating and integrating tensor operations, which are considered in Section 1.2.

The covariant components of a tensor of any rank are produced from the contravariant ones by the metric tensor as per (1.35). In the general case, the mixed tensor depends on the place, first or second, occupied by the upper and lower symbols, that is, Tαβ ≠ Tβα. The contraction operation, decreasing the rank of any tensor by 2, is defined as summation over one upper and one lower indices, for instance,

(1.53)

– the covariant vector, and so on.

Problems

1.42. Express the components of a metric tensor through the components of the orthogonal Cartesian orts sr20h.gif specified in a certain curvilinear system of coordinates.

1.43. Show that the functional determinant (1.39) is expressed through the determinant of a metric tensor sr21h.gif .

Hint: Following from equality (1.42), express the determinant g through the determinants of the matrices found in the second member of the equality.

1.44. Write down the square of the length of the vector A² and the cosine of the angle between two vectors in a arbitrary curvilinear system of coordinates.

1.45. Transform the antisymmetric unit tensor eαβγ in an curvilinear system of coordinates.

1.46. The metric tensor gαβ determining the square of the small element of length in curvilinear nonorthogonal coordinates, in accordance with formulas (1.41), is known. Three curvilinear coordinate lines may be drawn through each point of the space, only one coordinate q¹, q², or q³ changing along each of these lines, whereas the other two remain constant.

1. Find the connection between the element of length of a coordinate line and the differential of the respective coordinate.

2. Indicate the three basic vectors tangent to the coordinate curves at the specified point.

3. Find the cosines of the angles between the coordinate curves at that point.

4. Indicate the properties the metric tensor must have to make the curvilinear system orthogonal.

1.47. Write down the covariant and contravariant components of a metric tensor for a spherical and a cylindrical system of coordinates (see the drawing in the solution of Problem 1.18). Also, write down the vectors of the covariant and contravariant bases, expressing them through the basic orts considered in Problem 1.18.

1.48. Show that the volume element in curvilinear coordinates has the following form:

(1.54) 18.1.gif

where g is the determinant of a metric tensor. Find the volume element in spherical and cylindrical coordinates.

Hint: The volume element sought is the volume of an oblique-angled parallelepiped built on the elementary lengths dl¹ dl², and dl³ of the curvilinear coordinate axes. It may be found with the use of the results obtained in Problems 1.40 and 1.46.

Recommended literature:

Borisenko and Tarapov (1966); Arfken (1970); Rashevskii (1953); Lee (1965); Mathews and Walker (1964). See also Ugarov (1997, Addendum I).

1.2 Vector and Tensor Calculus

Scalar or vector functions representing the distribution of various physical quantities in three-dimensional space are sometimes called the fields of those quantities. This is how one may speak of fields of temperatures T(x, y, z) or pressures p(x, y, z) in the atmosphere, the fields of speeds in moving fluids or gases u(x, y, z), the electromagnetic vector field, and so on. Derivatives and integrals from such scalar and vector functions have certain common mathematical properties, which are very important for physical applications. One should become familiar and comfortable with these properties in advance. Only then, may such areas of physics as the theory of electromagnetic phenomena, the mechanics of fluids, gases, and solid bodies, quantum physics, and quantum field theory be successfully learned and fully understood.

1.2.1 Gradient and Directional Derivative. Vector Lines

We encounter the concept of the gradient of a scalar function in classical mechanics when learning about the properties of potential forces. Let us say there is a differentiable function U(x, y, z) whose partial derivatives are equal to the components of the vector of the force F(x, y, z), which, in this case, is called a potential.

(1.55)

where

(1.56)

is Hamilton’s operator¹¹) (nabla).

(1.57)

is called the gradient of the scalar function U(x, y, z). The necessary and sufficient conditions for the representation of the vector as a scalar function come in the form of equalities:

(1.58)

They follow from the equality of cross-derivatives, for example,

So far, we have been using only Cartesian coordinates. A generalization to include oblique nonorthogonal coordinates will be made in the closing part of this section (also see Problem 1.50 and later).

It is important to understand that a gradient is always directed toward increasing U, along a normal to the surface of the constant value of the scalar field U(x, y, z) = const. This follows from our obtaining, when differentiating the latter equality, dr · ∇U = 0. Since dr is here a tangent to the surface U = const, the gradient is perpendicular to that surface.

Example 1.4

Show that the derivative of the scalar function, along the direction determined by the unit vector l, is equal to the projection of the gradient onto that direction:

(1.59) 20.1.gif

Solution. Label the derivative, along the specified direction l, as ∂U/∂l. When displaced from the point with radius vector r to a distance s along the direction l, the function will take the value of U(x + lxs, y + lys, z + lzs). The derivative in the specified direction is the derivative at distance s:

Expression (1.58) also makes sense when applied to an arbitrary vector A(x, y, z): the quantity (l · ∇)A(x, y, z) is a derivative of vector A in direction l. This follows from the condition that the operator (l · ∇) must be applied to every projection of A and will produce the required derivatives, whereas their combination must be construed as a derivative of the whole vector in the specified direction.

A vivid conception of the structure of the vector field A is provided by vector lines.¹²) These are lines tangents to which, at any point, indicate the direction of vector A at that point. It is easy to write a system of equations in order to find the vector lines of the specified field A(x, y, z). The condition of the small element dl = (dx, dy, dz) being parallel to the vector line and vector A may be written as A × dl = 0. Having written this vector equality in projections on the respective axes, we get differential equations for two families of surfaces whose intersect lines are exactly the vector lines sought.

For instance, using Cartesian coordinates, we will have

(1.60)

Figure 1.4 The independence of work done by a potential force from the shape of the path of a material point.

The vector lines of any potential vector are perpendicular to the equipotential surfaces U(x, y, z) = const. This follows from the properties of the gradient of a scalar function.

The loop integral of the scalar product of a potential vector and the vector element of the length of the loop has an important property:

(1.61)

where the vector ds has constituents dx, dy, and dz, that is, the differentials of the coordinates are not independent and are just increments along the loop. Such integrals express work done by the force F on a material point moving along a specified trajectory from A to B and many other physical quantities. If the vector is a potential vector, then

(1.62)

is the complete differential of the function U(x, y, z). The computation of the integral gives us

(1.63) 21.4.gif

where dU is the increase of the function along the small segment ds and sr22h.gif is the full increase along the distance AB.

In this case, integration along the loop does not depend on the form of the curve, and only depends on the start and end points of the integration (Figure 1.4).

Integrating along a closed loop (Figure 1.5), we get the following:

(1.64) 21.6.gif

Figure 1.5 Diagram for the computation of the circulation of a vector along a closed loop.

Closed-loop integration over F · ds is called the circulation of vector F along the loop. The circulation of a potential vector along any closed loop equals zero (however, an arbitrary vector has no such property!).

It is important, however, to note that the condition of the representation of a vector as (1.55) is necessary but not sufficient for equalities (1.63) and (1.64) to be valid. It is also necessary for the potential function U(r) to be the unambiguous function of a point. Otherwise, for instance, after the circulation of the loop and return to point A, the potential U may take a different value, and equality (1.64) will be no longer valid.

Problems

1.49. Show that when a Cartesian system of coordinates is rotated, Hamilton’s operator (∇) (1.56) is transformed in accordance with rule (1.2) of vector transformation.

1.50. Find the potential energy that corresponds to the force Fx(x, y) = x + y, Fy(x, y) = x – y². Find the work R done by this force between points (a, b)

1.51. Show that in cylindrical and spherical systems of coordinates, Hamilton’s operator ∇ is expressed, respectively, as

(1.65) 22.2.gif

(1.66)

For that purpose, consider the elementary lengths in the directions of the respective coordinate orts and use formula (1.59), which connects the gradient with the directional derivative.

1.52. Use Cartesian spherical and cylindrical coordinates (see (1.56), (1.65), and (1.66)) to find grad (l · r), (l · ∇)r, where r is a radius vector and l is a constant vector.

1.53. Show that

1.54. Write down a system of equations determining the vector lines in cylindrical and spherical coordinates, respectively.

1.55. Find

1.56. Use spherical coordinates to draw a family of lines tangent to vector

1.57. Write down the cyclic components of a gradient in spherical coordinates. Find the definition of the cyclic components in the situation in Problem 1.17.

1.2.2 Divergence and Curl. Integral Theorems

Now, we will consider the effect of the ∇ operator on an arbitrary vector A. As is known, two vectors may produce two types of products: a scalar

(1.67)

and a vector

(1.68)

Both of these quantities are extremely important for vector calculus and are called the divergence (scalar!) and the curl (vector!). The left-hand-side members of the equalities contain the respective lettering. The right-hand-side members contain their explicit expressions in Cartesian coordinates only. For you to better realize their mathematical and physical meanings, we give other definitions of these important quantities, less formal and more obvious, if somewhat more complex. Yet the latter disadvantage is also an advantage in that the definitions in questions, unlike (1.67) and (1.68), do not depend on the selection of a system of coordinates. We will begin with divergence.

Select point M where you would like to define the divergence of vector field A(r). Surround that point with a closed smooth surface, enclosing a certain volume ∇V and find at every point of the surface an outside normal n. We will call the product ndS the vector element of the surface. The integral over the closed surface sr23h.gif A · dS produces the flux of the vector A through the surface S. Now, we will define divergence in a way different from (1.67):

(1.69) 23.5.gif

It is presumed here that the volume ΔV shrinks into point M. The little circle on the integral sign means a closed surface.

Example 1.5

Make sure that the definitions (1.67) and (1.69) are equivalent when Cartesian coordinates are used. In order to do that, select volume ΔV = dV = dxdydz forming a small rectangular parallelepiped with edges dx,dy,dz and find the boundary (1.69).

Solution. Making use of the smallness of the ribs of the parallelepiped, write down the approximate expression for the surface integral:

The mean value theorem was used when evaluating the integrals over the six separate edges, the quantities sr24h.gif , and sr_z1.gif being the values of the coordinates at a certain point of a respective edge. Also considered was the fact that normals are directed oppositely at the opposite edges and that when the volume shrinks to point M, all the coordinates take the values they must have at that point. Using the latter result, make sure that the definition of divergence (1.69), when Cartesian coordinates are used, leads to formula (1.67). box.gif

This means that the divergence at a certain point is other than zero if there is a nonzero vector flux through a closed surface surrounding the point in question. Inside the surface, there must be a source of a vector field that creates the flux. This is to say that divergence characterizes the density of field sources.

The above method of computing an integral over a small surface may be used to obtain explicit expressions of divergence in the most often used systems of coordinates, such as spherical, cylindrical, and so on. The shape of the volume should be selected each time so that one of the coordinates remains constant on each of its side surfaces.

Example 1.6

On the basis of the definition of divergence (1.67), produce a relation connecting the integral from div A over a certain volume with vector flux A through the surface bounding the volume in question.

Solution. Select any finite volume V bounded by a smooth closed surface S. Divide it into small cells ΔVi, each bounded by a respective surface ΔSi. The surfaces bounding the cells adjacent to the outside surface S will partially coincide with S. All other portions of the surfaces Si will be shared by pairs of adjacent cells. Making use of the smallness of each cell, use relation (1.69), giving it an approximate form:

(1.70) 25.1.gif

Now sum the first and second members of the latter approximate equality over i and pass to a limit, reducing the volume of each cell to zero and expanding the number of cells to infinity. The first member of the equality will now become an integral over the full volume V of divergence A: ∫V div AdV. In the second member of the equality, the integrals over the inner portions of the surface will cancel each other, the outer normals to each pair of adjacent cells being oppositely directed. Only the integral over the outside surface S bounding the full volume V remains. As a result, you will have an exact integral relation,

(1.71) 25.2.gif

called the Gauss–Ostrogradskn theorem¹³) (in Western literature, the name Ostrogradskii is omitted).

The Gauss–Ostrogradskii theorem is applicable to any tensor of rank s ≥ 1, for instance,

(1.72) 25.3.gif

(for the proof, refer to Problem 1.70*). box.gif

The curl of a vector field allows a definition similar to that of divergence (1.69). At point M, specify a unit vector n, that is, a direction. Make up a small flat area ΔS containing a point M and perpendicular to n. Then define the direction of tracing the loop l that bounds the area, coordinated with the direction n as per the right-screw rule. The projection of the rotor onto direction n at point M is defined as follows:

(1.73) 25.4.gif

where the integral represents the circulation of the vector A along the closed loop l.

Example 1.7

Make sure that the definitions of (1.68) and (1.73) are equivalent when Cartesian coordinates are used. For that purpose, find the projections of the curl on Cartesian axes using (1.73) and by selecting a rectangular area with sides parallel to the coordinate axes.

Solution. Direct n along the Oz axis, select a rectangular area ΔS = dS = dxdy, and use, as in the previous integral calculation, the mean value theorem to get the following:

After inserting this result into (1.73) and passing to a limit, we get the exact expression for curlzA in Cartesian coordinates, coinciding with (1.68). In the same way, one may find other projections of the curl. The curl will be other than zero if the lines of vector A curved, having either closed or spiral configurations. box.gif

Example 1.8

Using the definition of the curl (1.73), find the integral relation that connects the circulation of any vector along a closed loop with the curl flux of that vector through a nonclosed surface bounded by that loop.

Solution. Find an arbitrary three-dimensional nonclosed surface S bounded by loop l and, at every point of the surface, find normal n. Divide the surface into small portions ΔSi, each bounded by loop li. On the basis of (1.73), an approximate value may be written for every such area:

(1.74) 26.2.gif

After summing the two members of the approximate equality over i and passing to a limit of the infinitely small areas, we get the exact equality (Stokes theorem¹⁴)):

(1.75) 26.3.gif

An integral over the outer loop that bounds area S remains in the second member. All integrals over inner loops are canceled. Stokes theorem connects the integral over the curl flux through the surface with the circulation of the vector along the loop that bounds that surface. box.gif

1.2.3 Solenoidal and Potential (Curl-less) Vectors

Let us say that vector field H(r), over the whole space, satisfies the condition

(1.76) 27.1.gif

(in this case, vector H is called a solenoidal vector). This, for instance, is a property of a magnetic field. It is possible to prove (we will, for now, abstain from doing that) that condition (1.76) is necessary and sufficient for vector H to be represented as the curl of another vector A(r):

(1.77) 27.2.gif

Using the rules of vector differentiation, we can easily make sure that condition (1.76) is satisfied whatever the value of A is:

As noted previously, a potential vector is a vector that may be represented as the gradient of a certain scalar function:

(1.78) 27.4.gif

The necessary and sufficient conditions of the potentiality of a vector are expressed by equalities of the kind in (1.58), which, in their vector form, give the following:

(1.79) 27.5.gif

Using the definition of the potential vector (1.78) and expressing the curl operation through the ∇ operator, we make sure that equality (1.79) is equally valid for any U(r) functions that have second derivatives.

1.2.4 Differential Operations of Second Order

Differential operations of second order appear when the ∇ operator is applied to expressions of the kind ∇U, ∇ · A, and ∇ × A that already contain this operator. Using the rules of vector algebra, we find that, in Cartesian coordinates, the Laplace operator¹⁵)

(1.80)

Δ = ∇², has the following form:

(1.81) 28.2.gif

This is a very important operator used in just about all problems when complex physical phenomena have to be described in the language of mathematics.

Further,

(1.82) 28.3.gif

Even though such a combination of derivatives is hardly rare, no more compact letter notation has been devised for it.

The last operation of this kind is called a double vortex. It is transformed with the use of the following vector algebra formula (one should remember to place the differentiable vector function to the right of any operators that may affect it):

(1.83)

We see, therefore, that all the differential operations involving scalar and vector functions are expressed through the ∇ operator.

Problems

1.58. Show that div A (1.67) and the Laplace operator (1.81) are invariant with respect to rotations of Cartesian systems of coordinates and that curl A (1.68) is transformed as an antisymmetric tensor of rank 2 or as a vector that is dual to it.

1.59. Find ∇ · r, ∇ × r, ∇ · [ω × r], and ∇ × [ω × r], where ω is a constant vector.

1.60. Find

Build vector lines for vector H (draw a picture).

1.61. Using the rules of vector algebra and calculus and without making projections onto the coordinate axes, prove the following important identities frequently used in practical calculations:

(1.84) 28.6.gif

(1.85) 29.1.gif

(1.86) 29.2.gif

(1.87)

(1.88)

(1.89)

Here, φ and ψ are the scalar and A, B vector functions of the coordinates.

1.62. Prove the following identities:

(1.90)

(1.91)

(1.92) 29.8.gif

(1.93)

(1.94)

(1.95)

1.63. Find grad φ(r), div φ(r)r, curl φ(r)r, and (l · ∇)φ(r)r.

1.64. Find a function φ(r) that satisfies the condition div φ(r)r = 0.

1.65. Find the divergences and curls of the following vectors:

where a and b are constant vectors.

1.66. Find grad r · A(r), grad A(r) · B(r), div φ(r)A(r), curl φ(r)A(r), and (l · ∇)φ(r)A(r).

1.67. Prove that

1.68. Transform the integral over volume ∫V(grad φ · curl A)dV into the integral over the surface.

1.69. Express the integrals over the closed surface sr23h.gif r(a · dS) and sr23h.gif (a · r)dS in terms of the volume bounded by that surface. Here a is a constant vector.

Hint: Multiply each integral by the arbitrary constant vector b and use the Gauss–Ostrogradskii theorem

1.70*. Transform the integrals over a closed surface

into integrals over the volume bounded by that surface. Here b is a constant vector and n is the ort of the normal.

1.71. Using one of the identities proven in the previous problem, formulate the Archimedean law by summing pressures applied to the elements of the surface of a submerged body.

1.72*. Prove the identity

(1.96)

1.73. Inside volume V, vector A satisfies the condition div A = 0 and at the boundary of the volume (surface S) the condition An = 0. Prove that ∫V AdV = 0.

1.74*. Prove that

where A(r) is the vector defined in the previous problem.

1.75. Prove the Green’s identities¹⁶)

(1.97)

(1.98)

where φ an ψ are scalar differentiable functions.

1.76. Transform the integral over the closed loop ϕi ud f into the integral over a surface bounded by that loop.

1.77*. Prove the integral identities

(1.99) 30.6.gif

(1.100) 31.1.gif

(1.101) 31.2.gif

Here n is the ort of the normal to the surface, φ and A are functions of the coordinates, l is a closed loop, and S is a nonclosed surface bounded by that loop. These identities may be regarded as special cases of the generalized Stokes theorem

(1.102) 31.3.gif

where the symbol (…) labels a tensor of any rank.

1.78. Show that if the scalar function ψ is a solution of the Helmholtz equation¹⁷) Δψ + k²ψ = 0 and a is a certain constant vector, then the vector functions L = ∇ψ, M = ∇ × (aψ), and N = ∇ × M satisfy the Helmholtz vector equation ΔA + k²A = 0.

1.2.5 Differentiating in Curvilinear Coordinates

Unlike in Cartesian rectangular coordinates, when we use curvilinear nonorthogonal coordinates qα(α = 1, 2, 3), xβ(β = 1, 2, 3), the derivative over coordinates from a tensor of rank s ≥ 1 does not produce any tensor, which we will see later. This is due to the local nature of the definition of the tensor (1.51) applicable to a certain point. In the meantime, a derivative is defined through the difference of the values of two vectors at close but still different points. In order to define a covariant derivative from a tensor of any rank, that is, such a differential operation that increases the rank of a tensor by one, we will, for simplicity, consider a tensor of rank 1 (vector) and expand it in basic vectors of the curvilinear system of coordinates in question:

(1.103) 31.4.gif

Differentiate the equalities in (1.103) and form the covariant derivatives:

(1.104)

(1.105)

The first members of the equalities use the notation commonly accepted for covariant derivatives of covariant and contravariant vector components, respectively. The sign of the identity is followed by their definitions. The second members include derivatives of the components of the vector and basic vectors. In curvilinear systems of coordinates, unlike in Cartesian coordinates, derivatives of basic vectors are not equal to zero.

Differentiating equality (1.48) over the coordinate, we find that

(1.106) 32.1.gif

Now, add the Christoffel symbols of the second kind to our consideration:¹⁸)

(1.107) 32.2.gif

They allow us to write covariant derivatives in a more compact form:

(1.108)

Christoffel symbols are not tensors since they do not satisfy the applicable rules of transformation. They are symmetric as to the two lower symbols: sr25h.gif . The latter property follows from the representation of basic vectors (1.46):

(1.109) 32.4.gif

The rules (1.108) of computing a covariant derivative of a tensor of rank 1 are generalized, in an obvious way, to include tensor T of any rank. Besides the derivative over the coordinate from the tensor in question, one needs to add as many terms with a plus sign as the tensor has upper symbols and as many terms with a minus sign as the tensor has lower symbols.

Example 1.9

Express the Christoffel symbols (1.107) through the components of metric tensor gμv.

Solution. The definition (1.107) of Christoffel symbols allows us to write the following relation:

(1.110) 32.5.gif

It follows from the equality sr26h.gif which follows from the representations of basic vectors (1.46) and (1.49).

If we use the relation,

(1.111) 33.1.gif

Also consider Christoffel symbols of the first kind, Γv,μα

As follows from (1.110) and (1.111), Christoffel symbols of the first and second kinds may be regarded as the coefficients of the expansion of the quantity ∂eμ/∂qα in vectors of covariant and contravariant bases.

Using (1.48), we find from (1.111) that

(1.112) 33.2.gif

Multiplying (1.110), in a scalar way, by eλ and (1.111) by eλ and using (1.50), we find the connection between Christoffel symbols of the first and second kinds:

(1.113) 33.3.gif

Then, sequentially using the symmetry of the two symbols separated by a comma and relations (1.109), (1.112), and (1.113), find the following: box.gif

(1.114)

(1.115)

Example 1.10

Find the rules of the transformation of Christoffel symbols of the first and second kinds when they are transferred to another curvilinear coordinate system.

Solution. Do the sequential computations

(1.116)

(1.117)

Only the first terms in the second members of the resulting expressions conform to the rules of the transformation of tensors. The second terms violate the said rules, which means that Christoffel symbols are not tensors. box.gif

Example 1.11

Prove that the covariant derivatives of the vectors Av;α and sr27h.gif are transformed as covariant and mixed tensors, respectively, of rank 2.

Solution. Using the definition of covariant derivative (1.104) and the rule of transformation (1.116), sequentially find the following:

(1.118)

It has been proven that the quantity in question is transformed as a covariant tensor of rank 2. When considering the second tensor, one must use the following equality:

(1.119) 34.4.gif

It follows from differentiating over the coordinate of an equality such as (1.43). box.gif

Problems

1.79. Show that a derivative of a coordinate of the scalar (gradient) ∂S/∂qμ = S:μ, is a covariant vector.

1.80. Show that a covariant curl coincides with a proper curl:

1.81*. Show that the covariant divergence of a covariant vector (scalar) may be written as

(1.120) 35.1.gif

1.82. In curvilinear coordinates, write down the Laplace operator influencing a scalar function.

1.83. Write down covariant divergence Tμv;μ for any tensor of rank 2.

1.84. Do the same for the antisymmetric tensor Aμv = –Avμ.

1.85. Prove the following relation for the covariant components of the antisymmetric tensor Aμv = –Avμ:

1.86. Find the covariant derivatives of the metric tensor gμv;λ and gvμ;λ.

1.87. Prove the identity sr28h.gif

1.2.6 Orthogonal Curvilinear Coordinates

Orthogonal curvilinear coordinates in which gμv = 0 while μ ≠ v (see Problem 1.46) are practically used very frequently. In those cases, the following notation is used: sr29h.gif (no summing over μ is necessary). The element of length is written as

(1.121)

where, in accordance with (1.46), values hμ (Lamé coefficients)¹⁹) have the following form:

(1.122)

Since sr30h.gif = h1 h2 h3, the invariant volume element (1.54) assumes the following form:

(1.123) 36.1.gif

The characteristic peculiarity of an orthogonal basis is that the vectors of the original and mutual bases have the same directions but different sizes and physical dimensions (because the coordinates xα and qβ may have different dimensions). This is why the dimensions of different components of the same vector, expanded in the vectors of those bases, may also be different, which creates a certain inconvenience when physical problems are being solved. This is why the introduction of an orthogonal basis of unit vectors ea*, eα* · eβ* = δαβ is useful (we will label them with lower indices and an asterisk) and through which, in accordance with (1.50), the covariant and contravariant bases will be expressed in the following way:

(1.124) 36.2.gif

The expansion of an arbitrary vector A in orts eβ* assumes the following form:

(1.125) 36.3.gif

where the physical components of the vector Aμ* now have the same dimensionality matching that of A, that is, the physical quantity in question, and are connected to its covariant and contravariant components by the following relations:

(1.126) 36.4.gif

Since the use of the basis eβ* is convenient, hereafter we will use that basis everywhere, omitting the asterisk.

Using relations (1.120)–(1.126), and also (1.25), write down the principal operations of differentiation in orthogonal curvilinear coordinates:

(1.127)

(1.128)

(1.129)

(1.130)

Problems

1.88. From the common expressions (1.27)–(1.29), derive the basic differential operations below in the (r, a, z) cylindrical coordinate system where x = r cos α, y = r sin α, and z = z:

(1.131) 37.2.gif

(1.132)

(1.133)

(1.134)

1.89. Do the same for the (r, sr_v.gif , α) spherical coordinate system where x = r sin sr_v.gif cos α, y = r sin sr_v.gif sin α, and z = r cos sr_v.gif :

(1.135)

(1.136)

(1.137)