Mathematical Handbook for Scientists and Engineers

1. The vectors of each class permit a reciprocal one-to-one representation by translations (displacements, directed line segments) in the geometrical space. This representation preserves the results of vector addition, multiplication by scalars, and scalar multiplication of vectors (and thus magnitudes and relative directions of vectors; see also Secs. 12.1-6 and 14.2-1 to 14.2-7).

2. In most applications, vectors appear as functions of position in geometrical space, so that the vectors are associated with geometrical points (vector point functions, Sec. 5.4-1).

Vectors, such as velocities or forces, are usually first introduced in geometrical language as “quantities possessing magnitude and direction” or, somewhat more precisely, as quantities which can be represented by directed line segments subject to a “parallelogram law of addition.” Such a geometrical approach, common to most elementary courses, is employed in Secs. 5.2-1 and 5.2-8 to introduce the principal vector operations. Refer to Secs. 12.4-1 and Chap. 14 for a discussion of vectors from a much more general point of view.

Vector analysis is the study of vector (and scalar) functions. Each vector may be specified by a set of numerical functions (vector components) in terms of a suitable reference system (Secs. 5.2-2, 5.2-3, 5.4-1, and 6.3-1).

NOTE : The description of a physical situation in terms of vector quantities should not be regarded as merely a kind of shorthand summarizing sets of component equations by single equations, but as an instance of a mathematical model (Sec. 12.1-1) whose essential “building blocks” are not restricted to numbers. Note also that a class of objects admitting a one-to-one reciprocal correspondence with a class of directed line segments is not necessarily a vector space unless it has the algebraic properties outlined above (EXAMPLES: finite rotations, directed metal rods).

5.2. VECTOR ALGEBRA

5.2-1. Vector Addition and Multiplication of Vectors by (Real) Scalars. The operation (vector addition) of forming the vector sum a + b of two Euclidean vectors a and b of a suitable class is a vector corresponding to the geometrical addition of the corresponding displacements (parallelogram law). The product of a Euclidean vector a by a real number (scalar) a is a vector corresponding to a displacement a times as long as that corresponding to a, with a reversal in direction if a is negative. The null vector 0 of each class of vectors corresponds to a displacement of length zero, and a + 0 = a. With these geometrical definitions, vector addition and multiplication by scalars satisfy the relations

Refer to Secs. 14.2-1 to 14.2-4 for a more general discussion of vector algebra.

5.2-2. Representation of Vectors in Terms of Base Vectors and Components. m vectors a₁, a₂, . . . , a_m are linearly independent if and only if λ₁a₁ + λ₂a₂ + . . .+ λ_ma_m = 0 implies λ₁ = λ₂ = . . . = λ_m = 0; otherwise the set of vectors is linearly dependent (see also Sees. 1.9-3 and 14.2-3). Every vector a of a three-dimensional vector space can be represented as a sum

in terms of three linearly independent vectors e₁, e₂, e₃. The coefficients α₁, α₂, α₃ are the components^* of the vector a with respect to the

reference system defined by the base vectors e₁, e₂, e₃ (see also Sec. 14.2-4). Given a suitable reference system, the vectors a, b, . . . are thus represented by the respective ordered sets (α₁ α₂, α₃), (β₁,β₂, β₃), . . . of their components; note that a + b and αa are respectively represented by (α₁ + β₁, α₂ + β₂, α₃ + β₃) and (αα₁, αα₂, α α₃). Vector relations can, then, be expressed (represented) in terms of corresponding (sets of) relations between vector components.

Vectors belonging to two-dimensional vector spaces (e.g., plane displacements) are similarly represented by sets of two components.

The system of base vectors chosen for the representation of vectors defined at a given point of geometrical space (Sec. 5.4-1) is usually simply related to the coordinate system used for the description of the geometrical space. Chapter 6 deals specifically with the representation of vector relations in terms of “local” base vectors directed along, and perpendicular to, the coordinate lines of curvilinear coordinate systems at each point; the magnitudes and directions of the local base vectors are, in general, different at different points. Transformation equations relating vector components associated with different reference systems are given in Table 6.3-1 and in Sec. 14.6-1.

5.2-3. Rectangular Cartesian Components of a Vector. Given a right-handed rectangular cartesian-coordinate system (Sec. 3.1-4) in the geometrical space, the unit vectors (Sec. 5.2-5) i, j, k respectively directed along the positive x axis, the positive y axis, and the positive z axis form a convenient system of base vectors at each point. The components α _x, α_y, α_z of a vector

are the (right-handed) rectangular cartesian components of a.

Note that

(Sec. 5.2-6) are direction numbers of the vector a; the components u . i, u . j, u . k of any unit vector u are its direction cosines (Sec. 3.1-8).

5.2-4. Vectors and Physical Dimensions. Euclidean vectors may also be multiplied by scalars which are not themselves real numbers but are suitably labeled by real numbers (quantities isomorphic with the field of real numbers, Sec. 12.1-6; thus one multiplies a constant velocity vector by a time interval to obtain a displacement vector). If a vector (2) or (3) is a physical quantity, one usually associates its physical dimension with the components rather than with the base vectors. The latter are then regarded as dimensionless and may be used to define a common scheme of reference systems (scheme of measurements, see also Sec. 16.1-4) for various classes of vectors having different physical dimensions (e.g., displacements, velocities, forces, etc.; see also Sec. 16.1-4).

5.2-5. Absolute Value (Magnitude, Norm) of a Vector . The absolute value (magnitude, norm) |a| of a Euclidean vector a is a scalar proportional to the length of the displacement corresponding to a (Sec. 5.1-1; see also Secs. 14.2-5 and 14.2-7 for an abstract definition). Absolute values of vectors satisfy the relations (1.1-3). A vector of magnitude 1 is a unit vector . The (mutually perpendicular) base vectors i, j, k (Sec. 5.2-3) are defined to be unit vectors, so that |a_xi| = |a_x|, |a_yj| = |a_y|, |a_zk| = |a_z|, and

5.2-6. Scalar Product (Dot Product, Inner Product) of Two Vectors. The scalar product (dot product, inner product) a • b [alternative notation (ab)] of two Euclidean vectors a and b is the scalar

where γ is the angle (see also Secs. 14.2-6 and 14.2-7 for an abstract definition). If a and b are physical quantities, the physical dimension of the scalar product a . b must be observed (see also Sec. 5.2-4). Table 5.2-1 summarizes the principal relations involving scalar products. Two nonzero vectors a and b are perpendicular to each other if and only if a . b = 0.

Table 5.2-1. Relations Involving Scalar Products

5.2-7. The Vector (Cross) Product. The vector (cross) product a X b (alternative notation [ab]) of two vectors a and b is the vector of magnitude

whose direction is perpendicular to both a and b and such that the axial motion of a right-handed screw turning a into b is in the direction of a X b . Two vectors are linearly dependent (Sec. 5.2-2) if and only if their vector product is zero. Table 5.2-2 summarizes the principal relations involving vector products. Refer to Sec. 16.8-4 for a more general definition of the vector product and to Secs. 3.1-10 and 17.3-3c for the representation of plane areas as vectors.

Table 5.2-2. Relations Involving Vector (Cross) Products

5.2-8. The Scalar Triple Product (Box Product).

In terms of any basis e₁, e₂, e₃ (Sec. 5.2-2; see also Secs. 5.2-3 and 6.3-4)

In terms of right-handed rectangular cartesian components (Sec. 5.2-3)

5.2-9. Other Products Involving More Than Two Vectors.

5.2-10. Representation of a Vector a as a Sum of Vectors Respectively along and Perpendicular to a Given Unit Vector u.

5.2-11. Solution of Equations.

(c) ax + by + cz + d = 0 implies

(d) (b × c)x + (c × a)y + (a × b)z + d = 0 implies

5.3. VECTOR CALCULUS: FUNCTIONS OF A SCALAR PARAMETER

5.3-1. Vector Functions and Limits. A vector function v = v(t) of a scalar parameter t associates one (single-valued function) or more (multiple-valued function) “values” of the vector v with every value of the scalar parameter t (independent variable) for which v(t) is defined (see also Sec. 4.2-1). In terms of rectangular cartesian components

A vector function v(t) is bounded if |v(t)| is bounded. v(t) has the limit (see also Secs. 4.4-1 and 12.5-3) if and only if for every positive number є there exists a number δ > 0 such that |t — t₁| < δ implies |v₁ — v(t)| < є. If exists,

Formulas analogous to those of Sec. 4.4-2 (limits of sums, products, etc.) apply to vector sums, scalar products, and vector products. v(t) is continuous for t = t₁ if and only if(see also Secs. 4.4-6 and 12.5-3).

5.3-2. Differentiation. A vector function v(t) is differentiable for t = t₁ if and only if the derivative

exists and is unique for t = t₁ (see also Sec. 4.5-1). If the derivative d²v(t)/dt² of dv(t)/dt exists, it is called the second derivative of v(t), and so forth. Table 5.3-1 summarizes the principal differentiation rules.

Table 5.3-1. Differentiation of Vector Functions with Respect to a Scalar Parameter

Analogous rules apply to the partial derivatives ∂v/∂t₁ ≡ v_t1, ∂v/∂t₂ ≡ v_t2, . . . of a vector function v = v(t₁, t₂, . . .) of two or more scalar parameters t₁, t₂, . . .

NOTE: If u(t) is a unit vector (of constant magnitude but variable direction) and v(t) = v(t)u(t),

ω is directed along the axis about which u(t) [and thus also v(t)] turns as t varies, so that a right-handed screw turning with u(t) would be propelled in the direction of ω. Its magnitude is equal to the angular rate of turn of u(t) [and thus also of v(t)] with respect to t (EXAMPLE: angular velocity vector in physics; see also Sec. 17.2-3). Equation (4) describes the separate contributions of changes in the magnitude and direction of v(t).

5.3-3. Integration and Ordinary Differential Equations. The indefinite integral V (t) = ∫v(t) dt of a suitable vector function v(t) is defined as the solution of the vector differential equation (see also Sec. 9.1-1)

which may be replaced by a set of differential equations for the components of V(t). Other ordinary differential equations involving differentiation of vectors with respect to a scalar parameter are treated similarly. The definite integral

(see also Sec. 4.6-1) may be treated in terms of components:

5.4. SCALAR AND VECTOR FIELDS

5,4-1. Introduction. The remainder of this chapter deals specifically with scalar and vector functions of position in three-dimensional Euclidean space. Unless the contrary is stated, the scalar and vector functions of position are assumed to be single-valued, continuous, and suitably differentiable functions of the coordinates, and thus of the position vector r ≡ xi + yj + kz. In Secs. 5.4-2 to 5.7-3 relations involving scalar and vector functions are stated

1. In coordinate-free (invariant) form, and

2. In terms of vector components along right-handed rectangular cartesian coordinate axes (Sec. 5.2-3), so that ^*

The relations to be described are independent of the coordinate system used to specify position in space. The representation of vector relations in terms of vector components along, or perpendicular to, suitable curvilinear coordinate lines (and thus along different directions at different points) is treated in Chap. 6.

5.4-2. Scalar Fields. A scalar field is a scalar function of position (scalar point function) Φ(r) ≡ Φ(x, y, z) together with its region of definition. The surfaces

(Sec. 3.1-14) are called level surfaces of the field and permit its geometrical representation.

5.4-3. Vector Fields. A vector field is a vector function of position (vector point function) F(r) = F(x, y, z) together with its region of definition. The field lines (streamlines) of the vector field defined by F(r) have the direction of the field vector F(r) at each point (r) and are specified by the differential equations

A vector field may be represented geometrically by its field lines, with the relative density of the field lines at each point (r) proportional to the absolute value |F(r)| of the field vector.

5.4-4. Vector Path Element and Arc Length (see also Sec. 4.6-9). (a) The vector path element (vector element of distance) dr along a curve C described by

is defined at every regular point (r) ≡ [x(t), y(t), z(t)] as

dr is directed along the tangent to C at each regular point (see also Sec. 17.2-2).

(b) The arc length s on a rectifiable curve (4) (Sec. 4.6-9) is given by

The sign of ds is assigned arbitrarily, e.g., so that ds/dt > 0.

5.4-5. Line Integrals (see also Sec. 4.6-10). Given a rectifiable arc C represented by Eq. (4), the scalar line integrals

can be denfined directly as limits of sums in the manner of Sec. 4.6-10; it is, however, more convenient to substitute the functions x(t), y(t), z(t), dx/dt, dy/dt and dz/dt obtained from Eq. (4) into Eq. (7) and to integrate over t.

One similarly defines the vector line integrals

Unless special conditions are satisfied (Sec. 5.7-1), the value of a scalar or vector line integral depends on the path of integration C.

Refer to Secs. 6.2-3a and 6.4-3a for the use of curvilinear coordinates.

NOTE: It is often useful to introduce the arc length s as a new parameter into the expressions (7) to (9) by means of Eq. (6).

5.4-6. Surface Integrals (see also Secs. 4.6-12 and 17.3-3c). (a)At each regular point of a two-sided surface represented by r = r(u, v) (Sec. 3.1-14), it is possible to define a vector element of area

at each surface point (u, v). In the case of a closed surface, the sense and order of the surface coordinates u, v are customarily chosen so that the direction of dA (direction of the positive surface normal, Sec. 17.3-2) is outward from the bounded volume.

The scalar element of area at the surface point (u, v) is defined as

The sign of dA may be arbitrarily assigned (see also Secs. 4.6-11, 4.6-12, 6.4-3b, and 17.3-3C).

In particular, for u = x, v = y, z = z(x, y),

(b) In the following it will be assumed that the area ∫_s |dA| (Sec.4.6-11) of each surface region S under consideration exists; in this case Eq. (10) defines dA almost everywhere on S (Sec. 4.6-14b). The scalar surface integrals

and the vector surface integrals

of suitable field functions Φ(r) and F(r) may then be defined directly as limits of sums in the manner of Secs. 4.6-1, 4.6-10, and 5.3-3. One can, instead, employ Eq. (10) to express each surface integral as a double integral over the surface coordinates u and v (see also Sec. 6.4-3b).

Note

In the first integral u = y, v = z are independent variables; in the second integral u = z, v = x, etc. [see also Eq. (12)]. Equation (15) must not be interpreted to imply dA = i dy dz + j dx dz + k dx dy without such qualifications on the meaning of dx, dy, and dz.

5.4-7. Volume Integrals (see also Sec. 4.6-12). Given a simply connected region V of three-dimensional Euclidean space, the scalar volume integral

and the vector volume integral

may be defined as limits of sums in the manner of Sec. 4.6-1, or they may be expressed directly in terms of triple integrals over x, y, and z. Refer to Sees. 6.2-3b and 6.4-3c for the use of curvilinear coordinates.

5.5. DIFFERENTIAL OPERATORS

5.5-1. Gradient, Divergence, and Curl: Coordinate-free Definitions in Terms of Integrals. The gradient grad Φ(r) ≡ ∇Φ of a scalar point function Φ(r) ≡ Φ(x, y, z) is a vector point function denfined at each point (r) ≡ (x, y, z) where Φ(r) is suitably differentiate. In coordinate-free form,

where V₁ is a region containing the point (r) and bounded by a closed surface S₁ such that the greatest distance between the point (r) and any point of S₁ is less than δ > 0 (see also Sec. 4.3-5c).

Given a suitably differentiable vector point function F(r) = F(x, y, z) it is similarly possible to define a scalar point function, the divergence of F(r) at the point r,

and a vector point function, the curl (rotational) of F(r) at the point r,

NOTE: At each point where the vector grad Φ ≡ ∇Φ exists, it has the magnitude

of, as well as the direction associated with, the greatest directional derivative dΦ/ds (Sec. 5.5-3c) at that point. ∇Φ defines a vector field whose field lines are specified by the differential equations

The gradient lines defined by Eq. (5) intersect the level surfaces (5.4-2) perpendicularly.

5.5-2. The Operator ∇. In terms of rectangular cartesian coordinates, the linear operator ∇ (del or nabla) is defined by

Its application to a scalar point function Φ(r) or a vector point function F(r) corresponds formally to a noncommutative multiplication operation with a vector having the rectangular cartesian “components” ∂/∂x, ∂/∂y, ∂/∂z; thus, in terms of right-handed rectangular cartesian coordinates x, y, z,

Table 5.5-1 summarizes a number of rules for operations with the operator ∇.

Table 5.5-1. Rules for Operations Involving the Operator ∇

Note that vector equations involving ∇Φ, ∇ . F, and/or ∇ × F have a meaning independent of the coordinate system used. Refer to Chap. 6 and Sec. 16.10-7 for transformations expressing ∇Φ, ∇ . F, and ∇ × F in terms of different coordinate systems.

5.5-3. Absolute Differential, Intrinsic Derivative, and Directional Derivative. (a) The change (absolute differential) dΦ of a scalar point function Φ(r) associated with a change dr = i dx + j dy + k dz in position is (see also Sec. 4.5-3a)

(b) The intrinsic (absolute) derivative (see also Table 4.5-2a) dΦ/dt of Φ(r) along the curve r = r(t) is, at each point (r) of the curve, the rate of change of Φ(r) with respect to the parameter t as r varies as a function of t:

NOTE: If Φ depends explicitly on t [Φ = Φ(r, t)], then

(c) The directional derivative dΦ/ds of Φ(r) at the point (r) is the rate of change of Φ(r) with the distance s from the point (r) as a function of direction. The directional derivative of Φ(r) in the direction of the unit vector u ≡ i cos α_x + j cos α_y + k cos α_z defined by the direction cosines (Sec. 3.1-8a) cos α_x cos α_y, cos α_z is

dΦ/ds is the intrinsic derivative of Φ(r) with respect to the path length s along a curve directed along u = dr/ds.

(d) The absolute differential, intrinsic derivative, and directional derivative of a vector point function F(r) are defined in a manner analogous to that for a scalar point function. Thus

5.5-4. Higher-order Directional Derivatives. Taylor Expansion. The n^th-order directional derivative of Φ or F in the u direction is defined by

respectively. For suitably differentiable functions, one has, if the series in question converges (see also Sec. 4.10-5),

and

where each directional derivative is taken in the direction of Δr.

5.5-5. The Laplacian Operator. The Laplacian operator ∇² ≡ (∇ . ∇) (sometimes denoted by Δ), expressed in terms of rectangular cartesian coordinates by

(see Chap. 6 and Sec. 16.10-7 for other representations), may be applied to both scalar and vector point functions by noncommutative scalar “ multiplication,” so that

Note

5.5-6. Repeated Operations. Note the following rules for repeated operations with the operator ∇:

5.5-7. Operations on Special Functions. A number of results of differential operations on scalar and vector functions of the position vector r = (x, y, z) are tabulated in Tables 5.5-2 and 5.5-3, respectively. Additional formulas may be derived with the aid of Table 5.5-1. Note also

where a is a constant vector.

Table 5.5-2. Operations on Scalar Point Functions (r ≡ |r|; a is a constant vector; n = 0, ±1, ±2, . . .)

5.5-8. Functions of Two or More Position Vectors. Many problems involve scalar or vector functions of two or more position vectors (functions depending on

Table 5.5-3. Operations on Vector Point Functions(r = |r|; a is a constant vector; n = 0, ±1, ±2, . . .)

the positions of two or more points). In the typical case of two position vectors, r ≡ (x, y, z) and = (ξ, η, ζ), say, functions like Φ(r, ) ≡ Φ(x, y, z; ξ, η, ζ)and F(r, ) ≡ F(x, y, z; ξ, η, ζ) may be operated on by two different ∇ operators, described in terms of right-handed rectangular cartesian “components” by

Note in particular

5.6. INTEGRAL THEOREMS

5.6-1. The Divergence Theorem and Related Theorems. (a) Table 5.6-1 summarizes a number of important theorems relating volume integrals over a region V to surface integrals over the boundary surface S of the region V. In the formulas of Table 5.6-1, volume integrals are taken over a bounded, simply connected open region V bounded by a (two-sided) regular closed surface S (Sec. 3.1-14). All functions are assumed to be single-valued throughout V and on S. The existence of the (proper or improper) volume integrals is assumed. All theorems hold for unbounded regions V as well as for bounded regions if the integrands of the surface integrals are 0(l/r³) in absolute value as r—> ∞ (Sec. 4.4-3). Refer to Chap. 6 and Sec. 17.3-3 for formulas expressing surface and volume elements in terms of curvilinear coordinate systems; see also Secs. 15.6-5 and 15.6-10 for applications.

(b) Normal-derivative Notation. The normal derivative of a scalar function Φ(r) at a regular point of the surface S is the directional derivative of Φ(r) in the direction of the positive normal (usually the outward normal, Sec. 17.3-2), and thus in the direction of the vector dA.

Table 5.6-1. Theorems Relating Volume Integrals and Surface Integrals (see also See. 5.6-1)

The normal derivative is customarily denoted by ∂Φ/∂n, so that

5.6-2. Stokes' Theorem and Related Theorems. Given a vector function F(r) single-valued and differentiable with continuous partial derivatives throughout a finite region V containing a simply connected regular (one-sided) surface segment S bounded by a regular closed curve C,

i.e., the line integral of F(r) around C equals the flux of ∇ × F through the surface bounded by C.

Under the same conditions as above,

and for a scalar point function Φ(r) single-valued and differentiable with continuous partial derivatives throughout V

Equations (1), (2), and (3) apply to unbounded regions V if the integrands of the line integrals on the right are 0(1 /r²) in absolute value as r → ∞ (Sec. 4.4-3).

5.6-3. Fields with Surface Discontinuities (see also Sec. 15.6-5b). Let the scalar field Φ(r) or the components of F(r) be continuously differentiable on either side of a regular surface element S but discontinuous on S so that

on the positive side of S (Sec. 17.3-2), while

on the negative side of S. At each point (r) of S one defines the functions

where N(r) is the positive unit normal vector of S at the point (r) (Sec. 17.3-2). The definitions (4) permit one to extend the integral theorems of Table 5.6-1 to functions with surface discontinuities.

5.7. SPECIFICATION OF A VECTOR FIELD IN TERMS OF ITS CURL AND DIVERGENCE

5.7-1. Irrotational Vector Fields. A vector point function F(r) (as well as the field described by it) is called irrotational (lamellar) throughout a region V if and only if, for every point of V,

This is true if and only if – F(r) is the gradient ∇Φ(r) of a scalar point function Φ(r) at every point of V [see also Eq. (5.5-19)]; in this case

is an exact differential (Sec. 4.5-3a). Φ(r) is often called the scalar potential of the irrotational vector field.

If V is simply connected (Sec. 4.3-6b), Φ(r) is a single-valued function uniquely determined by F(r) except for an additive constant, and the line integral

is independent of the path of integration C if the latter comprises only points of V; the line integral around any closed path C [“ circulation” of F(r) around C] in V is zero. If V is multiply connected, Φ(r) may be a multiple-valued function.

As a special case,

is a necessary and sufficient condition that the line integral

is independent of the path of integration, i.e., that the integrand is an exact differential. ^*

5.7-2. Solenoidal Vector Fields. A vector point function F(r) (as well as the field described by it) is called solenoidal throughout a region V if and only if, for every point of V,

This is true if and only if F(r) is the curl ∇ × A(r) of a vector point function A(r) [see also Eq. (5.5-19)], the vector potential of the vector field described by F(r).

5.7-3. Specification of a Vector Point Function in Terms of Its Divergence and Curl. (a) Let V be a finite open region of space, bounded by a regular surface S (Sec. 3.1-14) whose positive normal is uniquely defined and varies continuously at every surface point. If the divergence and curl of a vector point function F(r) are given at every point (r) of V, then F(r) may be expressed throughout V as the sum of an irrotational vector point function F₁(r) and a solenoidal vector point function F₂(r),

(Helmholtz's Decomposition Theorem). F(r) is uniquely defined throughout V if, in addition, the normal component F(r) . dA/|dA| of F(r) is given at every surface point (Uniqueness Theorem).

The problem of actually computing F(r) from these data involves the solution of partial differential equations subject to certain boundary conditions. The important special case in which ∇ . F(r) = ∇ × F(r) = 0 and thus F(r) = - ∇Φ(r), ∇²Φ(r) = 0 throughout V forms the subject matter of potential theory as discussed in Sees. 15.6-1 to 15.6-10.

(b) If suitable functions

are given for every point (r) of space, then Eq. (6) defines F₁(r) and F₂(r), and hence F(r), uniquely except for additive functions F₀(r) such that ∇²F₀(r) = 0. One has

provided that the integrals on the right (scalar and vector potentials) exist (see also Sec. 15.6-5); the integration extends over all points ().

5.8. RELATED TOPICS, REFERENCES, AND BIBLIOGRAPHY

5.8-1. Related Topics. The following topics related to the study of vector analysis are treated in other chapters of this handbook:

Algebra of more general vector spaces Chaps. 12, 14

Applications of vector algebra to analytic geometry Chap. 3

Applications of vector calculus to the geometry of curves and surfaces (differential geometry) Chap. 17

Two-dimensional fields Chap. 10

Potential theory Chap. 10

General transformation properties of vector point functions Chap. 16

Functions of a complex variable Chap. 7

5.8-2. References and Bibliography (see also Sec. 6.6-2).

        5.1. Brand, L.: Vector and Tensor Analysis, Wiley, New York, 1947.

        5.2. : Vector Analysis, Wiley, New York, 1957.

        5.3. Dörrie, H.: Vektoren, Edwards, Ann Arbor, Mich., 1946.

        5.4. Halmos, P. R.: Finite Dimensional Vector Spaces, Princeton University Press, Princeton, N.J., 1942.

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

        5.6. Lass, H.: Vector and Tensor Analysis, McGraw-Hill, New York, 1950.

        5.7. McQuistan, R. B.: Scalar and Vector Fields, Wiley, New York, 1965.

        5.8. Sokolnikoff, I. S.: Tensor Analysis, 2d ed., Wiley, New York, 1964.

        5.9. Weatherburn, C. E.: Elementary Vector Analysis, Open Court, LaSalle, Ill., 1948.

        5.10. : Advanced Vector Analysis, Open Court, LaSalle, Ill., 1948.

^*Some authors refer to the component vectors α₁e₁, α₂e₂, α₃e₃ as components.

^* Throughout Chaps. 5 and 6, the subscripts in F_x, F_v, F_z, . . . do not indicate differentiation with respect to x, y, z, . . . ; in fact, no scalar function F(x, y, z) is introduced.

^*See footnote to Sec. 5.4-1.