Mathematical Handbook for Scientists and Engineers

(Sec. 2.1-9) in terms of suitably differentiable functions, the tangent to C at the point P₁ ≡ (x₁, y₁) is defined as the limit of a straight line (secant) through P₁ and a neighboring point P₂ as P₂ approaches P₁. The curve (1) has a unique tangent described by

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at every regular point (x₁, y₂) where it is possible to choose a parameter t so that x(t) and y(t) have unique continuous derivatives not both equal to zero; or, equivalently, where φ(x,y) has unique continuous first partial derivatives not both equal to zero. The slope (Sec. 2.2-1) of the tangent (2) is

17.1-2. Normal to a Plane Curve. The normal to the curve (1) at a regular point P₁ ≡ (x₁ , y₁) is the straight line through P₁ and perpendicular to the tangent at P₁:

The direction of the positive normal is arbitrarily fixed with respect to the common positive direction of curve and tangent. The positive direction on a curve (1) is arbitrarily fixed by some convention (e.g., direction of increasing t, increasing x, etc.; see also Sec. 2.2-1).

17.1-3. Singular Points. Given a curve (l b) such that the n^th-order derivatives of φ (x, y) exist and do not vanish simultaneously at P₁, let all (n – l)^nt-order derivatives of φ(x, y) be equal to zero at P₁. Then the curve has ntangents at P₁; two or more of these tangents may coincide, and an even number of them may be imaginary. Thus if all first-order derivatives but not all second-order derivatives of φ(x,y)vanish at P₁ ≡ (x₁, y₁),the slopes dy/dx of the two tangents are obtained as roots of the quadratic equation

The roots of Eq. (5), and hence the two tangents, may be real and different (double point), coincident (cusp, or self-osculation point,see also Sec. 17.1-5), or imaginary (isolated point).The properties of a curve at a singular point can be similarly described in terms of discontinuous or multiple-valued derivatives of f(x) or of x(t) and y(t).

17.1-4. Curvature of a Plane Curve. The circle of curvature (osculating circle) of a plane curve C at the point P₁ is the limit of a circle through P₁ and two other distinct points P₂ and P₃ of C as P₂ and P₃ approach P₁. The center of this circle (center of curvature of C corresponding to the curve point P₁) is located on the normal to C at P₁. The coordinates of the center of curvature are

where all derivatives are computed for x= x₁ (t= t₁); dots indicate differentiation with respect to t.* The radius ρ_k of the circle of curvature (radius of curvature of C at P₁) equals the reciprocal of the curvature _k of C at P₁ defined as the rate of turn of the tangent with respect to the arc length s along C (Sec. 4.6-9):

* Equations (6) and (7) may be rewritten in terms of the partial derivatives of φ(x,y) with the aid of Eq. (4.5-16).

where all derivatives are computed for x = x₁ (t = t₁).* A given curve C is, respectively, concave or convex in the direction of the positive y axis wherever d²y/dx² and thus _k is positive or negative. Many authors introduce |_k| rather than _k as the curvature, as in Sec. 17.2-3.

In terms of polar coordinates r, φ (Sec. 2.1-8), the distance element ds and the angle μ between the tangent to a curve r = r (φ) and the radius vector at each regular point P₁ ≡ (r₁, φ₁) are given by

17.1-5. Osculation. A point P₁ ≡ (x₁, y₁)is an osculation point (point of contact) of order n of two curves described by suitably differentiable equations y = f(x) and y = g(x) if and only if

Such a point may be regarded as the limit of n + 1 real or imaginary points of intersection approaching one another. At every osculation point the tangents to the two curves coincide; the curves intersect (cross) if and only if n is even. A point where a curve intersects its own tangent is a point of inflection. At every point of inflection k = 0.

17.1-6. Asymptotes. A straight line is an asymptote of the given curve C (C approaches the straight line asymptotically) if and only if the distance between the straight line and a point P= (x, y)on the curve tends to zero as x²+ y² → ∞. If C is a regular arc the asymptote is the limiting case of the tangent at P.

17.1-7. Envelope of a Family of Plane Curves. The envelope of a suitable one-parameter family of plane curves described by

osculates, or contains a singular point of, every curve (11). One obtains the equation of the envelope by elimination of the parameter λ from Eq.(11)and

*Equations (6) and (7) may be rewritten in terms of the partial derivaties of φ(x,y) with the aid of Eq. (4.5-16).

The envelope exists in a region of values of x, y, and λ such that

Equations (11) and (12) define the locus of limiting points of intersectionof φ(x, y,λ₁) = 0 and φ(x, y,λ₂) = 0 as λ₁ → λ₂. This locus (λ discriminant) includes loci of cusps and nodes as well as envelopes.

17.1-8. Isogonal Trajectories. The family of curves intersecting a given family φ(x, y, λ)= 0 at a given angle γ is described by the differential equation

For γ = π/2, Eq. (14) yields orthogonal trajectories.

17.2. CURVES IN THREE-DIMENSIONAL EUCLIDEAN SPACE

17.2-1. Introduction (see also Sec. 3.1-13). Sections 17.2-1 to 17.2-6 deal with the geometry of a regular arc C described by

where the functions (1) have unique continuous first derivatives and dr/dt ≠ 0 for t₁ ≤ t ≤ t₂. Higher-order derivatives will be assumed to exist as needed. It is convenient to introduce the arc length (Sec. 5.4-4) as a new parameter; the sign of ds is arbitrarily fixed to determine the positive direction of curve and tangents (see also Secs. 17.2-2 and 17.2-3). Differentiation with respect to s will be indicated by primes, so that, for example,

The representation of curves in terms of curvilinear coordinates (Chap. 6) is briefly discussed in Sec. 17.4-1.

17.2-2. The Moving Trihedron (see also Secs. 17.2-3 and 17.2-4). (a) Tangent to a Curve. The tangent to a regular arc C at the point P₁≡ (r₁ ≡ (x₁, y₁, z₁) is the limit of a straight line (secant) through P₁ and another point P₂ of C as P₂ approaches P₁. A unique tangent exists at every point of a regular arc. The positive tangent direction coincides with the positive direction of C at P₁.

(b) Osculating Circle and Plane. Principal Normal. The osculating circle or circle of curvature of C at the curve point P₁ is the limit of a circle through P₁ and two other distinct points P₂ and P₃ of C as P₂ and P₃ approach P_l. The plane of this circle (osculating plane of C at P₁) contains the tangent to C at P₁. The directed straight line from the curve point P₁ to the center of the osculating circle (center of curvature) is perpendicular to the tangent and is called the principal normal of C at P₁.

(c) Binormal. Normal and Rectifying Planes. The binomial of C at P₁ is the directed straight line through P₁ such that the positive tangent, principal normal, and binormal form a system of right-handed rectangular cartesian axes(Sec. 3.1-3). These axes determine the “moving trihedron” comprising the osculating plane and the normal and

rectifying planes respectively normal to the tangent and to the principal normal of C at P₁ (Fig. 17.2-1).

(d) The unit vectors t, n, and b respectively directed along the positive tangent, the principal normal, and the binormal are given by

at each suitable point of the curve. The vector kn = r" is called the curvature vector; k is the curvaturefurther discussed in Sec. 17.2-3.

17.2-3. Serret-Frenet Formulas. Curvature and Torsion of a Space Curve (see also Secs. 17.2-4 and 17.2-5). (a) The unit vectors (2) satisfy the relations

at each curve point P₁. As s increases, the point P₁ moves along the curve C and

1. The tangent rotates about the instantaneous binormal direction at the (positive) angular rate k (curvature of C at P₁).

2. The binormal rotates about the instantaneous tangent direction at the angular rate (torsion of C at P₁; is positive wherever the curve turns in the manner of a right-handed screw).

3. The entire moving trihedron rotates about the instantaneous direction of the Darboux vector Ω = rt + _kb at the (positive) angular rate (total curvature of C at P₁).

The instantaneous rotation of t, n, and b becomes more evident on rewriting the Serret-Frenet formulas as follows (see also Secs. 5.3-2 and 14.10-5):

ρ_k = 1/_k is the radius of the circle of curvature (radius of curvature) of C at P₁; ρ = 1/ is called the radius of torsion.

(b) The scalar functions _k = _k(S) and = (S) together define the curve C uniquely, except for its position and orientation in space (intrinsic equations of a space curve). C is a plane curve if and only if its torsion vanishes identically, and a straight lineif and only if its curvature _k vanishes identically.

where the dots indicate differentiations with respect to t. Note also

(decomposition of the acceleration of a moving point into tangential and normal components,see also Sec. 5.3-2).

17.2-4. Equations of the Tangent, Principal Normal, and Binormal, and of the Osculating, Normal, and Rectifying Planes, (a) The tangent, principal normal, and binormal of C at the point P₁ ≡(r₁) ≡ (x₁, y₁, z₁) are respectively described by

where u is a variable parameter. The equations of the osculating, normal, and rectifying planes are, respectively,

(b) The right-handed rectangular cartesian components of the unit vectors (2) are

Substitution of the correct direction cosines (9) in

yields the equations of the tangent, principal normal, and binormal in terms of rectangular cartesian coordinates, and

represents the osculating, normal, and rectifying planes.

17.2-5. Additional Topics. (a) The center of curvature associated with the curve point P₁ has the position vector

(b) The limit of a sphere through four distinct points P₁, P₂, P₃, P₄of C as P₂, P₃, and P₄ approach P₁ is the osculating sphere of C at P₁. Its center lies on the directed straight line in the positive binormal direction through the center of curvature (axis of curvature or polar line of C at P₁). The radius ρ₈ of the osculating sphere and the position vector r₈ of its center are

C lies on a sphere of radius R if and only if ρ ≡ R.

The polar lines of C are tangent to the polar curve defined as the locus of the centers of the osculating spheres of C. The polar surface (polar developable) of C is the ruled surface (Sec. 3.1-15) generated by the polar lines.

(c)Involutes and Evolutes. The tangents of C generate a ruled surface (tangent surface, tangential developable) of two sheets, which are tangent to one another at the given curve. The involutes of the given curve C are those curves on the tangent surface which are orthogonal to the generating tangents. Given the position vector r = r(s) of a moving point P_s on C, the points P_I = () of every involute of C Are given by

Each involute corresponds to a specific value of d. Note that d is the constant sum of s and the tangent P _s P_I (string property of involutes),

A curve C' is an evolute of C if the tangents of C' are normal to C, i.e., if C is an involute of C' . The evolutes of C lie on its polar surface (see also Ref. 17.7).

17.2-6. Osculation (see also Sec. 17.1-5). A point P₁≡ (r₁) ≡ (x₁, y₁, z₁) is an osculation point (point of contact) of order nof two regular arcs represented by r = f(s) and r = g(s) if and only if at P₁ one has s = s such that

17.3. SURFACES IN THREE-DIMENSIONAL EUCLIDEAN SPACE

17.3-1. Introduction (see also Sec. 3.1-14; see Sec. 3.5-10 for examples). Sections 17.3-1 to 17.3-14 deal with the geometry of a regular surface element S described by

in some region of values of the parameters (surface coordinates) u, v (Sec. 3.1-14). The functions (1) are to have continuous first partial derivatives such that the rank of the matrix

equals 2 (Sec. 13.2-7), i.e., such that the three functional determinants (x, y)/(u, v)_y (y, z) /(u, v), (z, x)/(u, v) do not vanish simultaneously, or

Higher partial derivatives will be assumed to exist as needed.

The conditions listed above ensure the existence and linear independence of the vectors r_u and r_v directed, respectively, along the tangents to the surface-coordinate lines u = constant and v = constant through the surface point (u, v). Surface points where the three determinants exist but vanish for every choice of the parameters u, v are singular points corresponding to edges, vertices, etc.

17.3-2. Tangent Plane and Surface Normal. (a) At every surface point P₁≡(r₁) ≡ (x₁, y₁ z₁) ≡ (u_l v₁) satisfying the conditions of Sec. 17.3-1 (regular pointof the surface) there exists a unique tangent plane defined as the limit of a plane through three distinct surface points P₁, P₂, P₃ as P₂ andP₃ approach P₁ along curves which do not have a common tangent at P₁. This tangent plane contains the tangents of all regular surface arcs through P₁; the equation of the tangent plane is

where all derivatives are taken for u = u₁, υ= υ₁.

(b) The surface normal of S at the regular surface point P₁ ≡ (r₁) ≡ (x₁, y₁, z₁) ≡ (u₁,υ₁) is the straight line through P₁ normal to the tangent plane at that point. This surface normal is described by the parametric equation

is the unit normal vector of S at the point P₁; all derivatives are taken for u = u₁ v = v₁.The direction of N is the direction of the positive normal at P₁; note that the positive udirection, the positive vdirection, and the positive normal form a right-handed system of axes (Sec. 3.1-3) directed along r_u, r_v, and N.

17.3-3. The First Fundamental Form of a Surface. Elements of Distance and Area. (a) The vector path element (Sec. 5.4-4) along a surface curve

and the square of the element of distance ds = |dr| on the surface (1) is given at each surface point (u,v) by

At every regular point of a real surface (1) described in terms of real surface coordinates u, v the quadratic form (9) is positive definite (Sec. 13.5-2), i.e.,

(b) The angle γ between two regular surface arcs

through the surface point (u, v) is given by

In particular, the angle γ₁ between the surface-coordinate lines u = constant and v = constant through the surface point (u, v) is given by

The surface coordinates u,v are orthogonal if and only if F ≡ 0 (see also Sees. 6.4-1 and 16.8-2).

(c) The vector element of area dA and the scalar element of area dA at the regular surface point (u, v) are denned by

The sign of dA may be fixed arbitrarily (see also Sees. 4.6-11, 5.4-6a, and 6.4-36).

17.3-4. Geodesic and Normal Curvature of a Surface Curve. Meusnier's Theorem, (a) At each point( u, v)of a regular surface arc C described by

the curvature vector r" =_k_n (Sec. 17.2-2d) has a unique component in the tangent plane (geodesic or tangential curvature vector) and a unique component along the surface normal (normal curvature vector), i.e.,

where the primes indicate differentiation with respect to the arc length s along C. At each point [u(s), v(s)],

is the curvature of the projection of C onto the tangent plane (see also Sec. 17.2-2d),and

is the curvature of the normal section (intersection of the surface and a plane containing the surface normal) through the tangent of C(see also Sec. 17.3-5).

(b) At a given surface point (u, v), the curvature k of every surface curve whose osculating plane (Sec.17.2-26) makes an angle a with the normal-section plane through the same tangent is

where k_N is the normal-section curvature given by Eq.(17) or (20) (Meusnier's Theorem). Equation (18) yields, in particular, the curvature k of any oblique surface section in terms of the curvature k_N of the normal section through the same tangent.

17.3-5. The Second Fundamental Form. Principal Curvatures, Gaussian Curvature, and Mean Curvature, (a) To write Eq. (17) in terms of the surface coordinates u, v, let dN = N_udu+ N_vdv. Then

where all derivatives are taken at the surface point (u, v); the box products can be expanded by Eq. (5.2-11). At a given surface point (r) ≡ (u,v) the curvature of the normal section containing the adjacent surface point(r + dr) ≡ (u+ du,v+ dv) is

(b) Unless k_N has the same value for every normal section through (u,v) (L:M:N = E:F:G, umbilic point), there exist two principal normal sections respectively associated with the largest value k₁ and the smallest value k₂ of k [principal curvatures of S at the surface point (u,v)]. The planes of the principal normal sections are mutually perpendicular; for any normal section through (u, v) whose plane forms the oblique angle with the plane of the first principal normal section,

k₁ and k₂ are eigenvalues of the type discussed in Sec. 14.8-7 (see also Sec. 14.8-8); they are obtainable as roots of the characteristic equation

The symmetric functions H(u, v)≡ 1/2(k₁+ k₂) and k(u,v)= _k₁_k₂ are respectively known as the mean curvature and the Gaussian curvature of the surface S at the point (u, v); note

(see also Sees. 17.3-8 and 17.3-13). k₁k₂, H, and K are surface point functions independent of the particular surface coordinates u, v used.

Depending on whether the quadratic form (19) is definite, semidefinite, or indefinite (Sec. 13.5-2) at the surface point (u, v),the latter is

An elliptic point with K= k₁k₂ > 0 (normal sections all convex or all concave;surface does not intersect its tangent plane. EXAMPLE: any point of an ellipsoid)

A parabolic point with K= k₁k₂ = 0 (e.g., any point of a cylinder)

A hyperbolic point (saddle point) with K =k₁k₂ < 0 (both convex and concave normal sections; surface intersects its tangent plane. EXAMPLE: any point of a hyperboloid of one sheet)

An umbilic point (k₁ = k₂, Sec. 17.3-56) is necessarily either elliptic or parabolic.

17.3-6. Special Directions and Curves on a Surface. Minimal Surfaces. (a) A line of curvature is a surface curve whose tangent at each point belongs to a principal normal section; the differential equation

defines the two mutually perpendicular lines of curvature v= v(u) through each point (u, v).

(b) An asymptotic line is a surface curve of zero normal curvature (20), so that

(EXAMPLE: any straight line on the surface). The tangent to an asymptotic line defines an asymptotic direction on the surface. Note that the asymptotic lines at elliptic surface points are imaginary curves v= v(u). The lines of curvature bisect the asymptotic directions.

(c) Two regular surface arcs u= U₁(t), v= V₁(t) and u= U₂(t), v= V₂(t) have mutually conjugate directions at the surface point (u, v)if and only if the tangent of one arc is the limiting intersection of the tangent planes at two points approaching (u, v) on the other arc, or

This relationship is necessarily reciprocal [EXAMPLE: directions of the lines of curvature at (u, v)].

(d) The parametric lines u = constant, v = constant are

Orthogonalif and only if F ≡ 0

Conjugateif and only if M ≡ 0

Lines of curvatureif and only if F ≡ M ≡ 0

Asymptotic linesif and only if L ≡ N ≡ 0

(e) A minimal surface is a surface such that H(u, v) ≡ 0; this is true if and only if the asymptotic lines form an orthogonal net.Surfaces having minimum area for a given reasonable boundary curve are minimal curves (Plateau's problem, soap-film models).

17.3-7. Surfaces as Riemann Spaces. Three-index Symbols and Beltrami Parameters. A regular surface element S with fundamental form (9) is a two-dimensional Riemann spaceof points (u,v)with metric-tensor components E, F, G(Sees. 16.7-1 and 17.4-1; see also Sec. 17.3-12); a(u, v)≡ EG — F²is the metric-tensor determinant (Sec. 16.7-1). One can define surface vectors and tensors, surface scalar products,and surface covariant differentiationon the Riemann space Sin the manner of Sees. 16.2-1, 16.8-1, and 16.10-1. The Christoffel three-index symbols of the second kind(Sec. 16.10-3) for the surface Stake the form

where the subscript S has been introduced to distinguish the surface Christoffel symbols (28) clearly from the Christoffel symbols {_i^k_j} associated with the surrounding space.

For suitably differentiable functions Φ(u, v), Ψ(u, v),the functions

are surface differential invariants respectively analogous to ∇Φ. ∇Ψ and ∇²Φ as defined in Table 16.10-1. Since the two-dimensional Riemann space Sis “embedded” in three-dimensional Euclidean space, one can interpret covariant differentiation on a curved surface by actual comparison of surface vectors at adjacent surface points.

17.3-8. Partial Differential Equations Satisfied by the Coefficients of the Fundamental Forms. Gauss's Theorema Egregium. (a) The following relations describe the change of the linearly independent vectors r_u, r_v, N (“local trihedron”) along the surface-coordinate directions:

(b) The following relations are compatibility relations (integrability conditions, Sec. 10.1-2c) ensuring that r_uuv = r_vuu, r_vuv = r_uvv:

(c) Equation (33) expresses K solely in terms of E, F, and G and their derivatives: the Gaussian curvature K(u, v) of a surface is a bending invariant unchanged by deformations which preserve the first fundamental form (Gauss's Theorema Egregium).

17.3-9. Definition of a Surface by E,F, (?, L, M,and N. Three given functions E(u, v) > 0, G(u, v) > 0, and F(u, v)such that EG — F² > 0 define the metric properties (intrinsic geometry) of the surface. Six functions E, F, G, L, M, N satisfying the above inequalities and thecompatibility conditions (32) and (33) uniquely define a corresponding real surfacer = r(u, v) except for its position and orientation in space (Fundamental Theorem of Surf ace Theory).

17.3-10. Mappings, (a) A suitably differentiable one-to-one transformation

maps the points (u, v) of a given regular surface element r = r(u, v) onto corresponding points (ū, ῡ)of another regular surface element r = ȓ (ū, ῡ). In the following, barred symbols refer to the second surface. The mapping is

Isometric (preserves all metric properties) if and only if (ū, ῡ) ≡ E(u, v), (ū, ῡ) ≡ F(u, v),(ū, ῡ)≡ G(u, v)

Conformal (preserves angles) if and only if (ū, ῡ) :(ū, ῡ) :(ū, ῡ)≡ E(u,v):F(u,v):G(u, v)

Equiareal (preserves areas) if and only if

(b) To obtain a conformal mapping (34), one maps each surface conformally onto a plane and relates the two planes by an analytic complex-variable transformation (Sec. 7.9-1).

To map the surface r = r(u, v)conformally onto a plane with rectangular cartesian coordinates ξ(u, υ), n (u, v),solve the differential equation

or, equivalently,

to obtain the complex surface curves (isotropic or minimal surface “curves” defined by ds² = 0)

Then the real orthogonal surface coordinates

(isometric or isothermic surface coordinates) reduce the first fundamental form of the surface to

which is proportional to the first fundamental form dξ²+dη² of a plane with rectangular cartesian coordinates , ξ,η.

17.3-11. Envelopes (see also Sees. 10.2-3 and 17.1-7). Let the equations

represent a one-parameter family of surfaces such that

in some region V of space. Then V contains a surface (envelope of the given surfaces) which touches (has a common tangent plane with) each surface (38) along a curve (characteristic) described by

Elimination of A from Eq. (40) yields the equation of the envelope.

If, in addition to the condition (39), [∇_φ∇_φ_λ∇_φ_λ_λ] ≠0 in V,then the envelope has an edge of regression

which touches each characteristic (40) at a focal point obtainable by eliminatioD of λ from Eq. (41).

Equation (40) defines the locus of the limiting curves of intersectionof φ(x, y, z,λ₁) = C and φ(x, y, z,λ₂)= 0 as λ ₂ → λ ₁. This locus (λ discriminant) may include loci of edges and nodes, as well as envelopes. An edge of regression is, similarly, a locus of limiting points of intersectionof three surfaces (38).

17.3-12. Geodesies (see also Sec. 17.4-3). A geodesic on the regular surface element Sis a regular arc whose geodesic curvature (Sec. 17.3-4a) is identically zero; a geodesic is either a straight line, or its principal normal coincides with the surface normal at each point. Every geodesic

satisfies the differential equations

These relations define a unique geodesic through any given point in any given direction.

Geodesies on a curved surface have many properties analogous to those of straight lines on a plane (see also Sec. 17.4-3). If there exists a surface curve of smallest or greatest arc length joining two given surface points, then that curve is a geodesic.The actual existence of a geodesic through two given surface points requires a separate investigation in each case.

17.3-13. Geodesic Normal Coordinates. Geometry on a Surface (see also Sec. 17.4-7). (a) For a system of geodesic normal coordinates u, v the u coordi-nate lines are orthogonal trajectories of a “field” of geodesies v= constant. The ucoordinate lines are then geodesic parallels cutting off equal increments of u= s on each geodesic υ =constant, and

In terms of geodesic normal coordinates u, v

(b) In the special case of geodesic polar coordinates, the geodesies v= constant intersect at a point (origin, pole),and vis the angle (Sec. 17.3-36) between the geodesic labeled by v and the geodesic v =0. Each u coordinate line is a geodesic circle of radius uintersecting all vgeodesies at right angles.

The “circular arc” of “radius” ucorresponding to an angular increment dvis

where K₀ is the Gaussian curvature at the origin. The quantity (445) is less than, equal to, or greater than u dvif, respectively, K₀ > 0, K₀ =0, or K₀ <0. The circumference C_G(u)and the area A_G(u) of a small geodesic circle of radius u about the origin are related to the circumference 2πu and the area πu² of a plane circle of equal radius by

(c) For any geodesic triangle on a surface of constant Gaussian curvature K,the excess of the sum of the angles A, B, Cover π and the triangle area S_T are related by

The resulting surface geometry is Euclidean for K= 0, elliptic for K > 0, and hyperbolic for K < 0. Surfaces of equal constant Gaussian curvature are isometric (Minding's Theorem).

EXAMPLES: On a sphere of radius R, K= 1/R²(see also Sec. B-6). A surface with constant K < 0 is obtained by rotation of the tractrix

(pseudosphere, Ref. 17.7).

17.3-14. The Gauss-Bonnet Theorem. Let K(u, v) be continuous on a simply connected surface region S whose boundary C consists of n regular arcs of geodesic curvature K_G(u, v). Then the sum Θ of the n exterior

angles of the boundary is related to theintegral curvature ∫_s K dA of the surface region S by

The first integral vanishes if all the boundary segments are geodesies; Eq. (47) is a special case of Eq. (48).

17.4. CURVED SPACES

17.4-1. Introduction. The theory of Sees. 17.4-2 to 17.4-7 treats geometrical concepts like distance, angle, and curvature in terms of general curvilinear coordinates and extends these concepts to a class of multidimensional spaces, viz., the Riemann spacesintroduced in Chap. 16. The tensor notation of Chap. 16 is employed.

17.4-2. Curves, Distances, and Directions in Riemann Space (see also Sees. 4.6-9, 5.4-4, and 6.2-3).* (a) The metric properties (see also Sec. 12.5-2) associated with an n-dimensional Riemann space of points (x¹, x²,. . . , xⁿ)are specified by its metric-tensor components g_i_k(x^l, x²,. . . , xⁿ)(Sec. 16.7.1) which define scalar products,and thus magnitudes and directions of vectors at each point (Sec. 16.8-1).

(b) A regular arc C is described by n parametric equations

with unique continuous derivatives dxⁱ/dt which do not vanish simultaneously. The components dxⁱ ≡ (dxⁱ/dt) dt represent a vector dr along (tangent to) the curve C at each of its points (x¹, x²,. . . , xⁿ), and the element of distance between two neighboring points (x^l, x²,. . . , xⁿ) and (x¹ + dx¹, x² + dx², . . . , xⁿ + dxⁿ) on C is defined as

The sign of ds is chosen so that ds ≥ 0 for dt > 0 (positive directionon the curve C). The arc length s along the curve C, measured from the curve point corresponding to t= t₀,is

The value of the integral (3) is independent of the particular parameter t used.

(c) The direction of the curve (1) at each of its points (x¹, x²,. . . , xⁿ) is that of the vector dr;i.e., the angle γbetween any vector a defined at (x;¹, x²,. . . , xⁿ)and the curve is given by cos γ =a . dr/ |a| |ds|(Sec. 16.8-1). In particular, the angle γ between two regular arcs xⁱ= X_i₁(t)and xⁱ= Xⁱ₂(t)is given by

* Equations (4) to (6) apply directly whenever ds ≠0. The case of null directions (dr ≠0, ds= |dr| = 0) in Riemann spaces with indefinite metric is briefly discussed in Sec. 17.4-4.

The point comman to n – 1 of the n coordinate hypersurfaces xⁱ ≡ constant (i = 1, 2, ... , n)through a given point (x¹, x², . .. , xⁿ) lie on a coordinate line associated with the n^th coordinate (see also Sec. 6.2-2). The cosine of the angle between the i^th and the k^th coordinate line through the point (x^l,x²,. . . , xⁿ) equals

(d) The unit vector dr/ds(represented by dx^l/ds)is the unit tangent vector of C at each curve point (x¹, x²,. . . , xⁿ).The first curvature vector

is perpendicular to the curve (principal-normal direction, see also Sec. 17.2-26); the absolute value

is the absolute geodesic curvature (absolute first curvature) of C at (x¹, x²,. . , xⁿ)(see also Sec. 17.3-7).

17.4-3. Geodesies (see also Sec. 17.3-12).* (a) A geodesic in a Rie-mann space is a regular arc whose geodesic curvature is identically zero, so that the unit tangent vector dr/ds is constant along the curve (parallel to itself, Sec. 16.10-9), i.e.,

The n second-order differential equations(6) define a unique geodesic xⁱ= xⁱ(s) through any given point [xⁱ= xⁱ(s₁)] in any given direction (given values of the dxⁱ/ds for 8= s₁).

More generally, the differential equations

(with suitable initial conditions on the xⁱ and dxⁱ/dt) define a geodesic xⁱ = xⁱ(t); the choice of the function

amounts to a choice of the parameter t used to describe the geodesic but does not affect the curve as such.

(b) Geodesies have many of the properties of the straight lines in Euclidean geometry (see also Sec. 17.4-6). If there exists a curve of

*See footnote to Sec. 17.4-2.

smallest or greatest arc length(3) joining two given points of a Riemann space, then that curve is a geodesic.More generally, the differential equations (6) or (7) of a geodesic may be regarded as Euler equations (Sees. 11.6-1 and 11.6-2) which ensure that the first variation of the arc length (3) along a curve joining the given points is equal to zero. The actual existence of a geodesic through two given points of a Riemann space requires a separate investigation in each case.

17.4-4. Riemann Spaces with Indefinite Metric. Null Directions and Null Geodesies. If the fundamental quadratic form g_i_k(x^l, x²,. . . , xⁿ) dxⁱ dx^kof a Riemann space is indefinite (Sec. 13.5-2) at a point (x¹, x², .. . , xⁿ),the square |a|² = g_i_kaⁱa^kof a vector a may be positive, negative, or zero, and |a| = 0 does not necessarily imply a = 0. At any point (x¹, x²,. . . , xⁿ)the direction of a vector a ≠ 0 such that |a|² = g_ikaⁱa^k= 0 is a null direction. For a vector displacement dr ≠ 0 in a null direction, ds = |dr| -0; note that the points (x¹, x²,. . . , xⁿ) and (x¹ + dx^l, x² + dx², . . . , xⁿ + dxⁿ) separated by such a null displacement dr are not identical. A curve xⁱ = xⁱ(t) such that

has a null direction at each of its points (curve of zero arc length, null curve, minimal curve, see also Sec. 17.3-106). A curve xⁱ= x^l(t)satisfying Eq. (7) as well as Eq. (9) is a null geodesic (geodesic null line). Every null direction defines a unique null geodesic through a given point (x^l, x², . .. , xⁿ) (application:light paths in relativity theory).

17.4-5. Specification of Space Curvature, (a) The Riemann-Christoffel curvature tensor of a given Riemann space is the absolute tensor of rank 4 defined by the mixed components

or by the covariant components

The components of the curvature tensor satisfy the following relations:

In an n-dimensional Riemann space there exist at most n²(n² – 1)/12 distinct non-vanishing covariant components R_ijkh.Note also

(b) The Ricci tensor of a Riemann space is the absolute tensor of rank 2 defined by the covariant components

or by the mixed components R_jⁱ ≡ g^ikR_kj. In an n-dimensional Riemann space there exist at most n(n +l)/2 distinct nonvanishing covariant components R_ij.

The eigenvector directions (Sec. 14.8-3) of the Ricci tensor are the Ricci principal directions at each point (x¹, x², . . . , xⁿ)of the Riemann space.

(c) The curvature invariant or scalar curvature of a Riemann space is the absolute scalar invariant

(d) The Einstein tensor of a Riemann space is the absolute tensor of rank 2 defined by the components

The divergence (Sec.16.10-7) G_j,iⁱ of the Einstein tensor vanishes identically.

(e) Refer to Sec. 17.3-7 for the special case of a two-dimensional Riemann space (curved surface in three-dimensional Euclidean space).

17.4-6. Manifestations of Space Curvature. Flat Spaces and Euclidean Spaces, (a) Parallel Propagation of a Vector. Parallel propagation of a vector a (Da_i = 0,Sec. 16.10-9) around the infinitesimal closed circuit (x¹, x²,. . . , xⁿ) → (x¹+ dx¹, x²+ dx²,. . . , xⁿ + dxⁿ) → (x¹ + dx¹ + dξ¹, x² + dx² + dξ²,. . . , xⁿ + dxⁿ + dξⁿ) → (x¹ + dξ¹, x² + dξ², . . . ,xⁿ + dξⁿ) → (x¹, x², . . . , xⁿ) changes each component aⁱ of a by

[see also Eq. (16.10-25) ].

(b) Geodesic Parallels. Geodesic Deviation. If a one-parameter family of geodesies x_i = x_i(s, λ) has orthogonal trajectories,* the latter are geodesic parallels cutting off equal increments of the arc length s on each geodesic of the given family (see also Sees. 17.3-12 and 17.4-7).

Adjacent geodesies xⁱ= xⁱ(s,λ) and xⁱ = xⁱ(s, λ + dλ)are then related by

The geodesic-deviation vector represented by the nⁱ(s, λ) is normal to the geodesic xⁱ(s, λ), or g_iknⁱp^k = 0, where the p^kare the unit-tangent

* Note that the notation xⁱ= xⁱ{s, λ) excludes null geodesies.

vector components of the geodesic xⁱ = xⁱ(s, λ). Along any one of the given geodesies(λ = constant), the nⁱ change with s so that

(c) Flat Spaces. A Riemann space is flat

1. If and only if all components Rⁱ_jkh or R_jkh of the curvature tensor vanish identically, so that successive covariant differentiations commute [Eq. (16.10-25)], and parallel propagation around infinitesimal circuits leaves all tensor components constant (Sees. 16.10-9 and 17.4-6a)

2. If and only if the Riemann space admits a system of rectangular cartesian coordinatesξ¹, ξ², . . . , ξⁿ such that, at every point of the space,

where each ∊_i is constant and equals either +1 or -1.

In every scheme of measurements defined by a cartesian coordinate system all Christoffel symbols (Sec. 16.10-3) vanish identically; covariant differentiation reduces to ordinary differentiation, and every geodesic or geodesic null line can be described by linearparametric equations ξ_i = aⁱt+ bⁱ

Equation (20) may be formally simplified by the introduction of homogeneous coordinates is imaginary if ∊_i = – 1 (this convention is used in relativity theory, Ref. 17.4).

(d) A Euclidean space is a flat Riemann space having a positive- definite metric, so that all ∊_i in Eq. (20) are equal to +1 (Euclidean geometry,see also Chaps. 2 and 3). Note that the topology (Sec. 12.5-1) of a Euclidean space may differ from the “usual” topology employed in elementary geometry (EXAMPLES: surfaces of cylinders and cones).

17.4-7. Special Coordinate Systems. Because of the invariance of tensor equations (Sees. 16.1-4, 16.4-1, and 16.10-76), it is often permissible to simplify mathematical arguments by using one of the following special coordinate systems.

(a) Not every Riemann space admits orthogonal coordinates {gⁱ^k= 0 for i ≠ k,Sec. 16.8-2 Sec. 16.8-2), but it is possible to choose oneof the coordinates, say xⁿ,so that its coordinate lines are normal to all others, or

at every point (x¹,x², . . . , xⁿ).It is always possible to choose such a coordinate system so that either g_nn ≡ 1 or g_nn ≡ –1; xⁿ is, then, measured by the arc length s along its coordinate lines, and the latter are geodesies normal to every hypersurface xⁿ = constant (geodesic normal coordinates, see also Sec. 17.3-13).

(b) Every Riemann space admits local rectangular cartesian coordinates ξ¹, ξ², . . . , ξⁿ such that the metric is given by Eq. (20) at any onegiven point (ξ¹, ξ², . . . , ξⁿ). Hence every Riemann space is “locally flat”; i.e., every sufficiently small portion of the space is flat (Euclidean if the metric is positive definite).

(c) Riemann coordinates with origin 0 are defined as xⁱ = spⁱ where the pⁱ are unit-tangent-vector components at 0 of the geodesic joining 0 and the point (x¹, x²,. . . , xⁿ), and s is the geodesic distance between these points. Every Riemann space admits Riemannian coordinates for any given origin 0;note that all Christoffel symbols vanish at 0.

Plane analytic geometry Chap. 2

Solid analytic geometry Chap. 3

Vector analysis Chap. 5

Curvilinear coordinates Chap. 6

Tensor analysis, Riemann spaces Chap. 16

17.5-2. References and Bibliography.

17.1. Blaschke, W., and H. Reichardt: Vorlesungen uber Differ entialgeometrie,2d ed., Springer, Berlin, 1960.

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

17.3. : Riemannian Geometry,Princeton, Princeton, N.J., 1949.

17.4. Guggenheimer, H. W.: Differential Geometry,McGraw-Hill, New York, 1963.

17.5. Kreyszig, E.: Differential Geometry,2d ed., University of Toronto Press, Toronto, Canada, 1963.

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

17.7. Sokolnikov, I. S.: Tensor Analysis,2d ed., Wiley, New York, 1964.

17.8. Spain, B.: Tensor Calculus,Oliver & Boyd, London, 1953.

17.9. Struik, D. J.: Lectures on Classical Differential Geometry,2d ed., Addison-Wesley, Reading, Mass., 1961.

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

17.11. Willmore, T. J.: Introduction to Differential Geometry,Oxford, Fair Lawn, N.J., 1959.

(See also the article by H. Tietz in vol. II of the Handbuch der Physik,Springer, Berlin, 1955.)