The intrinsic geometry of a surface studies geometric constructions and quantities related to the surface that can be determined solely from the first quadratic form.. On the opposite, t
Trang 19.2 T HEORY OF S URFACES 395
9.2.3 Intrinsic Geometry of Surface
9.2.3-1 Intrinsic geometry and bending of surface
Suppose that two surfaces U and U ∗ are given and there is one-to-one correspondence
between their points such that the length of each curve on U is equal to the length of the corresponding curve on U ∗ Such a one-to-one mapping of U into U ∗ is called a bending
of the surface U into the surface U ∗ , and the surfaces U and U ∗ are said to be applicable The correspondence between U and U ∗ is said to be isometric.
The intrinsic geometry of a surface studies geometric constructions and quantities related
to the surface that can be determined solely from the first quadratic form The notions of length of a segment, angle between two curves, and area of part of a surface all belong in intrinsic geometry
On the opposite, the curvature of a curve given on the surface by the equations
u = u(t), v = v(t)
cannot be found using only the first quadratic form, and hence it does not belong in intrinsic geometry
9.2.3-2 Index notation Surface as Riemannian space
From now on in this chapter, the following notation related to tensor analysis is used:
u1= u, u2= v;
g11= E, g12= g21= F , g22= G;
b11 = L, b12= b21= M , b22= N ;
r1 = ru, r2 = rv, r12= r21= rvu= ruv, r11= ruu, r22= rvv
In the new notation, the first fundamental quadratic form becomes
E du2+2F du dv+ G dv2=
2
α=1
2
β=1
g αβ du α du β = g
αβ du α du β,
and the second fundamental quadratic form is
L du2+2M du dv+ N dv2 =
2
α=1
2
β=1
b αβ du α du β = b αβ du α du β.
The expressions for the coefficients of the first and second quadratic forms in the new notation become
g ij = ri⋅rj, b ij = rij ⋅N,
where N is the unit normal vector to the surface; i and j are equal to either1or2
Trang 2396 DIFFERENTIALGEOMETRY
9.2.3-3 Derivation formula
The Christoffel symbolsΓk,ij of the first kind are defined to be the scalar products of the
vectors rkand rij; i.e.,
r ⋅rij =Γk,ij The Christoffel symbols of the first kind satisfy the formula
Γk,ij = 1
2
∂g
ki
∂u j +
∂g kj
∂u i –
∂g ij
∂u k
This is one of the basic formulas in the theory of surfaces; this formula means that the
scalar products of the second partial derivatives of the position vector r(u i , u j) by its partial derivatives can be expressed in terms of the coefficients of the first quadratic form (more precisely, in terms of their derivatives)
The Christoffel symbolsΓk
ij of the second kind are defined by the relations
Γk
ij =
n
t=1
g ktΓk,ij,
where g ktis given by
n
k=1
g kt g
ks=
1, if t = s,
0, if t≠s The Christoffel symbols of the second kind are the coefficients in the decomposition of
the vector rij in two noncollinear vectors r1and r2and the unit normal vector N:
rij =Γ1ijr1+Γ2ijr2+ b ijN (i, j =1,2) (9.2.3.1)
Formulas (9.2.3.1) are called the first group of derivation formulas (the Gauss derivation
formulas).
The formulas
N1= –b11r1– b21r2, N1= –b12r1– b22r2, (9.2.3.2)
where b j i = b iα g αj , are called the second group of derivation formulas (Weingarten formulas).
The formulas in the second group of derivation formulas express the partial derivatives of
the unit normal vector N in terms of the variables u1 and u2 in the decomposition in the
basis vectors r1, r2, and N.
Formulas (9.2.3.1) and (9.2.3.2) express the partial derivatives with respect to u1 and
u2 of the basis vectors, i.e., of the two tangent vectors r1 and r2 and the normal vector N,
at a given point on a surface These partial derivatives of r1, r2 and N are obtained as a decomposition in the vectors r1, r2and N themselves.
9.2.3-4 Gauss formulas Peterson–Codazzi formulas
If the first quadratic form of the surface is given, then the second quadratic form cannot be
chosen arbitrarily, since its discriminant (LN – M2) is completely determined by the Gauss
formula
b11b22– b212= ∂
2g12
∂u1∂u2 –
1 2
∂2g11
∂u2∂u2 –
1 2
∂2g22
∂u1∂u1 +Γγ12Γδ
12g γδ–Γα
11Γβ22g αβ, (9.2.3.3)
where γ, δ, α, and β are independent summation indices equal to either1or2
Trang 3R EFERENCES FOR C HAPTER 9 397
We use (9.2.3.3) to reduce relation (9.2.2.2) to the form
K (u, v) = k1k2 = LN – M
2
EG – F2 =
b11b22– b212
g11g22– g122 . (9.2.3.4)
In view of (9.2.3.4), the Gaussian curvature of the surface is completely determined
by the coefficients of the first quadratic form and by their first and second derivatives with
respect to u1and u2
The coefficients b11, b12, b22of the second quadratic form and their first derivatives are
related to the coefficients g11, g12, g22of the first quadratic form and their first derivatives (contained only inΓk
ij ) by differential equations known as the Peterson–Codazzi formulas:
∂b i1
∂u2 –Γ1i2b11–Γ2i2b21= ∂b ∂u i12 –Γ1i1b12–Γ2i1b22 (i =1,2)
The Gauss formula and the Peterson–Codazzi formulas are necessary and sufficient conditions for two analytically determined quadratic differential forms to be the fist and second quadratic forms of some surface
References for Chapter 9
Aubin, T., A Course in Differential Geometry, American Mathematical Society, Providendce, Rhoad Island,
2000.
Burke, W L., Applied Differential Geometry, Cambridge University Press, Cambridge, 1985.
Byushgens, S S., Differential Geometry [in Russian], Komkniga, Moscow, 2006.
Chern, S.-S., Chen, W.-H., and Lam, K S., Lectures on Differential Geometry, World Scientific Publishing
Co., Hackensack, New Jersey, 2000.
Danielson, D A., Vectors and Tensors in Engineering and Physics, 2nd Rep Edition, Westview Press, Boulder,
Colorado, 2003.
Dillen, F J E and Verstraelen, L C A., Handbook of Differential Geometry, Vol 1, North Holland,
Amsterdam, 2000.
Dillen, F J E and Verstraelen, L C A., Handbook of Differential Geometry, Vol 2, North Holland,
Amsterdam, 2006.
Guggenheimer, H W., Differential Geometry, Dover Publications, New York, 1977.
Kay, D C., Schaum’s Outline of Tensor Calculus, McGraw-Hill, New York, 1988.
Kobayashi, S and Nomizu, K., Foundations of Differential Geometry, Vol 1, Wiley, New York, 1996 Kreyszig, E., Differential Geometry, Dover Publications, New York, 1991.
Lang, S., Fundamentals of Differential Geometry, Springer, New York, 2001.
Lebedev, L P and Cloud, M J., Tensor Analysis, World Scientific Publishing Co., Hackensack, New Jersey,
2003.
Lovelock, D and Rund, H., Tensors, Differential Forms, and Variational Principles, Dover Publications, New
York, 1989.
O’Neill, V., Elementary Differential Geometry, Rev 2nd Edition, Academic Press, New York, 2006.
Oprea, J., Differential Geometry and Its Applications, 2nd Edition, Prentice Hall, Englewood Cliffs, New
Jersey, 2003.
Pogorelov, A V., Differential Geometry, P Noordhoff, Groningen, 1967.
Postnikov, M M., Linear Algebra and Differential Geometry (Lectures in Geometry), Mir Publishers, Moscow,
1982.
Pressley, A., Elementary Differential Geometry, Springer, New York, 2002.
Rashevsky, P K., A Course in Differential Geometry, 4th Edition [in Russian], URSS, Moscow, 2003 Simmonds, J G., A Brief on Tensor Analysis, 2nd Edition, Springer, New York, 1997.
Somasundaram, D., Differential Geometry: A First Course, Alpha Science International, Oxford, 2004 Spain, B., Tensor Calculus: A Concise Course, Dover Publications, New York, 2003.
Spivak, M., A Comprehensive Introduction to Differential Geometry Vols 1–5, 3rd Edition, Publish or Perish,
Houston, 1999.
Struik, D J., Lectures on Classical Differential Geometry, 2nd Edition, Dover Publications, New York, 1988 Temple, G., Cartesian Tensors: An Introduction, Dover Publications, New York, 2004.
Trang 5Chapter 10
Functions of Complex Variable
10.1 Basic Notions
10.1.1 Complex Numbers Functions of Complex Variable
10.1.1-1 Complex numbers
The set of complex numbers is an extension of the set of real numbers An expression of
the form z = x + iy, where x and y are real numbers, is called a complex number, and the symbol i is called the imaginary unit: i2= –1 The numbers x and y are called, respectively, the real and imaginary parts of z and denoted by
The complex number x + i0 is identified with real number x, and the number 0+ iy is denoted by iy and is said to be pure imaginary Two complex numbers z1 = x1+ iy1and
z2 = x2+ iy2are assumed to be equal if x1= x2and y1= y2.
The complex number¯z = x – iy is said to be conjugate to the number z.
The sum or difference of complex numbers z1= x1+ iy1and z2= x2+ iy2is defined to
be the number
z1 z2= x1 x2+ i(y1 y2). (10.1.1.2)
Addition laws:
1 z1+ z2= z2+ z1(commutativity)
2 z1+ (z2+ z3) = (z1+ z2) + z3(associativity)
The product z1z2 of complex numbers z1= x1+ iy1and z2= x2+ iy2is defined to be the number
z1z2= (x1x2– y1y2) + i(x1y2– x2y1). (10.1.1.3)
Multiplication laws:
1 z1z2= z2z1(commutativity).
2 z1(z2z3) = (z1z2)z3(associativity).
3 (z1+ z2)z3 = z1z3+ z2z3(distributivity with respect to addition)
The product of a complex number z = x + iy by its conjugate is always nonnegative:
z ¯z = x2+ y2 (10.1.1.4)
For a positive integer n, the n-fold product of z by itself is called the nth power of the number z and is denoted by z n A number w is called an nth root of a number z and is denoted by w = √ n
z if w n = z.
If z2 ≠ 0, then the quotient of z1and z2is defined as
z1
z2 =
x1x2+ y1y2
x2
2+ y22
+ i x2y1– x1y2
x2
2+ y22
Relation (10.1.1.5) can be obtained by multiplying the numerator and the denominator of
the fraction z1/z2by¯z2.
399
Trang 6400 FUNCTIONS OFCOMPLEXVARIABLE
10.1.1-2 Geometric interpretation of complex number
There is a one-to-one correspondence between complex numbers z = x + iy and points M with coordinates (x, y) on the plane with a Cartesian rectangular coordinate system OXY
or with vectors −−→ OM connecting the origin O with M (Fig 10.1) The length r of the vector
−−→
OM is called the modulus of the number z and is denoted by r =|z|, and the angle ϕ formed
by the vector −−→ OM and the positive direction of the OX-axis is called the argument of the number z and is denoted by ϕ = Arg z.
X
Y
r φ
M
O
Figure 10.1 Geometric interpretation of complex number.
The modulus of a complex number is determined by the formula
|z|=√
z ¯z = x2+ y2. (10.1.1.6)
The argument Arg z is determined up to a multiple of2π, Arg z = arg z +2kπ, where k is
an arbitrary integer and arg z is the principal value of Arg z determined by the condition –π < arg z≤π The principal value arg z is given by the formula
arg z =
⎧
⎪
⎪
⎪
⎪
arctan(y/x) for x >0,
π + arctan(y/x) for x <0, y ≥ 0,
–π + arctan(y/x) for x <0, y <0,
π/2 for x =0, y >0,
–π/2 for x =0, y <0
(10.1.1.7)
For z =0, Arg z is undefined.
Since x = r cos ϕ and y = r sin ϕ, it follows that the complex number can be written in the trigonometric form
z = x + iy = r(cos ϕ + i sin ϕ). (10.1.1.8)
For numbers z1= r1(cos ϕ1+ i sin ϕ1) and z2= r2(cos ϕ2+ i sin ϕ2), written in trigonometric form, the following rules of algebraic operations are valid:
z1z2= r1r2
cos(ϕ1+ ϕ2) + i sin(ϕ1+ ϕ2)
,
z1
z2 =
r1
r2
cos(ϕ1– ϕ2) + i sin(ϕ1– ϕ2)
In the latter formula, it is assumed that z≠ 0 For any positive integer n, this implies the de
Moivre formula
z n = r n (cos nϕ + i sin nϕ), (10.1.1.10)
as well as the formula for extracting the root of a complex number For z ≠ 0, there are
exactly n distinct values of the nth root of the number z = r(cos ϕ + i sin ϕ) They are
determined by the formulas
n
√
z= √ n
r
cos ϕ+2kπ
n + i sin ϕ+2kπ
n
(k =0,1,2, , n –1) (10.1.1.11)
Trang 710.1 B ASIC N OTIONS 401
Example Let us find all values of√3
i.
We represent the complex number z = i in trigonometric form We have r =1and ϕ = arg z = 12π The distinct values of the third root are calculated by the formula
ω k=√3 i
cos
π
2 + 2πk
3 + i sin
π
2 + 2πk
3
(k =0 , 1 , 2 ),
so that
ω0= cosπ6 + i sin
π
√
3
2 + i12,
ω1= cos 5π
6 + i sin
5π
√
3
1
2,
ω2= cos 3π
2 + i sin
3π
2 = –i.
The roots are shown in (Fig 10.2).
X
Y
i
i
ω
ω
ω
1
2
0
2
Figure 10.2 The roots of√3
i.
X
Y
O
z
z+z
z z z
1
1
1 2
2 2
Figure 10.3 The sum and difference of complex
numbers.
The plane OXY is called the complex plane, the axis OX is called the real axis, and the axis OY is called the imaginary axis The notions of complex number and point on the
complex plane are identical
The geometric meaning of the operations of addition and subtraction of complex
num-bers is as follows: the sum and the difference of complex numnum-bers z1and z2are the vectors
equal to the directed diagonals of the parallelogram spanned by the vectors z1 and z2
(Fig 10.3) The following inequalities hold (Fig 10.3):
|z1+ z2| ≤ |z1|+|z2|, |z1– z2| ≥|z1|–|z2|. (10.1.1.12) Inequalities (10.1.1.12) become equalities if and only if the arguments of the complex
numbers z1and z2coincide (i.e., arg z1= arg z2) or one of the numbers is zero
10.1.2 Functions of Complex Variable
10.1.2-1 Notion of function of complex variable
A subset D of the complex plane such that each point of D has a neighborhood contained
in D (i.e., D is open) and two arbitrary points of D can be connected by a broken line lying
in D (i.e., D is connected) is called a domain on the complex plane A point that does not itself lie in D but whose arbitrary neighborhood contains points of D is called a boundary
point of D The set of all boundary points of D is called the boundary of D, and the union
... (10.1.1.10)as well as the formula for extracting the root of a complex number For z ≠ 0, there are
exactly n distinct values of the nth root of the number z = r(cos ϕ + i sin... imaginary axis The notions of complex number and point on the
complex plane are identical
The geometric meaning of the operations of addition and subtraction of complex
num-bers... arbitrary neighborhood contains points of D is called a boundary
point of D The set of all boundary points of D is called the boundary of D, and the union