332 INTEGRALS 7.4.2-4. Necessary and sufficient conditions for a vector field to be potential. Let U be a simply connected domain in R 3 (i.e., a domain in which any closed contour can be deformed to a point without leaving U)andleta(x, y, z)beavectorfield in U. Then the following four assertions are equivalent to each other: (1) the vector field a is potential; (2) curla ≡ 0; (3) the circulation of a around any closed contour C U is zero, or, equivalently, C a ⋅ dr = 0; (4) the integral AB a ⋅ dr is independent of the shape of AB U (it depends on the starting and the finishing point only). 7.4.3. Surface Integral of the First Kind 7.4.3-1. Definition of the surface integral of the first kind. Let a function f(x, y,z)bedefined on a smooth surface D. Let us break up this surface into n elements (cells) that do not have common internal points and let us denote this partition by D n .Thediameter, λ(D n ), of a partition D n is the largest of the diameters of the cells (see Paragraph 7.3.4-1). Let us select in each cell an arbitrary point (x i , y i , z i ), i = 1, 2, , n, and make up an integral sum s n = n i=1 f(x i , y i , z i ) ΔS i , where ΔS i is the area of the ith element. If there exists a fi nite limit of the sums s n as λ(D n ) → 0 that depends on neither the partition D n nor the selection of the points (x i , y i , z i ), then it is called the surface integral of the first kind of the function f(x, y, z) and is denoted D f(x, y,z) dS. 7.4.3-2. Computation of the surface integral of the first kind. 1 ◦ . If a surface D is defined by an equation z = z(x, y), with (x, y) D 1 ,then D f(x, y,z) dS = D 1 f x, y, z(x, y) 1 +(z x ) 2 +(z y ) 2 dx dy. 2 ◦ . If asurface D isdefined by a vector equationr =r(x, y, z)=x(u, v) i+y(u, v) j+z(u, v) k, where (u, v) D 2 ,then D f(x, y,z) dS = D 2 f x(u, v), y(u, v), z(u, v) |n(u, v)|du dv, where n(u, v)=r u × r v is the unit normal to the surface D; the subscripts u and v denote the respective partial derivatives. 7.4. LINE AND SURFACE INTEGRALS 333 7.4.3-3. Applications of the surface integral of the first kind. 1 ◦ . Area of a surface D: S D = D dS. 2 ◦ . Mass of a material surface D with a surface density γ = γ(x, y, z): m = D γ(x, y, z) dS. 3 ◦ . Coordinates of the center of mass of a material surface D: x c = 1 m D xγ dS, y c = 1 m D yγ dS, z c = 1 m D zγ dS. To the uniform surface density there corresponds γ = const. 7.4.4. Surface Integral of the Second Kind 7.4.4-1. Definition of the surface integral of the second kind. Let D be an oriented surface defined by an equation r = r(u, v)=x(u, v) i + y(u, v) j + z(u, v) k, where u and v are parameters. The fact that D is oriented means that every point M D has an associated unit normal n(M)=n(u, v) continuously dependent on M. Two cases are possible: (i) the associated unit normal is n(u, v)=r u × r v or (ii) the associated unit normal is opposite, n(u, v)=r v × r u =–r u × r v . Remark. If a surface is defined traditionally by an equation z = z(x, y), its representation in vector form is as follows: r = r(x, y)=x i + y j + z(x, y) k. Let a vector field a(x, y, z)=P i + Q j + R k be defined on a smooth oriented surface D. Let us perform a partition, D n , of the surface D into n elements (cells) that do not have common internal points. Also select an arbitrary point M i (x i , y i , z i ), i = 1, 2, , n,for each cell and make up an integral sum s n = n i=1 a(x i , y i , z i ) ⋅ n ◦ i ΔS i ,whereΔS i is area of the ith cell and n ◦ i is the unit normal to the surface at the point M i , the orientation of which coincides with that of the surface. If there exists a fi nite limit of the sums s n as λ(D n ) → 0 that depends on neither the partition D n nor the selection of the points M i (x i , y i , z i ), then it is called the surface integral of the second kind (or the flux of the vector field a across the oriented surface D)andis denoted D a(x, y, z) ⋅ −→ dS,or D Pdydz+ Qdxdz+ Rdxdy. Note that the surface integral of the second kind changes its sign when the orientation of the surface is reversed. 334 INTEGRALS 7.4.4-2. Computation of the surface integral of the second kind. 1 ◦ . If a surface D is defined by a vector equation r = r(u, v), where (u, v) D 1 ,then D a(x, y, z) ⋅ −→ dS = D 1 a x(u, v), y(u, v), z(u, v) ⋅ n(u, v) du dv. The plus sign is taken if the unit normal associated with the surface is n(u, v)=r u ×r v ,and the minus sign is taken in the opposite case. 2 ◦ . If a surface D is defined by an equation z = z(x, y), with (x, y) D 2 , then the normal n(x, y)=r x × r y =–z x i – z y j + k orients the surface D “upward,” in the positive direction of the z-axis; the subscripts x and y denote the respective partial derivatives. Then D a ⋅ −→ dS = D 2 –z x P – z y Q + R dx dy, where P = P x, y, z(x, y) , Q = Q x, y, z(x, y) ,andR = R x, y, z(x, y) . The plus sign is taken if the surface has the “upward” orientation, and the minus sign is chosen in the opposite case. 7.4.5. Integral Formulas of Vector Calculus 7.4.5-1. Ostrogradsky–Gauss theorem (divergence theorem). Let a vector fielda(x, y, z)=P (x, y, z) i + Q(x, y, z) j + R(x, y, z) k be continuously differ- entiable in a finite simply connected domain V ⊂ R 3 oriented by the outward normal and let S denote the boundary of V . Then the Ostrogradsky–Gauss theorem (or the divergence theorem) holds: S a ⋅ −→ dS = V divadxdydz, where diva is the divergence of the vector a, which is defined as follows: diva = ∂P ∂x + ∂Q ∂y + ∂R ∂z . Thus, the flux of a vector field across a closed surface in the outward direction is equal to the triple integral of the divergence of the vector field over the volume bounded by the surface. In coordinate form, the Ostrogradsky–Gauss theorem reads S Pdydz+ Qdxdz+ Rdxdy = V ∂P ∂x + ∂Q ∂y + ∂R ∂z dx dy dz. 7.4.5-2. Stokes’s theorem (curl theorem). 1 ◦ . Let a vector field a(x, y, z) be continuously differentiable in a domain of the three- dimensional space R 3 that contains an oriented surface D. The orientation of a surface uniquely defines the direction in which the boundary of the surface is traced; specifically, the boundary is traced counterclockwise when looked at from the direction of the normal to REFERENCES FOR CHAPTER 7 335 the surface. Then the circulation of the vector field around the boundary C of the surface D is equal to the flux of the vector curl a across D: C a ⋅ dr = D curla ⋅ −→ dS. In coordinate notation, Stokes’s theorem reads C Pdx+Qdy+Rdz= D ∂R ∂y – ∂Q ∂z dy dz+ ∂P ∂z – ∂R ∂x dx dz+ ∂Q ∂x – ∂P ∂y dx dy. 2 ◦ . For a plane vector field a(x, y)=P (x, y) i + Q(x, y) j, Stokes’s theorem reduces to Green’s theorem: C Pdx+ Qdy = D ∂Q ∂x – ∂P ∂y dx dy, where the contour C of the domain D on the x, y plane is traced counterclockwise. 7.4.5-3. Green’s first and second identities. Gauss’s theorem. 1 ◦ .LetΦ = Φ(x, y, z)andΨ = Ψ(x, y, z) be twice continuously differentiable functions definedinafinite simply connected domain V ⊂ R 3 bounded by a piecewise smooth boundary S. Then the following formulas hold: V ΨΔΦ dV + V ∇Φ ⋅∇Ψ dV = S Ψ ∂Φ ∂n dS (Green’s first identity), V (ΨΔΦ – ΦΔΨ) dV = S Ψ ∂Φ ∂n – Φ ∂Ψ ∂n dS (Green’s second identity), where ∂ ∂n denotes a derivative along the (outward) normal to the surface S,andΔ is the Laplace operator. 2 ◦ . In applications, the following special cases of the above formulas are most common: V ΦΔΦ dV + V |∇Φ| 2 dV = S Φ ∂Φ ∂n dS (first identity with Ψ = Φ), V ΔΦ dV = S ∂Φ ∂n dS (second identity with Ψ = 1). References for Chapter 7 Adams, R., Calculus: A Complete Course, 6th Edition, Pearson Education, Toronto, 2006. Anton, H., Calculus: A New Horizon, 6th Edition, Wiley, New York, 1999. Anton, H., Bivens, I., and Davis, S., Calculus: Early Transcendental Single Variable, 8th Edition, John Wiley & Sons, New York, 2005. Aramanovich, I. G., Guter, R. S., et al., Mathematical Analysis (Differentiation and Integration), Fizmatlit Publishers, Moscow, 1961. Borden, R. S., A Course in Advanced Calculus, Dover Publications, New York, 1998. Brannan, D., A First Course in Mathematical Analysis, Cambridge University Press, Cambridge, 2006. Bronshtein, I. N. and Semendyayev, K. A., Handbook of Mathematics, 4th Edition, Springer-Verlag, Berlin, 2004. 336 INTEGRALS Browder, A., Mathematical Analysis: An Introduction, Springer-Verlag, New York, 1996. Clark, D. N., Dictionary of Analysis, Calculus, and Differential Equations, CRC Press, Boca Raton, 2000. Courant, R. and John, F., Introduction to Calculus and Analysis, Vol. 1, Springer-Verlag, New York, 1999. Danilov, V. L., Ivanova, A. N., et al., Mathematical Analysis (Functions, Limits, Series, Continued Fractions) [in Russian], Fizmatlit Publishers, Moscow, 1961. Dwight, H. B., Tables of Integrals and Other Mathematical Data, Macmillan, New York, 1961. Edwards, C. H., and Penney, D., Calculus, 6th Edition, Pearson Education, Toronto, 2002. Fedoryuk, M. V., Asymptotics, Integrals and Series [in Russian], Nauka Publishers, Moscow, 1987. Fikhtengol’ts, G. M., Fundamentals of Mathematical Analysis, Vol. 2, Pergamon Press, London, 1965. Fikhtengol’ts, G. M., A Course of Differential and Integral Calculus, Vol. 2 [in Russian], Nauka Publishers, Moscow, 1969. Gradshteyn, I. S. and Ryzhik, I. M., Tables of Integrals, Series and Products, 6th Edition, Academic Press, New York, 2000. Kaplan, W., Advanced Calculus, 5th Edition, Addison Wesley, Reading, Massachusetts, 2002. Kline, M., Calculus: An Intuitive and Physical Approach, 2nd Edition, Dover Publications, New York, 1998. Landau, E., Differential and Integral Calculus, American Mathematical Society, Providence, 2001. Marsden, J. E. and Weinstein, A., Calculus, 2nd Edition, Springer-Verlag, New York, 1985. Mendelson, E., 3000 Solved Problems in Calculus, McGraw-Hill, New York, 1988. Polyanin, A. D., Polyanin, V. D., et al., Handbook for Engineers and Students. Higher Mathematics. Physics. Theoretical Mechanics. Strength of Materials, 3rd Edition [in Russian], AST/Astrel, Moscow, 2005. Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I., Integrals and Series, Vol. 1, Elementary Functions, Gordon & Breach, New York, 1986. Silverman, R. A., Essential Calculus with Applications, Dover Publications, New York, 1989. Strang, G., Calculus, Wellesley-Cambridge Press, Massachusetts, 1991. Taylor, A. E. and Mann, W. R., Advanced Calculus, 3rd Edition, John Wiley, New York, 1983. Thomas, G. B. and Finney, R. L., Calculus and Analytic Geometry, 9th Edition, Addison Wesley, Reading, Massachusetts, 1996. Widder, D. V., Advanced Calculus, 2nd Edition, Dover Publications, New York, 1989. Zorich, V. A., Mathematical Analysis, Springer-Verlag, Berlin, 2004. Zwillinger, D., CRC Standard Mathematical Tables and Formulae, 31st Edition, CRC Press, Boca Raton, 2002. Chapter 8 Series 8.1. Numerical Series and Infinite Products 8.1.1. Convergent Numerical Series and Their Properties. Cauchy’s Criterion 8.1.1-1. Basic definitions. Let {a n } be a numerical sequence. The expression a 1 + a 2 + ···+ a n + ···= ∞ n=1 a n is called a numerical series (infinite sum, infinite numerical series), a n is the generic term of the series,and s n = a 1 + a 2 + ···+ a n = n k=1 a k is the nth partial sum of the series. If there exists a finite limit lim n→∞ s n = S,theseries is called convergent,andS is called the sum of the series. In this case, one writes ∞ n=1 a n = S. If lim n→∞ s n does not exist (or is infinite), the series is called divergent.The series a n+1 + a n+2 + a n+3 + ··· is called the nth remainder of the series. Example 1. Consider the series ∞ n=1 aq n–1 = a + aq +aq 2 + ··· whose terms form a geometric progression with ratio q. This series is convergent for |q| < 1 (its sum has the form S = a 1–q ) and is divergent for |q| ≥ 1. 8.1.1-2. Necessary condition for a series to be convergent. Cauchy’s criterion. 1. A necessary condition for a series to be convergent. For a convergent series ∞ n=1 a n , the generic term must tend to zero, lim n→∞ a n = 0. If lim n→∞ a n ≠ 0, then the series is divergent. Example 2. The series ∞ n=1 cos 1 n is divergent, since its generic term a n =cos 1 n does not tend to zero as n →∞. The above necessary condition is insufficient for the convergence of a series. Example 3. Consider the series ∞ n=1 1 √ n . Its generic term tends to zero, lim n→∞ 1 √ n = 0, but the series ∞ n=1 1 √ n is divergent because its partial sums are unbounded, s n = 1 √ 1 + 1 √ 2 + ···+ 1 √ n > n 1 √ n = √ n →∞ as n →∞. 337 338 SERIES 2. Cauchy’s criterion of convergence of a series.Aseries ∞ n=1 a n is convergent if and only if for any ε > 0 there is N = N (ε) such that for all n > N and any positive integer k, the following inequality holds: |a n+1 + ···+ a n+k | < ε. 8.1.1-3. Properties of convergent series. 1. If a series is convergent, then any of its remainders is convergent. Removal or addition of finitely many terms does not affect the convergence of a series. 2. If all terms of a series are multiplied by a nonzero constant, the resulting series preserves the property of convergence or divergence (its sum is multiplied by that constant). 3. If the series ∞ n=1 a n and ∞ n=1 b n are convergent and their sums are equal to S 1 and S 2 , respectively, then the series ∞ n=1 (a n b n ) are convergent and their sums are equal to S 1 S 2 . 4. Terms of a convergent series can be grouped in successive order; the resulting series has the same sum. In other words, one can insert brackets inside a series in an arbitrary order. The inverse operation of opening brackets is not always admissible. Thus, the series (1 –1)+(1 –1)+··· is convergent (its sum is equal to zero), but, after removing the brackets, we obtain the divergent series 1 – 1 + 1 – 1 + ··· (its generic term does not tend to zero). 8.1.2. Convergence Criteria for Series with Positive (Nonnegative) Terms 8.1.2-1. Basic convergence (divergence) criteria for series with positive terms. 1. The first comparison criterion.If0 ≤ a n ≤ b n (starting from some n), then conver- gence of the series ∞ n=1 b n implies convergence of ∞ n=1 a n ; and divergence of the series ∞ n=1 a n implies divergence of ∞ n=1 b n . 2. The second convergence criterion. Suppose that there is a finite limit lim n→∞ a n b n = σ, where 0 < σ < ∞.Then ∞ n=1 a n is convergent (resp., divergent) if and only if ∞ n=1 b n is convergent (resp., divergent). Corollary. Suppose that a n+1 /a n ≤ b n+1 /b n starting from some N (i.e., for n > N). Then convergence of the series ∞ n=1 b n implies convergence of ∞ n=1 a n , and divergence of ∞ n=1 a n implies divergence of ∞ n=1 b n . 3. D’Alembert criterion. Suppose that there exists the limit (finite or infinite) lim n→∞ a n+1 a n = D. . 1988. Polyanin, A. D., Polyanin, V. D., et al., Handbook for Engineers and Students. Higher Mathematics. Physics. Theoretical Mechanics. Strength of Materials, 3rd Edition [in Russian], AST/Astrel,. that do not have common internal points and let us denote this partition by D n .Thediameter, λ(D n ), of a partition D n is the largest of the diameters of the cells (see Paragraph 7.3.4-1). Let. the subscripts u and v denote the respective partial derivatives. 7.4. LINE AND SURFACE INTEGRALS 333 7.4.3-3. Applications of the surface integral of the first kind. 1 ◦ . Area of a surface D: S D = D dS. 2 ◦ .