Handbook of mathematics for engineers and scienteists part 53 pot

Definition of the surface integral of the second kind.. Note that the surface integral of the second kind changes its sign when the orientation of the surface is reversed... Computation

332 INTEGRALS

7.4.2-4 Necessary and sufficient conditions for a vector field to be potential

Let U be a simply connected domain inR3(i.e., a domain in which any closed contour can

be deformed to a point without leaving U ) and let a(x, y, z) be a vector field in U Then the

following four assertions are equivalent to each other:

(1) the vector field a is potential;

(2) curl a≡0;

(3) the circulation of a around any closed contour C  U is zero, or, equivalently, 8

C a⋅dr=0;

(4) the integral

AB a⋅dr is independent of the shape ofAB  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) be defined on a smooth surface D Let us break up this surface into

nelements (cells) that do not have common internal points and let us denote this partition

byDn The diameter, λ( Dn), of a partitionDnis 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

whereΔS i is the area of the ith element.

If there exists a finite limit of the sums s n as λ( Dn) → 0that depends on neither the partitionDn 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)

D1, then



D f(x, y, z) dS =



D1

f x, y, z(x, y)

1+ (z x )2+ (z y )2dx dy.

2◦ If a surface D is defined by a vector equationr =r(x, y, z) = x(u, v)i+y(u, v)j+z(u, v) k, where (u, v)D2, then



D f (x, y, z) dS =



D2

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

7.4.3-3 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):



D γ(x, y, z) dS.

3◦ Coordinates of the center of mass of a material surface D:

xc = 1

m



D xγ dS, yc=

1

m



D yγ dS, zc =

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) = xi + y j + z(x, y) k.

Let a vector field a(x, y, z) = Pi + Qj + Rk 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=1a(x i , y i , z i)⋅ n ◦

i ΔS i, whereΔS iis 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 finite limit of the sums s n as λ( D n) → 0that depends on neither the partitionD 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) and is

D a(x, y, z)⋅dS, −→ or



D P dy dz + Q dx dz + R dx dy.

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)

D1, then



D a(x, y, z)⋅dS −→=



D1

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)

D2, 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 −→=



D2

–z x P – z y Q + R

dx dy, where P = P x, y, z(x, y)

, Q = Q x, y, z(x, y)

, and R = 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 field a(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 ⊂ R3 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 div a dx dy dz, where div a is the divergence of the vector a, which is defined as follows:

div a = ∂P

∂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 P dy dz + Q dx dz + R dx dy =



V



∂P

∂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 R3 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

the surface Then the circulation of the vector field around the boundaryC of the surface D

is equal to the flux of the vector curl a across D:

8

C a⋅dr=



D curl a⋅dS −→

In coordinate notation, Stokes’s theorem reads

8

C P dx+Q dy+R dz =



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: 8

C P dx + Q dy =



D



∂Q

∂y



dx dy,

where the contourC 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◦ Let Φ = Φ(x, y, z) and Ψ = Ψ(x, y, z) be twice continuously differentiable functions

defined in a finite simply connected domain V ⊂ R3 bounded by a piecewise smooth

boundary S.

Then the following formulas hold:



V ΨΔΦ dV +



V ∇Φ⋅ ∇Ψ dV =



SΨ∂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|2dV =



SΦ∂Φ

∂n dS (first identity withΨ = Φ),



V ΔΦ dV =



S

∂Φ

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.

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

a1+ a2+· · · + 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 = a1+ a2+· · · + 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, the series

is called convergent, and S is called the sum of the series In this case, one writes

∞



n=1a 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=1aq

n–1= a + aq + aq2+· · · whose terms form a geometric progression

with ratio q This series is convergent for|q| < 1(its sum has the form S = 1a–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=1a 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

√

2+· · · +

1

√

n > n √1

n =√

n → ∞ as n → ∞.

337

338 SERIES

2 Cauchy’s criterion of convergence of a series A series ∞

n=1a nis convergent if and

only if for any ε >0there 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=1a nand

∞



n=1b n are convergent and their sums are equal to S1and S2,

respectively, then the series∞

n=1(a n b n ) are convergent and their sums are equal to S1 S2.

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 series1–1+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 If 0 ≤a n≤b n (starting from some n), then

conver-gence of the series ∞

n=1b nimplies convergence of

∞



n=1a n; and divergence of the series

∞



n=1a n

implies divergence of∞

n=1b 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=1a n is convergent (resp., divergent) if and only if

∞



n=1b 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=1b n implies convergence of

∞



n=1a n, and divergence of

∞



n=1a nimplies divergence of

∞



n=1b n.

3 D’Alembert criterion Suppose that there exists the limit (finite or infinite)

lim

n→∞

a n+1

a n = D.

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],... not have common internal points and let us denote this partition

byDn The diameter, λ( Dn), of a partitionDnis the largest of the diameters of the cells (see

Paragraph... Mathematical Analysis, Cambridge University Press, Cambridge, 2006 Bronshtein, I N and Semendyayev, K A., Handbook of Mathematics, 4th Edition, Springer-Verlag, Berlin,

2004.