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(BQ) Part 1 book Advanced engineering mathematics has contents: Linear second order equations, first order differential equations, the laplace transform, series solutions, approximation of solutions, vectors and vector spaces, matrices and linear systems,... and other contents.
This page intentionally left blank Guide to Notation L[ f ] Laplace transform of f L[ f ](s) Laplace transform of f evaluated at s L−1 [F] inverse Laplace transform of F H (t) Heaviside function f ∗g often denotes a convolution with respect to an integral transform, such as the Laplace transform or the Fourier transform δ(t) delta function < a, b, c > vector with components a, b, c + bj + ck standard form of a vector in 3-space V norm (magnitude, length) of a vector V F · G dot product of vectors F and G F × G cross product of F and G n-space, consisting of n-vectors < x1 , x2 , · · · , xn > Rn [ai j ] matrix whose i, j-element is j If the matrix is denoted A, this i, j element may also be denoted Ai j Onm n × m zero matrix n × n identity matrix In transpose of A At reduced (row echelon) form of A AR rank(A) rank of a matrix A [A B] augmented matrix inverse of the matrix A A−1 |A| or det(A) determinant of A pA (λ) characteristic polynomial of A often denotes the fundamental matrix of a system X = AX T often denotes a tangent vector N often denotes a normal vector n often denotes a unit normal vector κ curvature ∇ del operator ∇ϕ or grad ϕ gradient of ϕ Du ϕ(P) directional derivative of ϕ in the direction of u at P f d x + g dy + h dz line integral C F · dR another notation for C f d x + g dy + h dz with F = f i + gj + hk C C1 C2 · · · Cn join of curves C1 , C2 , · · · , Cn f (x, y, z) ds line integral of f over C with respect to arc length C Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it October 14, 2010 15:48 THM/NEIL Page-1 27410_00_IFC_p01-02 Guide to Notation ∂( f, g) Jacobian of f and g with respect to u and v ∂(u, v) f (x, y, z) dσ surface integral of f over f (x0 −), f (x0 +) left and right limits, respectively, of f (x) at x0 F[ f ] or fˆ Fourier transform of f ˆ F[ f ](ω) or F(ω) Fourier transform of f evaluated at ω −1 inverse Fourier transform F Fourier cosine transform of f FC [ f ] or fˆC inverse Fourier cosine transform FC−1 or fˆC−1 Fourier sine transform of f F S [ f ] or fˆS inverse Fourier sine transform F S−1 or fˆS−1 D[u] discrete N - point Fourier transform (DFT) of a sequence u j windowed Fourier transform fˆwin often denotes the characteristic function of an interval I χI σ N (t) often denotes the N th Cesàro sum of a Fourier series Z (t) in the context of filtering, denotes a filter function Pn (x) nth Legendre polynomial (x) gamma function B(x, y) beta function Bessel function of the first kind of order ν Jν γ depending on context, may denote Euler’s constant Bessel function of the second kind of order ν Yν modified Bessel functions of the first and second kinds, respectively, of order zero I0 , K Laplacian of u ∇ 2u Re(z) real part of a complex number z Im(z) imaginary part of a complex number z z complex conjugate of z |z| magnitude (also norm or modulus) of z arg(z) argument of z f (z) dz integral of a complex function f (z) over a curve C C f (z) dz integral of f over a closed curve C C Res( f, z ) residue of f (z) at z f : D → D∗ f is a mapping from D to D ∗ Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it October 14, 2010 15:48 THM/NEIL Page-2 27410_00_IFC_p01-02 This is an electronic version of the print textbook Due to electronic rights restrictions, some third party content may be suppressed Editorial review has deemed that any suppressed content does not materially affect the overall learning experience The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it This page intentionally left blank A D VA N C E D ENGINEERING M AT H E M AT I C S Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it October 15, 2010 17:43 THM/NEIL Page-i 27410_00_fm_pi-xiv Advanced Engineering Mathematics Seventh Edition Peter V O’Neil Publisher, Global Engineering: Christopher M Shortt Senior Acquisitions Editor: Randall Adams Senior Developmental Editor: Hilda Gowans c 2012, 2007 Cengage Learning ALL RIGHTS RESERVED No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means 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1-111-42741-0 Proofreader: Martha McMaster Indexer: Shelly Gerger-Knechtl Compositor: Integra Senior Art Director: Michelle Kunkler Cover Designer: Andrew Adams Cover Image: Shutterstock/IuSh Internal Designer: Terri Wright Senior Rights, Acquisitions Specialist: Mardell Glinski-Schultz Cengage Learning 200 First Stamford Place, Suite 400 Stamford, CT06902 USA Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, Including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan Locate your local office at: international.cengage.com/region Text and Image Permissions Researcher: Kristiina Paul Cengage Learning products are represented in Canada by Nelson Education Ltd First Print Buyer: Arethea L Thomas For course and learning solutions, visit www.login.cengage.com Purchase any of our products at your local college store or at our preferred online store www.cengagebrain.com Printed in the United States of America 13 12 11 10 Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it October 15, 2010 17:43 THM/NEIL Page-iv 27410_00_fm_pi-xiv A D VA N C E D ENGINEERING M AT H E M AT I C S 7th Edition PETER V O’NEIL The University of Alabama at Birmingham Australia · Brazil · Japan · Korea · Mexico · Singapore · Spain · United Kingdom · United States Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it October 15, 2010 17:43 THM/NEIL Page-iii 27410_00_fm_pi-xiv This page intentionally left blank 12.9 Stokes’s Theorem 411 For the converse, it is enough to show that, if F has curl zero, then C F · d R is independent of path, since then we can define a potential function by choosing P0 and setting (x,y,z) ϕ(x, y, z) = F · dR P0 To show this independence of path, let C and K be paths in D from P0 to P1 Form a closed path L = C (−K ) Since D is simply connected, there is a piecewise smooth surface in D having C as boundary By Stokes’s theorem, F · dR = L F · dR − C = F · dR K (∇ × F) · n dσ = 12.9.2 Maxwell’s Equations The theorems of Gauss and Stokes are used in the analysis of vector fields We will illustrate this with electric and magnetic fields and Maxwell’s equations To begin, we will use the following standard notation and relationships: E = electric intensity = permitivity of the medium J = current density σ = conductivity μ= permeability D = E = electric flux density Q = charge density B = μH = magnetic flux density H= magnetic intensity Q d V = total charge in a region V q= V ϕ= B · n dσ = magnetic flux across i= J · n dσ = flux of current across In these, flux is computed using an outer unit normal to the closed surface We also have the following relationships, which have been observed and verified experimentally E · dR = − Faraday’s law C ∂ϕ ∂t Here C is any piecewise smooth closed curve in the medium We may think of this as saying that the rate of change of the magnetic flux across is the negative of the measure of the tangential component of the electric intensity around any closed curve bounding Ampère’s law H · dR = i This says that the measure of the tangential component of the magnetic intensity about C is the current flowing through any surface bounded by C Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it October 14, 2010 14:53 THM/NEIL Page-411 27410_12_ch12_p367-424 412 CHAPTER 12 Vector Integral Calculus D · n dσ = q and Gauss’s laws B · n dσ = These say that the measure of the normal component of the electric flux density across equals the total charge in the bounded region, and that the measure of the normal component of the magnetic flux density across is zero We will now carry out arguments similar to that used to derive the heat equation using the divergence theorem Begin by applying Stokes’s theorem to Faraday’s law to obtain E · dR = ∇ × E · n dσ = − C =− ∂ ∂t ∂ϕ ∂t B · n dσ = − ∂B · n dσ ∂t Then ∇ ×E+ ∂B · n dσ = ∂t Since this holds for any piecewise smooth closed surface must be zero, leading to ∇ ×E+ within the medium, then the integrand ∂B = ∂t A similar analysis, using Ampère’s law, yields ∇ × H = J Maxwell had observed that J=σE+ ∂E ∂t Then ∇ ×H=σE+ ∂E ∂t Now start on a new tack Apply Gauss’s theorem to Gauss’s law q = D · n dσ = (∇ · D) d V = q = V D · n dσ to obtain Q d V V Again falling back on the arbitrary nature of , we conclude from this that ∇ · D = Q Now go back to ∇ × E = −∂B/∂t and take the curl of both sides: ∇ × (∇ × E) = ∇ × − ∂B ∂ = − (∇ × B) ∂t ∂t We were able to interchange ∇ and ∂/∂t here because the curl involves only the space variables Since B = μH, then ∂ ∂ (∇ × μH) = −μ (∇ × H) ∂t ∂t It is a routine calculation to verify that this is the same as ∇ × (∇ × E) = − ∇(∇ · E) − (∇ · ∇)E = −μ ∂ (∇ × H) ∂t Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it October 14, 2010 14:53 THM/NEIL Page-412 27410_12_ch12_p367-424 12.9 Stokes’s Theorem 413 In this, ∇ ·∇ = ∂2 ∂2 ∂2 + + 2 ∂x ∂y ∂z Since ∂E , ∂t ∇ ×H=σE+ we have finally ∇(∇ · E) − (∇ · ∇)E = −μ ∂ ∂t σE+ ∂E ∂t In practice, it is often the case that Q = In this event, Q = ∇ · D = ∇ · (cE) = ∇ · E = 0, hence, ∇ · E = We can then further conclude that ∂ 2E ∂E +μ ∂t ∂t This is Maxwell’s equation for the electric intensity field By a similar analysis we obtain Maxwell’s equation for the magnetic intensity field: (∇ · ∇)E = μσ ∂ 2H ∂H +μ ∂t ∂t In the case of a perfect dielectric, σ = 0, and Maxwell’s equations become (∇ · ∇)H = μσ ∂ (H ) ∂ (E) and (∇ · ∇)H = μ ∂t ∂t If, instead of σ = 0, we have = 0, then we have (∇ · ∇)E = μ ∂E ∂H and (∇ · ∇)H = μσ ∂t ∂t These are vector forms of the three-dimensional heat equation (∇ · ∇)E = μσ PROBLEMS SECTION 12.9 In each of Problems through 5, use Stokes’s theorem (∇ × F) · n dσ , whichever to evaluate C F · dR or appears easier F = yx i − x y j + z k with z = 4, z ≥ the hemisphere x + y + F = x yi + yzj + x zk with for x + y ≤ the paraboloid z = x + y 2 2 F = zi + xj + yk with ≤ z ≤ the cone z = x + y for F = z i + x j + y k with the part of the paraboloid z = − x − y above the x, y - plane F = x yi + yzj + x yk with the part of the plane 2x + 4y + z = in the first octant Calculate the circulation of F = (x − y)i + x yj + ax zk counterclockwise about the circle x + y = Here a is any positive number Hint: Use Stokes’s theorem with any smooth surface having the circle as boundary Use Stokes’s theorem to evaluate C F · T ds, where C is the boundary of the part of the plane x + 4y + z = 12 lying in the first octant, and F = (x − z)i + (y − x)j + (z − y)k Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it October 14, 2010 14:53 THM/NEIL Page-413 27410_12_ch12_p367-424 CHAPTER 12 414 12.10 Vector Integral Calculus Curvilinear Coordinates Thus far, we have done vector algebra and calculus in rectangular coordinates For some settings, other coordinate systems may be more convenient Spherical coordinates are natural when dealing with spherical surfaces, cylindrical coordinates for cylinders, and sometimes we invent systems to deal with other settings we may encounter Begin with the usual rectangular coordinate system with axes labeled x, y and z Suppose we have some other coordinate system with coordinates labeled q1 , q2 and q3 We assume that the two systems are related by equations x = x(q1 , q2 , q3 ), y = y(q1 , q2 , q3 ), z = z(q1 , q2 , q3 ) (12.11) We also assume that these equations are invertible and can be solved for q1 = q2 (x, y, z), q2 = q2 (x, y, z), q3 = q3 (x, y, z) In this way we can convert the coordinates of points back and forth from one system to the other Finally, we assume that each point in 3-space has exactly one set of coordinates (q1 , q2 , q3 ), as it does in rectangular coordinates We call (q1 , q2 , q3 ) a system of curvilinear coordinates EXAMPLE 12.28 Cylindrical Coordinates As shown in Figure 12.27, a point P having rectangular coordinates (x, y, z) can be specified uniquely by a triple (r, θ, z), where (r, θ) are polar coordinates of the point (x, y) in the plane, and z is the same in both rectangular and cylindrical coordinates (the distance from the x, y-plane to the point) These coordinate systems are related by x = r cos(θ ), y = r sin(θ ), z = z with ≤ θ < 2π, r ≥ and z any real number With some care in using the inverse function tangent function, these equations can be inverted to write y , z = z r = x + y , θ = arctan x z P z θ y r x Cylindrical coordinates of Example 12.28 FIGURE 12.27 Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it October 14, 2010 14:53 THM/NEIL Page-414 27410_12_ch12_p367-424 12.10 Curvilinear Coordinates 415 z φ ρ P y θ x FIGURE 12.28 Spherical coordinates of Example 12.29 EXAMPLE 12.29 Spherical Coordinates Any point P having rectangular coordinates (x, y, z) also has unique spherical coordinates (ρ, θ, ϕ) Here ρ is the distance from the origin to P, θ is the angle of rotation from the origin to the line from (0, 0) to (x, y) in the x,y-plane, and ϕ is the angle of declination from the positive z-axis to the line from the origin to P These are indicated in Figure 12.28 Thus, ρ ≥ 0, ≤ θ < 2π and ≤ ϕ ≤ π Rectangular and spherical coordinates are related by x = ρ cos(θ ) sin(ϕ), y = ρ sin(θ ) sin(ϕ), z = ρ cos(ϕ) Again with care in using the inverse trigonometric functions, these equations can be inverted to read ρ= x + y + z , θ = arcsin ϕ = arccos y x + y2 + z2 y x + y2 + z2 The coordinate systems of these examples may appear quite dissimilar, but they share a common feature if we adopt a particular point of view Let P0 : (x0 , y0 , z ) be a point in rectangular coordinates Observe that P0 is the point of intersection of the planes x = x0 , y = y0 , and z = z , which are called coordinate surfaces for rectangular coordinates Now suppose P0 has cylindrical coordinates (r0 , θ0 , z ) Look at the corresponding coordinate surfaces for these coordinates In 3-space, the surface r = r0 is a cylinder of radius r0 about the origin The surface θ = θ0 is a half-plane with edge on the z-axis and making an angle θ0 with the positive x-axis And the surface z = z is the same as in rectangular coordinates, a plane in 3-space parallel to the x, y-plane The point P0 : (r0 , θ0 , z ) is the intersection of these three cylindrical coordinate surfaces Spherical coordinates can be viewed in the same way Suppose P0 has spherical coordinates (ρ0 , θ0 , ϕ0 ) The coordinate surface ρ = ρ0 is a sphere of radius ρ0 about the origin The surface θ = θ0 is a half-plane with one edge along the z-axis, as in cylindrical coordinates And the surface ϕ = ϕ0 is an infinite cone with vertex at the origin and making an angle ϕ with the z-axis These surfaces intersect at P0 (Figure 12.29) Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it October 14, 2010 14:53 THM/NEIL Page-415 27410_12_ch12_p367-424 416 CHAPTER 12 Vector Integral Calculus z ∇q3(P0) Half-plane θ = θ0 Cone φ = φ q2 = k2 ρ0 q1 = k1 y θ0 ∇q1(P0) q3 = k3 ∇q2(P0) x Intersection of surfaces in spherical coordinates FIGURE 12.29 FIGURE 12.30 coordinate Coordinate surfaces in curvilinear coordinates In general curvilinear coordinates, which need not be any of these three systems, we similarly specify a point ((q1 )0 , (q2 )0 , (q3 )0 ) as the intersection of the three coordinate surfaces q1 = (q1 )0 , q2 = (q2 )0 and q3 = (q3 )0 (Figure 12.30) In rectangular coordinates, the coordinate surfaces are planes x = x0 , y = y0 , z = z , which are mutually orthogonal Similarly, in cylindrical and spherical coordinates, the coordinate surfaces are mutually orthogonal, in the sense that their normal vectors are mutually orthogonal at any point of intersection Because of this, we refer to these coordinate systems as orthogonal curvilinear coordinates EXAMPLE 12.30 We will verify that cylindrical coordinates are orthogonal curvilinear coordinates In terms of rectangular coordinates, cylindrical coordinates are given by x + y2, y θ = arctan x z = z, r= except at the origin, which is called a singular point of these coordinates Suppose P0 is the point of intersection of the cylinder r = r0 , the half-plane θ = θ0 and the half-plane z = z To verify that these surfaces are mutually orthogonal, we will show that their normal vectors are mutually orthogonal Compute these normal vectors using the gradient in rectangular coordinates: ∇r = ∇θ = x + y2 (xi + yj), (−yi + xj), ∇z = k x + y2 Now it is routine to verify that ∇r · ∇θ = ∇r · ∇z = ∇θ · ∇z = Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it October 14, 2010 14:53 THM/NEIL Page-416 27410_12_ch12_p367-424 12.10 Curvilinear Coordinates 417 A similar, but more complicated, calculation shows that spherical coordinates are orthogonal curvilinear coordinates In rectangular coordinates, the differential element ds of arc length is given by (ds)2 = (d x)2 + (dy)2 + (dz)2 (12.12) We assume that this differential element of arc length is given in terms of the orthogonal curvilinear coordinates q1 , q2 , q3 by the quadratic form 3 (ds)2 = h i2j dqi dq j i=1 j=1 The numbers h i j are called scale factors for the curvilinear coordinate system We want to determine these scale factors so that we can compute such quantities as arc length, area, volume, gradient, divergence and curl in curvilinear coordinates Begin by differentiating equations (12.11): ∂x ∂x ∂x dq1 + dq2 + dq3 , ∂q1 ∂q2 ∂q3 ∂y ∂y ∂y dq1 + dq2 + dq3 , dy = ∂q1 ∂q2 ∂q3 ∂z ∂z ∂z dq1 + dq2 + dq3 dz = ∂q1 ∂q2 ∂q3 dx = Substitute these into equation (12.12) This is a long calculation, but after collecting the coefficients of (dq1 )2 , (dq2 )2 and (dq3 )2 (the terms in the double sum with i = j), and leaving the cross product terms involving dqi dq j with i = j within the summation notation, we obtain ∂x ∂q1 (ds)2 = ∂y ∂q1 + ∂x ∂q2 + ∂x ∂q3 + ∂z ∂q1 + ∂y ∂q2 + ∂y ∂q3 + (dq1 )2 ∂z ∂q2 + ∂z ∂q3 + (dq2 )2 (dq3 )2 3 + h i j dqi dq j i=1 j=1, j =i 3 = (h 11 )2 (dq1 )2 + (h 22 )2 (dq2 )2 + (h 33 )2 (dq3 )2 + h i j dqi dq j i=1 j=1, j =i 3 i=1 j=1 = h i j dqi dq j Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it October 14, 2010 14:53 THM/NEIL Page-417 27410_12_ch12_p367-424 418 CHAPTER 12 Vector Integral Calculus In this equation, equate coefficients of (dqi )2 for i = 1, 2, to obtain ∂x ∂q1 h 211 = ∂x ∂q2 h 222 = ∂x ∂q3 h 233 = ∂y ∂q1 + ∂y ∂q2 + ∂y ∂q3 + ∂z ∂q1 + ∂z ∂q2 + ∂z ∂q3 + , , We left the cross product terms within the summation because all such terms are zero for orthogonal coordinates For example, h 212 = ∂x ∂x ∂y ∂y ∂z ∂z + + ∂q1 ∂q2 ∂q1 ∂q2 ∂q1 ∂q2 = ∇x(q1 , q2 , q3 ) · ∇ y(q1 , q2 , q3 ) = by virtue of the orthogonality of the curvilinear coordinates Similarly, each h i j = for i = j To simplify the notation, write h ii = h i for i = 1, 2, Finally we can write (ds)2 = (h dq1 )2 + (h dq2 )2 + (h dq3 )2 (12.13) with h , h , h given in terms of partial derivatives of x, y, and z in terms of q1 , q2 and q3 EXAMPLE 12.31 We will put these ideas into the context of cylindrical coordinates Now q1 = r , q2 = θ , and q3 = z Compute ∂x ∂r hr = ∂x ∂θ hθ = ∂x ∂z hz = ∂y ∂r + ∂y ∂θ + ∂y ∂z + ∂z ∂r + ∂z ∂θ + ∂z ∂z + = 1, = r, = In the plane, cylindrical coordinates are polar coordinates and the differential element d x d y of area in rectangular coordinates corresponds to d x d y = ds1 ds2 = h r h θ dr dθ = r dr dθ This accounts for the change of variables formula for transforming a double integral from rectangular to polar coordinates: f (x, y) d x d y = D f (r cos(θ ), r sin(θ ))r dr dθ D We can also recognize r as the Jacobian ∂(x, y) cos(θ ) −r sin(θ ) = = r sin(θ ) r cos(θ ) ∂(r, θ) In 3-space, d x d y dz = h r h θ h z dr dθ dz = r dr dθ dz Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it October 14, 2010 14:53 THM/NEIL Page-418 27410_12_ch12_p367-424 12.10 Curvilinear Coordinates 419 This is the reason for the formula for converting a triple integral from rectangular to cylindrical coordinates: f (x, y, z) d x d y dz = M f (r cos(θ ), r sin(θ ), z)r dr dθ dz M Again, in 3-space we can recognize the factor of r as the Jacobian cos(θ ) −r sin(θ ) ∂(x, y, z) = sin(θ ) r cos(θ ) = r ∂(r, θ, z) 0 EXAMPLE 12.32 In spherical coordinates, q1 = ρ, q2 = θ , and q3 = ϕ From Example 12.29 we know x, y and z in terms of ρ, θ and ϕ, so compute the partial derivatives to obtain ∂x ∂ρ hρ = ∂x ∂θ hθ = ∂x ∂ϕ hϕ = ∂y ∂ρ + ∂y ∂θ + ∂y ∂ϕ + ∂z ∂ρ + ∂z ∂θ + + ∂z ∂ϕ = 1, = ρ sin(θ ), = ρ Therefore, in spherical coordinates, the differential element of arc length, squared, is (ds)2 = (dρ)2 + ρ sin2 (ϕ)(dθ )2 + ρ (dϕ)2 In general, if dsi is the differential element of arc length along the qi axis (in the qi direction), then dsi = h i dqi Therefore the differential elements of area are dsi ds j = h i h j dqi dq j The differential element of volume is ds1 ds2 ds3 = h h h dq1 dq2 dq3 The formula for a differential volume element in spherical coordinates is dsρ dsθ dsϕ = ρ sin(ϕ) dρ dθ dϕ This should look familiar In calculus, we are told that when we convert a triple integral from rectangular to spherical coordinates we obtain f (x, y, z) d x d y dz = M F(ρ, θ, ϕ)ρ sin(ϕ) dρ dθ dϕ, Mρ,θ,ϕ in which F(ρ, θ, ϕ) is obtained by substituting for x, y, z in terms of spherical coordinates in f (x, y, z), and Mρ,θ,ϕ is the region M defined in spherical coordinates Notice that the ρ sin(ϕ) has shown up in the differential element of volume That is, in terms of differentials, d x d y dz = ρ sin(ϕ) dρ dθ dϕ Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it October 14, 2010 14:53 THM/NEIL Page-419 27410_12_ch12_p367-424 420 CHAPTER 12 Vector Integral Calculus We can also recognize ρ sin(ϕ) as the Jacobian ∂(x, y, z) ∂(ρ, θ, ϕ) seen in the general expression for transformation of triple integrals Now let ui be a unit vector in the direction of increasing qi at the point (x(q1 , q2 , q3 ), y(q1 , q2 , q3 ), z(q1 , q2 , q3 )) In cylindrical coordinates, these unit vectors can be written in terms of the standard i, j, and k as ur = cos(θ )i + sin(θ )j, uθ = − sin(θ )i + cos(θ )j, uz = k In spherical coordinates, uρ = cos(θ ) sin(ϕ)i + sin(θ ) sin(ϕ)j + cos(ϕ)k, uθ = − sin(θ )i + cos(θ )j, uϕ = cos(θ ) cos(ϕ)i + sin(θ ) cos(ϕ)j − sin(ϕ)k Unlike rectangular coordinates, where the standard unit vectors are constant, with orthogonal curvilinear coordinates, the vectors u1 , u2 , u3 are generally functions of the point A vector field in curvilinear coordinates has the form F(q1 , q2 , q3 ) = F1 (q1 , q2 , q3 )u1 + F2 (q1 , q2 , q3 )u2 + F3 (q1 , q2 , q3 )u3 We want to write expressions for the gradient, Laplacian, divergence, and curl operations in curvilinear coordinates Gradient Let ψ(q1 , q2 , q3 ) be a scalar-valued function At any point, we want ∇ψ to be normal to the level surface ψ = constant passing through that point, and we want this gradient to have magnitude equal to the greatest rate of change of ψ from that point Thus, the component of ∇ψ normal to q1 = constant must be ∂ψ/∂s1 , or ∂ψ h ∂q1 Arguing similarly for the other components, we have ∂ψ ∂ψ ∂ψ ∇ψ(q1 , q2 , q3 ) = u1 + u2 + u3 h ∂q1 h ∂q2 h ∂q3 Divergence We will use the flux interpretation of divergence to obtain an expression for the divergence of a vector field in curvilinear coordinates First write F = F1 u1 + F2 u2 + F3 u3 Referring to Figure 12.31, the flux across the face abcd is approximately F(q1 + ds1 , q2 , q3 ) · h (q1 + ds1 , q2 , q3 )h (q1 + ds1 , q2 , q3 ) dq2 dq3 Across face e f gk the flux is F(q1 , q2 , q3 ) · u1 h (q1 , q2 , q3 )h (q1 , q2 , q3 ) dq2 dqq Across both of these faces the flux is approximately ∂ (F1 h h ) dq1 dq2 dq3 ∂q1 Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it October 14, 2010 14:53 THM/NEIL Page-420 27410_12_ch12_p367-424 12.10 Curvilinear Coordinates 421 u3 c k g ds3 = h3dq3 b d a c d e u2 ds2 = h2dq2 f e = (q1, q2, q3) a a = (q1 + ds1, q2, q3) b u1 b = (q1 + ds1, q2 + ds2, q3) Calculating the curl in curvilinear coordinates FIGURE 12.32 Calculating the divergence in curvilinear coordinates FIGURE 12.31 Similarly, the fluxes across the other two pairs of opposite faces are ∂ ∂ (F2 h h ) dq1 dq2 dq3 and (F3 h h ) dq1 dq2 dq3 ∂q2 ∂q3 We obtain the divergence, or flux per unit volume, at a point by adding these three expressions for the flux across pairs of opposite sides, and dividing by the volume h h h dq1 dq2 dq3 to obtain ∇ · F(q1 , q2 , q3 ) = h1h2h3 ∂ ∂ ∂ (F1 h h ) + (F2 h h ) + (F3 h h ) ∂q1 ∂q2 ∂q3 Laplacian Knowing the divergence, we immediately have the Laplacian, since ∇2 f = ∇ · ∇ f for a scalar field f Then ∇ f (q1 , q2 , q3 ) = ∇ · ∇ f (q1 , q2 , q3 ) = ∂ h h h ∂q1 h2h3 ∂ f h ∂q1 + ∂ ∂q2 h1h3 ∂ f h ∂q2 + ∂ ∂q3 h1h2 ∂ f h ∂q3 Curl For the curl in curvilinear coordinates, we will use the interpretation of (∇ × F) · n as the rotation or swirl of a fluid with velocity field F about a point in a plane having unit normal n At P, the component of ∇ × F in the direction u1 is lim A→0 A F, C where C may be taken as a rectangle about P in the u2 − u3 plane at P (Figure 12.32) Compute the integral over each side of this rectangle On side a, F ≈ F2 (q1 , q2 , q3 )h (q1 , q2 , q3 ) dq2 , a Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it October 14, 2010 14:53 THM/NEIL Page-421 27410_12_ch12_p367-424 422 CHAPTER 12 Vector Integral Calculus since u2 is tangent to a On side c, c F is approximately −F2 (q1 , q2 , q3 )h (q1 , q2 , q3 + dq3 ) dq3 The net contribution from sides a and c is approximately − ∂ (F2 h ) dq2 dq3 ∂q3 Similarly, from sides b and d, the net contribution is approximately ∂ (F3 h ) dq2 dq3 ∂q2 Then (∇ × F) · u1 = h h dq2 dq3 ∂ ∂ (F3 h ) − (F2 h ) dq2 dq3 ∂q2 ∂q3 Obtain the other components of ∇ × F in the same way We obtain ∇ ×F= ∂ ∂ (F3 h ) − (F2 h ) u1 ∂q2 ∂q3 h2h3 + h1h3 ∂ ∂ (F1 h ) − (F3 h ) u2 ∂q3 ∂q1 + h1h2 ∂ ∂ (F2 h ) − (F1 h ) u3 ∂q1 ∂q2 This can be written in a convenient determinant form: ∇ ×F= h u1 ∂/∂q1 h1h2h3 F h 1 h u2 ∂/∂q2 F2 h h u3 ∂/∂q3 F3 h We will apply these to spherical coordinates, recalling that h ρ = 1, h θ = ρ sin(ϕ), h ϕ = ρ If F = Fρ uρ + Fθ uθ + Fϕ uϕ , then divergence is given by ∇ ·F= 1 ∂ ∂ ∂ (ρ Fρ ) + (Fθ ) + (Fϕ sin(ϕ)) ρ ∂ρ ρ sin(ϕ) ∂θ ρ sin(ϕ) ∂ϕ The curl is obtained as uρ ∇ × F = ∂/∂ρ Fρ ρ sin(ϕ)uθ ∂/∂θ ρ sin(ϕ)Fθ ρuϕ ∂/∂ϕ ρ Fϕ The gradient of a scalar function f (ρ, θ, ϕ) is ∇f = 1 ∂f ∂f ∂f uρ + uθ + uϕ ∂ρ ρ sin(ϕ) ∂θ ρ ∂ϕ From this, we have the Laplacian ∇2 f = ∂f ∂ ρ2 ρ ∂ρ ∂ρ + 1 ∂ ∂f ∂2 f + sin(ϕ) 2 ∂θ ρ sin(ϕ) ∂ϕ ∂ϕ ρ sin (ϕ) Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it October 14, 2010 14:53 THM/NEIL Page-422 27410_12_ch12_p367-424 12.10 Curvilinear Coordinates 423 The Laplacian in various coordinate systems is often encountered in connection with diffusion problems, wave motion and potential theory SECTION 12.10 PROBLEMS (a) Sketch the coordinate surfaces u = constant, v = constant, and z = constant Are these coordinates orthogonal? Compute the scale factors for cylindrical coordinates Use them to compute ∇ · F and ∇ × F if F(r, θ, z) is a vector field in cylindrical coordinates If g(r, θ, z) is a scalar field, compute ∇g and ∇ g (b) Determine the scale factors h u , h v , h z Elliptic cylindrical coordinates are defined by (c) Determine ∇ f (u, v, z) in this system (d) Determine ∇ · F(u, v, z) and ∇ × F(u, v, z) in this system x = a cosh(u) cos(v), y = a sinh(u) sin(v), z = z, where u ≥ 0, ≤ v < 2π and z can be any real number (a) Sketch the coordinate surfaces u = constant, v = constant, and z = constant (e) Determine ∇ f (u, v, z) Parabolic cylindrical coordinates are defined by x = uv, y = (u − v ), z = z, with u ≥ and v and z any real numbers (b) Determine the scale factors h u , h v , h z (c) Determine ∇ f (u, v, z) in this system (d) Determine ∇ · F(u, v, z) and ∇ × F(u, v, z) in this system (a) Sketch the coordinate surfaces u = constant, v = constant, and z = constant (e) Determine ∇ f (u, v, z) (b) Determine the scale factors h u , h v , h z Bipolar coordinates are defined by (c) Determine ∇ f (u, v, z) in this system (d) Determine ∇ · F(u, v, z) and ∇ × F(u, v, z) in this system a sin(u) a sinh(v) ,y= , z = z, x= cosh(v) − cos(u) cosh(v) − cos(u) (e) Determine ∇ f (u, v, z) with u and z any real numbers and ≤ v < 2π Copyright 2010 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it October 14, 2010 14:53 THM/NEIL Page-423 27410_12_ch12_p367-424 This page intentionally left blank This page intentionally left blank ... October 15 , 2 010 17 :43 THM/NEIL Page-vii 27 410 _00_fm_pi-xiv viii Contents CHAPTER 12 Vector Integral Calculus 367 12 .1 12.2 12 .3 12 .4 12 .5 12 .6 12 .7 12 .8 12 .9 12 .10 PART Line Integrals 367 12 .1. 1 Line... October 14 , 2 010 14 :9 THM/NEIL Page-4 27 410 _ 01_ ch 01_ p 01- 42 1. 1 Terminology and Separable Equations 1. 5 0.5 –0.6 –0.4 –0.2 x 0.2 0.4 0.6 –0.5 1 FIGURE 1. 1 Some integral curves from Example 1. 1 1 ,... CHAPTER 17 The Heat Equation 586 611 17 .1 Initial and Boundary Conditions 611 17 .2 The Heat Equation on [0, L] 612 17 .2 .1 Ends Kept at Temperature Zero 612 17 .2.2 Insulated Ends 614 17 .2.3 Radiating