Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

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276 Multiple and Line Integrals [Ch. 7 Example 3. (4x + 2y) dx+ (2x 6y) dy = dU. Find U. Solution. Let A- O = O, y = Q. Then or U(x, y) = where C U (0, 0) is an arbitrary constant. Y Uo --4- XQ 2 + 2y) dx + C= 3// Fig. 102 /I X -i-c, 4. Green's formula for a plane. If C is the boundary of a region S and the functions P (x, y) and Q (x, y) are continuous together with their firstorder partial derivatives in the closed region S-j-C, then Green's formula holds: (S) here t'ê circulation about the contour C is chosen so that the region S should remain to the left. 5. Applications of line integrals. 1 ) An area bounded by the closed contour C is = (p y dx= (p xdy (the direction of circulation of the contour is chosen counterclockwise). The fo lowing formula for area is more convenient for application: 2) The work of a force, having projections X = X(x, y, 2), Y = Y (x, y, z), Z (x, y t z) (or, accordingly, the work of a force field), along a path C is
Sec. 9]_Line Integrals__277 expressed by the integral A= ^Xdx + Ydy + Zdz. c If the force has a potential, i.e., if there exists a function U = U (x, y, z) (a potential function or a force function) such that dV dU dU - = A , TT j , -7 L , dx dy dz then the work, irrespective of the shape of the path C, is equal to A = (*. //2, *j) (*J, '/J. Z.) Xdx + Ydy + Zdz^ dU = U(x 2t y tt J (*|. {/,, Zi) (*i, J l/ lt 2-J z 2)-t/ (x lt y, z } ), where (v l5 f/ 1? Zj) is the initial and (x 2 , 0). s x\ + \ y\ = a 2294. \ r ,. --^= , where C is a segment of the straight line 2 A: |- if \- \ c K connecting the points 0(0, 0) and A (I, 2). 2295. ^xyds, where C is a quarter of the ellipse + = l î fi> c lying in the first quadrant. 2296. ifds, where C is the first arc of the cycloid x = a (t sin /)> c a (1 cos /). 2 2297. }x + y 2 ds, where C is an c circle x arc of the involute of the ---- a (cos / (-/sin/), y = a(smt tcost) I0^/=^2ji]. 2298. ^ (x 2 -\- y 2 2 ) c d$, where C is an arc of the logarithmic spi- ral râe m v(m>Q) from the point A (0, a) to the point 0( oo, 0). 2299. JU + y) rfs where C is the right-hand loop c niscate r 2 = a 2 cos2
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TEXT FLY WITHIN THE BOOK ONLY
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OUP-43 30-1.71 5,000 OSMANIA UNIVER
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T. C. BapaneHKoe. C. M. Koean, r Jl
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TO THE READER MIR Publishers would
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Contents Sec. 6. Integrating Certai
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8 Chapter Contents Sec. 13. Linear
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Sec. 1. Functions Chapter I INTRODU
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Sec 1] Functions 13 8. Find the rat
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Sec. 1]_Functions_15 34. Prove that
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Sec. 2] Graphs of Elementary functi
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Sec. 2] Graphs of Elementary functi
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Sec. 2] Graphs of Elementary Functi
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Sec. 3] Limits 23 For the existence
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Sec. 3] 179. Hm (Vn + 1 \f~n). n -+
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Sec. #] Limits 27 209. lim 210. lim
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^Sec 3] Limits Therefore, Example 9
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Sec. 31 Limits 31 269. a) 270. lim-
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Sec. 4] Infinitely Small and Large
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Sec. 4] Infinitely Small and Large
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Sec. 5] Continuity of Functions 37
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Sec. 5]_Continuity of Functions_39
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Sec. 5] Continuity of Functions 334
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Sec. 1] Calculating Derivatives Dir
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Sec. 1] Calculating Derivatives Dir
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Sec. 2]__Tabular Differentiation_47
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Sec. 2]_Tabular Differentiation_49
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Sec. 2] Tabular Differentiation_ 51
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Sec. 2] Tabular Differentiation 504
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Sec. 2] Tabular Differentiation 55
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Sec. 3] The Derivatives of Function
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Sec. 3] The Derivatives of Function
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Sec 4] Geometrical and Mechanical A
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Sec. 4\ Geometrical and Mechanical
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Sec. 4] Geometrical and Mechanical
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Sec. 5]_Derivatives of Higher Order
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Sec. 5]_Derivatives of Higher Order
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Sec. 6] Differentials of First and
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Sec. 6}_Differentials of First and
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Sec. 7]_Mean-Value Theorems_75 745.
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Sec. 8] Taylor's Formula 77 765. a)
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Sec. 9]_ V Hospital-Bernoulli Rule
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Sec. 9] L' Hospital-Bernoulli Rule
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Chapter III THE EXTREMA OF A FUNCTI
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Sec. 1] The Extrema of a Function o
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Sec. 2]_The Direction of Concavity.
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Sec. 3]_Asymptotes_93 Example 2. Fi
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Sec. 3]_Asymptotes_95 Solution. Sin
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Sec 4] Graphing Functions by Charac
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CM CM -h CO O non ' > C " -~ a - >
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Sec. />] Differential of an Arc. Cu
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Sec 5] Differential of an Arc. Curv
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Sec. 5]_Differential of an Arc. Cur
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Sec. 1. Direct Integration 1. Basic
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Sec 1] Direct Integration_109 1048*
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Sec. 2] Integration by Substitution
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Sec. 2] Integration by Substitution
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Sec. 3\_Integration by Parts_H7 Exa
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Sec 4] Standard Integrals Containin
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Sec. 5] Integration of Rational Fun
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Sec. 5] In t egration of Rational F
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Sec. 6] Integrating Certain Irratio
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Sec. 6] Integrating Certain Irratio
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Sec. 7]_Integrating Trigonometric F
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Sec. 7]_Integrating Trigonometric F
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Sec. 8] Integration of Hyperbolic F
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Sec. 11] Using Reduction Formulas 1
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Sec. 12] Miscellaneous Examples on
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Sec. 1] The Definite Integral as th
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Sec. 2]_Evaluating Definite Integra
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Sec. 3]_Improper Integrals_143 Sec.
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Sec. 3] Improper Integrals 145 Solu
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Sec. 4] Change of Variable in a Def
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Sec. 5] Integration by Parts 149 Bu
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Sec. 6] Mean-Value Theorem 151 In p
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Sec. 7] The Areas of Plane Figures
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Sec. 7] The Areas of Plane Figures
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Sec. 7\_The Areas of Plane Figures_
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Sec. 8] The Arc Length of a Curve 1
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Sec, 9] Volumes of Solids 161 1673.
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Sec. 9] Volumes of Solids 163' Solu
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Sec. 9]_Volumes of Solids_165 1693.
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Sec. 10] The Area of a Surface of R
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Sec. II] Moments. Centres of Gravit
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Sec 11] Moments. Centres of Gravity
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&?c. 12] Applying Definite Integral
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Sec. 12] Applying Definite Integral
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Sec. 1) Basic Notions [SI Putting *
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Sec. 1]_Basic Notions_183 1790*. Le
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Sec. 3] Partial Derivatives 185 180
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Sec. 4]_Total Differential of a Fun
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Sec. 4}_Total Differential of a Fun
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Sec. 5]_Differentiation of Composit
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Sec. 6]_Derivative in a Given Direc
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Sec. 6] Derivative in a Given Direc
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Sec. 7]_Higher-Order Derivatives an
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Sec. 7] Higher-Order Derivatives an
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Sec. 7]_Higher-Order Derivatives an
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Sec. ti\_Integration of Total Diffe
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Sec. 9]_Differentiation of Implicit
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Sec. 9] Differentiation of Implicit
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Sec. 9]_Differentiation of Implicit
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Sec. 10]_Change of Variables_2H 196
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Sec. 10]_Change of Variables_213 or
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Sec. 10] Change of Variables 215 an
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Sec. 11]_The Tangent Plane and the
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Sec. 11]_The Tangent Plane and the
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Sec. 12] Taylor's Formula for a Fun
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Sec. 13]_The Extremum of a Function
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Page 233 and 234: Sec. 14] Finding the Greatest and S
Page 235 and 236: Sec. 14] Finding the Greatest and S
Page 237 and 238: Sec. 15] Singular Points of Plane C
Page 239 and 240: Sec. 16] Envelope 233 Solution. The
Page 241 and 242: Sec. 18] The Vector Function of a S
Page 243 and 244: Sec. 18]_The Vector Function of a S
Page 245 and 246: Sec. 19] The Natural Trihedron of a
Page 247 and 248: Sec. 19] The Natural Trihedron of a
Page 249 and 250: Sec. 20] Curvature and Torsion of a
Page 251 and 252: Sec. 20]_Curvature and Torsion of a
Page 253 and 254: Sec. 1] The Double Integral in Rect
Page 259 and 260: Sec. 2]_Change of Variables in a Do
Page 261 and 262: Sec. 2]_Change of Variables in a Do
Page 263 and 264: Sec 3]_Computing Areas_257 2177. Co
Page 265 and 266: Sec. 5]_Computing th Area* of Surfa
Page 267 and 268: Sec. 6]_Applications of the Double
Page 269 and 270: Sec. 7] Triple Integrals 2o3 Evalua
Page 271 and 272: Sec 7\ Triple Integrals 265 We ther
Page 273 and 274: gee. 7]_Triple Integrals_267 2251.
Page 275 and 276: Sec. 8] Improper Integrals Dependen
Page 277 and 278: Sec. 8]_Improper Integrals Dependen
Page 279 and 280: See. 9] Line Integrals 273 Test for
Page 281: Sec. 9]____Line Inlegtah____275 whe
Page 285 and 286: Sec. 9] Lim Integrals 279 (/-axis;
Page 287 and 288: Sec 9] Line Integrals 281 Evaluate
Page 289 and 290: Sec. 9] Line Integrals 2S& D. Appli
Page 291 and 292: Sec. 10] Surface Integrals 285 When
Page 293 and 294: ec. 11]_The Ostrogradsky-Gauss Form
Page 295 and 296: Sec. 12] Fundamentals of Field Theo
Page 297 and 298: Sec. 12] Fundamentals of Field Theo
Page 299 and 300: Sec. 1. Number Series Chapter VIII
Page 301 and 302: Sec. 1]_Number Series_295 c) D'Alem
Page 303 and 304: Sec /\_Number Series_297 Note. For
Page 305 and 306: 2418. + + 0419 J___1_ . 4- 4- 4- '
Page 307 and 308: Sec. 1] Number Series 2464. cos-2-)
Page 309 and 310: Sec. 1]_Number Series_303 - V r(2-Q
Page 311 and 312: Sec. 2]_Functional Series_305 In th
Page 313 and 314: Sec. 2]_Functional Series_307 [the
Page 315 and 316: Sec. 2] Functional Series 309 Deter
Page 317 and 318: Sec. 3]_Taylor's Series_311 v8 s r
Page 319 and 320: Sec. 3] Taylor's Series 313 Since a
Page 321 and 322: Sec 3} Taylor's Series 315 2615. ln
Page 323 and 324: Sec. 3] Taylor's Series 317 2651. F
Page 325 and 326: Sec. 4] Fourier Series 319 If a fun
Page 327 and 328: Sec. 4\__Fourier Series _321 2690.
Page 329 and 330: Sec 1 1 Verifying Solutions 323 It
Page 331 and 332: Sec. 21 First-Order Differential Eq
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Sec. 3] Differential Equations with
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Sec. 3\ Differential Equations with
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Sec. 4]_First-Order Homogeneous Dif
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Sec. 5] Bernoulli's Equation 333 Co
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Sec. 6] Exact Differential Equation
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Sec. 7] First-Order Differential Eq
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Sec 8]_The Lagrange and Clairaut Eq
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Sec. 9] Miscellaneous Exercises on
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Sec. 9] Miscellaneous Exercises on
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Sec. 10] Higher-Order Differential
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Sec. 10]__Higher-Order Differential
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Sec. 11]_Linear Differential Equati
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Sec. 12] Linear Differential Equati
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Sec. 14]_Euler's Equations__357 Fin
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Sec. 15]_Systems of Differential Eq
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.Sec. 16} Integration of Differenti
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Sec. 17] Problems on Fourier's Meth
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Sec. 17]_Problems on Fourier's Meth
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Chapter X APPROXIMATE CALCULATIONS
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Sec. 1] Operations on Approximate N
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Sec. /]_Operations on Approximate N
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Sec, 2] Interpolation of Functions
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Sec. 2] Interpolation of Functions
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Sec. 3]_Computing the Real Roots of
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Sec. 4] Numerical Integration of Fu
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Sec. 5] Numerical Integration of Or
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Table 3. Basic Table for Calculatin
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394 Approximate Calculations \Ch 10
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ANSWERS Chapter I 1. Solution. Sinc
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398 Answers Form a table of values:
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400 Answer* 271. Solution. If x & k
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402 Answers b) /'(!)= lim 1^15^= li
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404_Answers 494. , . 495. - - . 1 2
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-406 Answers 640. Hint. The equatio
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408_Answers_ 755. e xcosa sin(*sina
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410_Answers_4 where R is the radius
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412 Answers 1/~JT f Q \ J/max = TT=
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414 Answers 976. j/mi n =1.32 when
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416 Answers 1037. j/ 1040. 3x* 13 J
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418 1 1137. 4- 6 3a . In I ocot3x|
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420 Answers 4^8 Problem 1218*. 1221
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422 Answers X arctan(* + 2). 1294.
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424 Answers sin 5* , 1374. -!- 1377
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426 Answers 3 1442. In l-x2 ' '" 2
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428_Answers_ the function /(x) = --
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430 Answers 1651. 3na 2 . 1652. n(b
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432 Answers inertia /=Y 2jttf 1 r *
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434 Answers along the large lower s
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436 Answers cos^. ,810. * ox x* x d
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438_Answers 1905. 1914. u(*. !/) =
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440 Answers 1983. 3* + 4j/-H2z 169
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442 Answers is ~ |/2 + -~=, the alt
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444 Answers (binormal). 2103. bx *=
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444_Answers_x = \ ' > (principal no
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446 Answers 2 a a a 2139. ( dy(f(x,
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448_Answers_ r 4 **/* [ rj-cos*(p--
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450 Answers 2243. 2244. 1 i VT=T* ^
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452 Answers 2 t 2297. ) ~ll . |-[(I
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454_Answers__ TtD the flux, use the
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456__Answers_ and when x | ^ =- , t
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458 Answers
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460 Answers %-l n=o (n)=^=%. 2873.
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462 Answers Chapter IX 2704. Yes. 2
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464 Answers 2822. g-C*+; y=2*. 2824
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466 Answers x + C z . 2914. #=
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468_Answers_ A $007. 1) x = C cosco
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470 Answers 3081. x = C l e 3082. x
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472 Answers ,*. f (2"+ 1)JtJf dx. H
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474 Answers 3131. a) 0.211; 0.389;
Page 484 and 485:
476 Appendix III. Inverse Quantitie
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478 Appendix IV. Trigonometric Func
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480 Appendix VI. Some Curves (for R
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482 Appendix 11. Graphs //-=secx an
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486 Appendix 24. Strophoid, * u s^-
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Absolute error Absolute value 367 o
Page 496 and 497:
490 Index Differential equations of
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492 Index Infinite discontinuities
Page 500 and 501:
494 Index Operator Hamiltonian 288
Page 502:
496 Index Taylor's series Term 311,
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