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

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494 Index Operator Hamiltonian 288 Laplacian 289 Order of smallness 35 Orthagonal surfaces 219 Orthagonal trajectories 328 Osculating circle 103 Osculating plane 238 Ostrogradsky-Gauss formula 286-288 Ostrogradsky-Gauss theorem 291 Ostrogradsky method 123, 125 Parabola 17, 20, 104, 105, 480, 485 cubic 17, 105, 234 Niele's 18, 234, 481 safety 234 semicubical 18, 20, 234, 481 Parabolic formula 382 Parameters variation of 332, 349, 352 Parametric representation of a function 207 Partial derivative hirheg-order 197 "mixed" 197 second 197 Partial sum 293 Particular integral 322 Particular solution 339 Pascal's lima^on 158 Path of integration 273, 274, 280 Period of a function 14 Periodic function 14 Picard's method 384, 385 Plane normal 238 osculating 238 rectifying 238 tangent 217 Point bending 84 critical (of the second kind) 92 of discontinuity 37 double 230 extremal 84 of inflection 91 isolated 230 maximum 84 minimum 84 singular 230 stationary 196 of tangency 217 Points characteristic 96 critical 84 stationary 222, 225 Polar subnormal 61 Polar subtangent 61 Potential (of a field) 289 Potential vector field 289 Power series 305 Principal normal 238 Principle of equal effects 369 Runge 383, 386 of superposition of solutions 353 Probability curve 19, 484 Probability integral 144 Product of two convergent series 298 Progression geometric 293, 294 Proper rational fraction 121 Proportionate parts rule of 376 Quadratic interpolation 372 Quadratic trinomial 118, 119, 123 Quantity infinitely large 33 infinitely small 33 Radius of convergence 305 Radius of curvature 102, 243 Radius of second curvature 243 Radius of torsion 243 Rate of change of a function 43 mean 42 Ratio (of a geometric progression) 294 Real numbers 11 Rectangular hyperbola 480 Rectifying plane 238 Reduction formulas 130, 135, 150 Reduction method 123 Region of convergence 304 Region of integration 246-248 Relative error 367 Remainder 31 1 Remainder of a series 293, 304 Remainder term 311 Removable discontinuity 37 Right-hand derivative 44 Right horizontal asymptote 93 Right inclined asymptote 93 Rolle's theorem 75 Root multiplicities 121
Rose four-leafed 487 three-leafed 20, 487 Rotation (of a vector field) 288 Rule Leibniz 67, 269 1'Hospital-Bernoulli 78-82 of proportionate parts 376 Runge-Kutta method 385-387, 390 Runge principle 383, 386 Safety parabola 234 Scalar Scheme field 288 twelve-ordinate 393-395 Second curvature 243 Second derivative 66 Second deferential 198 Second-ordeP differential 72 Second partial derivative Segment of the normal 61 197 Segment of the polar normal 61 Segment of the polar tangent 61 Segment of a straight line 20 Segment of the tangent 61 Semicircle 20 Semicubical parabola Series 18, 20, 1M4, -181 absolutely convergent 296, with complex terms 297 297 conditionally (not convergent 296 convergent 293 Scries (cont) absolutely) Dirichlet 295, 296 divergent 293, 294 Fourier 318, 319 functional 304 harmonic 294, 296, 297 incomplete Fourier 318, 319 Maclaurin's 311, 313 number series 293 operations on 297 power 305 Taylor's 311, Serpentine 313 Newton's 18 Simpson's formula 382-384 Sine curve 481 Single-valued function 11 Singular integral 337 Singular point 230 Slope (of a tangent) 43 Smallest value 225, 227 Index 495 Solenoidal vector field 289 Solution (of an equation) general 323, 359 particular 339 Spiral 322 of Archimedes 20, 65, 66, 105, 487 hyperbolic 20, 105, 487 logarithmic 20, 21, Static moment 168 105, 106, 487 Stationary point 196, 222, 225 Stationary scalar or vector field Stokes' formula 285, 286, 289 288 Straight line 17, 20 Strophoid 157, Subnormal 61 232, 234, 486 polar 61 Substitutions hyperbolic 114, 116, 133 trigonometric 114, 115, 133 Subtangent 61 polar 61 Successive approximation method of 384, 385, 389 377, 385 Sufficient conditions (for an extremum) 222 Sum integral 138 partial 293 of of a series 293, 304 two convergent series 298 Superposition of solutions principle of 353 Surface integral of the first type 284 Surface integral of the second type 284 Surface integrals 284-286 Surfaces level 288 orthogonal 219 Table diagonal table 389 of standard integrals 107 Table interval 372 Tabular differentiation 46 Tacnode 230 Tangency point "of 217 Tangent 238 Tangent curve 481 Tangent plane 217 equation Tangents of 218 method of 377 Taylor's formula 77, 220
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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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Sec. 13] The Extremutn of a Functio
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Sec. 14] Finding the Greatest and S
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Sec. 14] Finding the Greatest and S
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Sec. 15] Singular Points of Plane C
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Sec. 16] Envelope 233 Solution. The
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Sec. 18] The Vector Function of a S
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Sec. 18]_The Vector Function of a S
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Sec. 19] The Natural Trihedron of a
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Sec. 19] The Natural Trihedron of a
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Sec. 20] Curvature and Torsion of a
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Sec. 20]_Curvature and Torsion of a
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Sec. 1] The Double Integral in Rect
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Sec. 2]_Change of Variables in a Do
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Sec. 2]_Change of Variables in a Do
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Sec 3]_Computing Areas_257 2177. Co
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Sec. 5]_Computing th Area* of Surfa
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Sec. 6]_Applications of the Double
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Sec. 7] Triple Integrals 2o3 Evalua
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Sec 7\ Triple Integrals 265 We ther
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gee. 7]_Triple Integrals_267 2251.
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Sec. 8] Improper Integrals Dependen
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Sec. 8]_Improper Integrals Dependen
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See. 9] Line Integrals 273 Test for
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Sec. 9]____Line Inlegtah____275 whe
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Sec. 9]_Line Integrals__277 express
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Sec. 9] Lim Integrals 279 (/-axis;
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Sec 9] Line Integrals 281 Evaluate
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Sec. 9] Line Integrals 2S& D. Appli
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Sec. 10] Surface Integrals 285 When
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ec. 11]_The Ostrogradsky-Gauss Form
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Sec. 12] Fundamentals of Field Theo
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Sec. 12] Fundamentals of Field Theo
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Sec. 1. Number Series Chapter VIII
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Sec. 1]_Number Series_295 c) D'Alem
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Sec /\_Number Series_297 Note. For
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2418. + + 0419 J___1_ . 4- 4- 4- '
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Sec. 1] Number Series 2464. cos-2-)
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Sec. 1]_Number Series_303 - V r(2-Q
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Sec. 2]_Functional Series_305 In th
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Sec. 2]_Functional Series_307 [the
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Sec. 2] Functional Series 309 Deter
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Sec. 3]_Taylor's Series_311 v8 s r
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Sec. 3] Taylor's Series 313 Since a
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Sec 3} Taylor's Series 315 2615. ln
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Sec. 3] Taylor's Series 317 2651. F
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Sec. 4] Fourier Series 319 If a fun
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Sec. 4\__Fourier Series _321 2690.
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Sec 1 1 Verifying Solutions 323 It
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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
Page 444 and 445:
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
Page 450 and 451: 444 Answers (binormal). 2103. bx *=
Page 452 and 453: 444_Answers_x = \ ' > (principal no
Page 454 and 455: 446 Answers 2 a a a 2139. ( dy(f(x,
Page 456 and 457: 448_Answers_ r 4 **/* [ rj-cos*(p--
Page 458 and 459: 450 Answers 2243. 2244. 1 i VT=T* ^
Page 460 and 461: 452 Answers 2 t 2297. ) ~ll . |-[(I
Page 462 and 463: 454_Answers__ TtD the flux, use the
Page 464 and 465: 456__Answers_ and when x | ^ =- , t
Page 466 and 467: 458 Answers
Page 468 and 469: 460 Answers %-l n=o (n)=^=%. 2873.
Page 470 and 471: 462 Answers Chapter IX 2704. Yes. 2
Page 472 and 473: 464 Answers 2822. g-C*+; y=2*. 2824
Page 474 and 475: 466 Answers x + C z . 2914. #=
Page 476 and 477: 468_Answers_ A $007. 1) x = C cosco
Page 478 and 479: 470 Answers 3081. x = C l e 3082. x
Page 480 and 481: 472 Answers ,*. f (2"+ 1)JtJf dx. H
Page 482 and 483: 474 Answers 3131. a) 0.211; 0.389;
Page 484 and 485: 476 Appendix III. Inverse Quantitie
Page 486 and 487: 478 Appendix IV. Trigonometric Func
Page 488 and 489: 480 Appendix VI. Some Curves (for R
Page 490 and 491: 482 Appendix 11. Graphs //-=secx an
Page 492 and 493: 486 Appendix 24. Strophoid, * u s^-
Page 494 and 495: Absolute error Absolute value 367 o
Page 496 and 497: 490 Index Differential equations of
Page 498 and 499: 492 Index Infinite discontinuities
Page 502: 496 Index Taylor's series Term 311,
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