| Preface |
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xi | |
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1 Revision of methods of differentiation |
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1 | (23) |
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1 | (1) |
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1.2 Differentiation of common functions |
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2 | (4) |
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1.3 Differentiation of a product |
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6 | (3) |
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1.4 Differentiation of a quotient |
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9 | (3) |
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1.5 Function of a function |
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12 | (4) |
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1.6 Successive differentiation |
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16 | (3) |
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19 | (5) |
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2 Solution of equations by iterative methods |
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24 | (13) |
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24 | (1) |
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2.2 An algebraic method of successive approximation |
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24 | (5) |
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2.3 The Newton--Raphson method |
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29 | (5) |
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34 | (3) |
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37 | (11) |
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37 | (1) |
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3.2 Denominator containing linear factors |
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38 | (3) |
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3.3 Denominator containing repeated linear factors |
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41 | (2) |
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3.4 Denominator containing a quadratic factor |
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43 | (2) |
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45 | (1) |
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46 | (2) |
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4 Matrix arithmetic and the determinant of a matrix |
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48 | (17) |
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48 | (1) |
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4.2 Addition, subtraction and multiplication of third order matrices |
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49 | (5) |
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4.3 The determinant of a 3 by 3 matrix |
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54 | (3) |
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4.4 The inverse or reciprocal of a matrix |
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57 | (4) |
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61 | (4) |
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5 The general properties of 3 by 3 determinants and the solution of simultaneous equations |
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65 | (14) |
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5.1 The properties of third order determinants |
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65 | (10) |
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5.2 The solution of simultaneous equations having three 70 unknowns |
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75 | (4) |
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6 Maclaurin's and Taylor's series |
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79 | (19) |
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79 | (8) |
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87 | (5) |
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6.3 Approximate values of definite integrals by using series expansions |
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92 | (2) |
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94 | (4) |
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98 | (28) |
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98 | (1) |
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7.2 Definition of a complex number |
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99 | (1) |
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100 | (1) |
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7.4 Operations involving cartesian complex numbers |
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101 | (5) |
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7.5 The polar form of a complex number |
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106 | (2) |
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7.6 Multiplication and division using complex numbers in polar form |
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108 | (4) |
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7.7 Applications of complex numbers |
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112 | (9) |
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121 | (5) |
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126 | (18) |
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8.1 The exponential form of a complex number |
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126 | (1) |
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8.2 De Moivre's theorem -- powers and roots of complex numbers |
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127 | (7) |
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8.3 Expressing cos nθ and sin nθ in terms of powers of cos θ and sin θ |
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134 | (3) |
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8.4 Expressing cosn θ and sinn θ in terms of sines and cosines of multiples of θ |
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137 | (3) |
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140 | (4) |
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144 | (22) |
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9.1 Definitions of hyperbolic functions |
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144 | (1) |
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9.2 Some properties of hyperbolic functions |
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145 | (1) |
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9.3 Evaluation of hyperbolic functions |
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146 | (1) |
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9.4 Graphs of hyperbolic functions |
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147 | (4) |
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9.5 Hyperbolic identities -- Osborne's rule |
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151 | (4) |
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9.6 Differentiation of hyperbolic functions |
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155 | (3) |
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9.7 Solution of equations of the form a cosh x + b sinh x = c |
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158 | (1) |
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9.8 Series expansions for cosh x and sinh x |
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159 | (2) |
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161 | (5) |
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10 The relationship between trigonometric and hyperbolic functions and hyperbolic identities |
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166 | (10) |
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10.1 The relationship between trigonometric and hyperbolic functions |
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166 | (2) |
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10.2 Hyperbolic identities |
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168 | (5) |
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10.3 Summary of trigonometric and hyperbolic identities |
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173 | (1) |
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174 | (2) |
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11 Differentiation of implicit functions |
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176 | (9) |
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176 | (1) |
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11.2 Differentiating implicit functions |
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176 | (6) |
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182 | (3) |
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12 Differentiation of functions defined parametrically |
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185 | (12) |
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12.1 Parametric representation of points |
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185 | (2) |
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12.2 Differentiation in parameters |
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187 | (7) |
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194 | (3) |
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13 Logarithmic differentiation |
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197 | (11) |
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13.1 The laws of logarithms applied to functions |
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197 | (4) |
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13.2 Logarithmic differentiation |
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201 | (4) |
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205 | (3) |
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14 Differentiation of inverse trigonometric and inverse hyperbolic functions |
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208 | (26) |
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208 | (1) |
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14.2 Differentiation of inverse trigonometric functions |
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208 | (9) |
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14.3 Logarithmic forms of the inverse hyperbolic functions |
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217 | (3) |
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14.4 Differentiation of inverse hyperbolic functions |
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220 | (9) |
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229 | (5) |
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15 Partial differentiation |
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234 | (12) |
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15.1 Differentiating a function having two variables |
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234 | (1) |
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15.2 First order partial derivatives |
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235 | (3) |
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15.3 Second order partial derivatives |
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238 | (4) |
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242 | (4) |
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16 Total differential, rates of change and small changes |
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246 | (12) |
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246 | (3) |
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249 | (2) |
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251 | (3) |
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254 | (4) |
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17 Revision of basic integration |
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258 | (14) |
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258 | (1) |
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17.2 The general solution of integrals of the form axn |
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259 | (1) |
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259 | (7) |
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266 | (3) |
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269 | (3) |
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18 Integration using substitutions |
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272 | (28) |
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272 | (1) |
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18.2 Integration using algebraic substitutions |
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272 | (5) |
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18.3 Integration using trigonometric and hyperbolic identities and substitutions |
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277 | (14) |
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18.4 Change of limits of integration by a substitution |
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291 | (2) |
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293 | (7) |
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19 Integration using partial fractions |
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300 | (9) |
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300 | (5) |
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305 | (4) |
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20 The t = tan θ/2 substitution |
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309 | (8) |
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309 | (6) |
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315 | (2) |
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317 | (9) |
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317 | (1) |
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21.2 Application of the integration by parts formula |
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317 | (7) |
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324 | (2) |
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22 First order differential equations by separation of the variables |
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326 | (14) |
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326 | (1) |
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22.2 Solution of differential equations of the form dy/dx = f(x) |
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327 | (2) |
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22.3 Solution of differential equations of the form dy/dx = f(y) |
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329 | (2) |
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22.4 Solution of `variables separable' type of differential equations |
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331 | (3) |
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334 | (6) |
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23 Homogeneous first order differential equations |
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340 | (6) |
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23.1 Solution of differential equations of the form P dy/dx = Q |
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340 | (4) |
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344 | (2) |
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24 Linear first order differential equations |
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346 | (8) |
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24.1 Solution of differential equations of the form dy/dx + Py = Q |
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346 | (5) |
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351 | (3) |
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25 The solution of linear second order differential equations of the form a(d2y/dx2) + b(dy/dx) + cy = 0 |
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354 | (17) |
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354 | (1) |
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25.2 Types of solution of second order differential equations with constant coefficients |
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355 | (6) |
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25.3 Summary of the procedure used to solve differential equations of the form a(d2y/dx2) + b(dy/dx) + cy = 0 |
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361 | (5) |
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366 | (5) |
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26 The solution of linear second order differential equations of the form a(d2y/dx2) + b(dy/dx) + cy= f(x) |
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371 | (19) |
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26.1 Complementary function and particular integral |
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371 | (1) |
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26.2 Methods of finding the particular integral |
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372 | (12) |
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26.3 Summary of the procedure to solve differential equations of the form α(d2y/dx2) + b(dy/dx) + cy = f(x) |
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384 | (1) |
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384 | (6) |
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27 The Fourier series for periodic functions of period 2π |
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390 | (16) |
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390 | (1) |
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391 | (3) |
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27.3 Evaluation of definite integrals involving trigonometric functions |
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394 | (2) |
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396 | (8) |
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404 | (2) |
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28 The Fourier series for a non-periodic function over range 2π |
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406 | (7) |
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28.1 Expansion of non-periodic functions |
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406 | (4) |
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410 | (3) |
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29 The Fourier series for even and odd functions and half range series |
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413 | (17) |
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29.1 Even and odd functions |
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413 | (3) |
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29.2 Fourier cosine and Fourier sine series |
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416 | (6) |
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29.3 Half range Fourier series |
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422 | (4) |
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426 | (4) |
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30 Fourier series over any range |
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430 | (11) |
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30.1 Expansion of a periodic function of period l |
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430 | (4) |
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30.2 Half range Fourier series for functions defined over range l |
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434 | (4) |
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438 | (3) |
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31 A numerical method of harmonic analysis |
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441 | (12) |
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441 | (1) |
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31.2 Harmonic analysis of data given in tabular or graphical form |
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441 | (5) |
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31.3 Complex waveform considerations |
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446 | (3) |
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449 | (4) |
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32 Introduction to Laplace transforms |
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453 | (11) |
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453 | (1) |
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32.2 Definition of the Laplace transform |
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453 | (1) |
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32.3 Linearity property of the Laplace transform |
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454 | (1) |
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32.4 Laplace transforms of elementary functions |
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454 | (8) |
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462 | (2) |
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33 Properties of Laplace transforms |
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464 | (13) |
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33.1 The Laplace transform of eatf(t) |
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464 | (1) |
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33.2 Laplace transforms of functions of the type eatf(t) |
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464 | (4) |
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33.3 The Laplace transforms of derivatives |
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468 | (3) |
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33.4 The initial and final value theorems |
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471 | (3) |
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474 | (3) |
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34 Inverse Laplace transforms and the use of Laplace transforms to solve differential equations |
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477 | (17) |
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34.1 Definition of the inverse Laplace transform |
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477 | (1) |
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34.2 Inverse Laplace transforms of simple functions |
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477 | (4) |
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34.3 Inverse Laplace transforms using partial fractions |
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481 | (3) |
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34.4 Use of Laplace transforms to solve second order differential equations with constant coefficients |
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484 | (6) |
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490 | (4) |
| Index |
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494 | |
Lisainfo e-raamatute kohta