## Basic Training in Mathematics: A Fitness Program for Science StudentsBased on course material used by the author at Yale University, this practical text addresses the widening gap found between the mathematics required for upper-level courses in the physical sciences and the knowledge of incoming students. This superb book offers students an excellent opportunity to strengthen their mathematical skills by solving various problems in differential calculus. By covering material in its simplest form, students can look forward to a smooth entry into any course in the physical sciences. |

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### Contents

DIFFERENTIAL CALCULUS OF ONE VARIABLE | 1 |

13 Exponential and Log Functions | 5 |

14 Trigonometric Functions | 19 |

15 Plotting Functions | 23 |

16 Miscellaneous Problems on Differential Calculus | 25 |

17 Differentials | 29 |

18 Summary | 30 |

INTEGRAL CALCULUS | 33 |

75 Scalar Field and the Gradient | 167 |

76 Curl of a Vector Field | 172 |

77 The Divergence of a Vector Field | 182 |

78 Differential Operators | 186 |

79 Summary of Integral Theorems | 188 |

711 Applications from Electrodynamics | 192 |

712 Summary | 202 |

MATRICES AND DETERMINANTS | 205 |

22 Some Tricks of the Trade | 44 |

23 Summary | 49 |

CALCULUS OF MANY VARIABLES | 51 |

32 Integral Calculus of Many Variables | 61 |

33 Summary | 72 |

INFINITE SERIES | 75 |

42 Tests for Convergence | 77 |

43 Power Series in x | 80 |

44 Summary | 87 |

COMPLEX NUMBERS | 89 |

52 Complex Numbers in Cartesian Form | 90 |

53 Polar Form of Complex Numbers | 94 |

54 An Application | 98 |

55 Summary | 104 |

FUNCTIONS OF A COMPLEX VARIABLE | 107 |

62 Analytic Functions Defined by Power Series | 116 |

63 Calculus of Analytic Functions | 126 |

64 The Residue Theorem | 132 |

65 Taylor Series for Analytic Functions | 139 |

66 Summary | 144 |

VECTOR CALCULUS | 149 |

72 Time Derivatives of Vectors | 155 |

73 Scalar and Vector Fields | 158 |

74 Line and Surface Integrals | 159 |

82 Matrix Inverses | 211 |

83 Determinants | 215 |

84 Transformations on Matrices and Special Matrices | 220 |

85 Summary | 227 |

LINEAR VECTOR SPACES | 229 |

92 Inner Product Spaces | 237 |

93 Linear Operators | 247 |

94 Some Advanced Topics | 252 |

95 The Eigenvalue Problem | 255 |

96 Applications of Eigenvalue Theory | 266 |

97 Function Spaces | 277 |

98 Some Terminology | 294 |

910 Summary | 300 |

DIFFERENTIAL EQUATIONS | 305 |

102 ODEs with Constant Coefficients | 307 |

First Order | 315 |

Second Order and Homogeneous | 318 |

105 Partial Differential Equations | 329 |

106 Greens Function Method | 345 |

107 Summary | 347 |

ANSWERS | 351 |

359 | |

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Basic Training in Mathematics: A Fitness Program for Science Students R. Shankar Limited preview - 1995 |

### Common terms and phrases

adjoint analytic function angle answer area vector arrows basis vectors boundary calculation called cartesian circle coefficients column vector complex conjugate complex numbers complex plane components Consider constant contour converges coordinates corresponding cosh cross product cube curl defined definition differential equation dimensions direction displacement divergence dot product eigenvalue eigenvectors equal evaluate example expansion exponential exponential function factor finite follows formula given gives gradient hermitian infinite infinitesimal inner product integrand inverse limit line integral linear combination linearly independent matrix multiply normal obey obtain operator origin orthogonal orthonormal basis partial derivatives path perpendicular pole power series Problem radial radius rate of change Recall relation result roots rotation scalar segment Show solution solve square stationary point surface integral Taylor series tensor Theorem unit unitary vanish variable vector field vector space velocity Verify write zero

### Popular passages

Page iii - ... at great cost to herself. This book is yet another example of what she has made possible through her tireless contributions as the family muse. It is dedicated to her and will hopefully serve as one tangible record of her countless efforts. NOTE TO THE INSTRUCTOR If you should feel, as I myself do, that it is not possible to cover all the material in the book in one semester, here are some recommendations. • To begin with, you can skip any topic in fine print.