## 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 | |

### Other editions - View all

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

Basic Training in Mathematics: A Fitness Program for Science Students R. Shankar Limited preview - 2013 |

### Common terms and phrases

analytic angle answer assume basis boundary calculation called charge circle closed coefficients column combination complex components Consider constant continuous contour contribution converges coordinates corresponding course defined definition depends derivative determinant differential dimensions direction discussed displacement divergence dot product eigenvalue eigenvectors equal equation evaluate example expansion exponential expression fact factor field Figure follows force function given gives independent infinite integral inverse length limit linear look mass matrix means multiply normal Note null vector obey obtain operator origin particle path plane positive Problem radius relation respect result roots rotation rule scalar Show sides simply solution solve space square surface Taylor series tells Theorem turn unit vanish variable vector 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.