## Calculus of Several VariablesThe present course on calculus of several variables is meant as a text, either for one semester following A First Course in Calculus, or for a year if the calculus sequence is so structured. For a one-semester course, no matter what, one should cover the first four chapters, up to the law of conservation of energy, which provides a beautiful application of the chain rule in a physical context, and ties up the mathematics of this course with standard material from courses on physics. Then there are roughly two possibilities: One is to cover Chapters V and VI on maxima and minima, quadratic forms, critical points, and Taylor's formula. One can then finish with Chapter IX on double integration to round off the one-term course. The other is to go into curve integrals, double integration, and Green's theorem, that is Chapters VII, VIII, IX, and X, §1. This forms a coherent whole. |

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### Contents

2 | |

17 | |

5 Parametric Lines | 32 |

CHAPTER II | 48 |

2 Length of Curves | 62 |

CHAPTER IV | 76 |

3 Differentiability and Gradient | 77 |

The Chain Rule and the Gradient | 87 |

CHAPTER X | 269 |

CHAPTER XI | 292 |

3 Center of Mass | 313 |

3 Surface Integrals | 333 |

6 Stokes Theorem | 355 |

2 Multiplication of Matrices | 372 |

CHAPTER XIV | 385 |

2 Linear Mappings | 392 |

3 Directional Derivative | 99 |

5 The Law of Conservation of Energy | 111 |

CHAPTER V | 123 |

3 Lagrange Multipliers | 135 |

CHAPTER VI | 143 |

2 The Quadratic Term at Critical Points | 149 |

3 Algebraic Study of a Quadratic Form | 155 |

4 Partial Differential Operators | 162 |

5 The General Expression for Taylors Formula | 170 |

CHAPTER | 183 |

3 An Important Special Vector Field | 194 |

5 Proof of the Local Existence Theorem | 201 |

1 Definition and Evaluation of Curve Integrals | 207 |

2 The Reverse Path | 217 |

3 Curve Integrals When the Vector Field Has a Potential Function | 220 |

CHAPTER IX | 233 |

3 Polar Coordinates | 252 |

3 Geometric Applications | 398 |

4 Composition and Inverse of Mappings | 404 |

CHAPTER XV | 412 |

3 Additional Properties of Determinants | 420 |

4 Independence of Vectors | 428 |

CHAPTER XVI | 434 |

3 The Chain Rule | 440 |

5 Implicit Functions | 446 |

CHAPTER XVII | 453 |

2 Dilations | 463 |

3 Change of Variables Formula in Two Dimensions | 469 |

5 Change of Variables Formula in Three Dimensions | 478 |

APPENDIX | 487 |

2 Computation of Fourier Series | 494 |

Answers | 1 |

108 | |

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### Common terms and phrases

3-space A X B Algebra angle assume boundary calculus centered chain rule Chapter circle of radius column constant continuous function cos(xy counterclockwise critical point curl F D2 f definition denote differentiable function disc of radius distance divergence theorem dot product dr d6 dy dx equal equation Example Exercise exists Figure Find the integral finite number function f given grad f grad f(X gradient graph Green's theorem Green’s Hence inequalities integral of F interval inverse mapping Jacobian matrix L.M. point Let f Let f(x linear map located vector maximum minimum multiplied n-tuples notation open set origin parallelogram parametric representation partial derivatives perpendicular polar coordinates potential function proof prove quadratic form rectangle respect satisfy set of points Show sin(xy spanned sphere of radius square Stokes theorem Suppose surface Theorem 2.1 unit vector vector field vector field F(x write