## Calculus: Multivariable, Volume 2Calculus is one of the milestones of human thought. Every well-educated person should be acquainted with the basic ideas of the subject. In todaya??s technological world, in which more and more ideas are being quantified, knowledge of calculus has become essential to a broader cross-section of the population. This Debut Edition of Calculus by Brian Blank and Steven G. Krantz is published in two volumes, Single Variable and Multivariable. Teaching and writing from the traditional point of view, these authors have distilled the lessons of reform and bring you a calculus book focusing on todaya??s best practices in calculus teaching. |

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### Contents

Vectors | 1 |

111 Vectors in the Plane | 2 |

112 Vectors in ThreeDimensional Space | 12 |

113 The Dot Product and Applications | 21 |

114 The Cross Product and Triple Product | 32 |

115 Lines and Planes in Space | 44 |

Summary of Key Topics | 58 |

Genesis Development | 62 |

Genesis Development | 226 |

Multiple Integrals | 231 |

141 Double Integrals over Rectangular Regions | 232 |

142 Integration over More General Regions | 240 |

143 Calculation of Volumes of Solids | 248 |

144 Polar Coordinates | 254 |

145 Integrating in Polar Coordinates | 263 |

146 Triple Integrals | 277 |

VectorValued Functions | 65 |

121 VectorValued Functions Limits Derivatives and Continuity | 66 |

122 Velocity and Acceleration | 77 |

123 Tangent Vectors and Arc Length | 87 |

124 Curvature | 97 |

125 Applications of VectorValued Functions to Motion | 107 |

Summary of Key Topics | 121 |

Genesis Development | 125 |

Functions of Several Variables | 129 |

131 Functions of Several Variables | 130 |

132 Cylinders and Quadric Surfaces | 141 |

133 Limits and Continuity | 150 |

134 Partial Derivatives | 156 |

135 Differentiability and the Chain Rule | 166 |

136 Gradients and Directional Derivatives | 178 |

137 Tangent Planes | 187 |

138 MaximumMinimum Problems | 198 |

139 Lagrange Multipliers | 212 |

Summary of Key Topics | 222 |

147 Physical Applications | 283 |

148 Other Coordinate Systems | 292 |

Summary of Key Topics | 298 |

Genesis Development | 304 |

Vector Calculus | 307 |

151 Vector Fields | 308 |

152 Line Integrals | 317 |

153 Conservative Vector Fields and PathIndependence | 328 |

154 Divergence Gradient and Curl | 340 |

155 Greens Theorem | 348 |

156 Surface Integrals | 358 |

157 Stokess Theorem | 369 |

158 Flux and the Divergence Theorem | 383 |

Summary of Key Topics | 392 |

Genesis Development | 396 |

ANSWERS TO SELECTED EXERCISES | 399 |

423 | |

### Common terms and phrases

acceleration angle approximation arc length arc length parameterization axis boundary Calculator/Computer Exercises Cartesian equation Chain Rule circle component constant continuous function continuously differentiable continuously differentiable function critical point cross product curl(F curvature cylinder defined denote determine directed curve directed line segment Divergence Theorem domain dot product double integral ellipse Example field F formula Further Theory given gradient graph Green's Theorem intersection iterated integral level curves level sets line integral mass normal vector obtain oriented paraboloid parallel parametric equations partial derivatives perpendicular planar region plane plot point P0 polar coordinates position vector Practice In Exercises Problems for Practice radius rectangle rectangular coordinates Riemann sum scalar-valued function Section shows Sketch slices solid space spherical coordinates Stokes's Theorem Suppose surface area symmetric tangent line tangent vector unit vector v x w v-axis v-simple variables vector-valued function Verify volume vv-plane vy-plane