Brian Erf Blank is an associate professor of mathematics at Washington University in St. Louis. He received his Ph.D. in 1980 at Cornell University, with Anthony Knapp as advisor. His 20th century work involved harmonic analysis. Steven George Krantz is an American scholar, mathematician, and writer. He has authored more than 235 research papers and more than 125 books. Additionally, Krantz has edited journals such as the Notices of the American Mathematical Society and The Journal of Geometric Analysis.

9.1 Vectors in the Plane 1

9.2 Vectors in Three-Dimensional Space 7

9.3 The Dot Product and Applications  12

9.4 The Cross Product and Triple Product 16

9.5 Lines and Planes in Space  22

10 Vector-Valued Functions 34

10.1 Vector-Valued Functions|Limits, Derivatives, and Continuity  34

10.2 Velocity and Acceleration  43

10.3 Tangent Vectors and Arc Length 53

10.4 Curvature  64

10.5 Applications of Vector-Valued Functions  74

11 Functions of Several Variables 86

11.1 Functions of Several Variables  86

11.2 Cylinders and Quadratic Surfaces  95

11.3 Limits and Continuity 103

11.4 Partial Derivatives 106

11.5 Dierentiability and the Chain Rule  114

11.6 Gradients and Directional Derivatives  123

11.7 Tangent Planes  129

11.8 Maximum-Minimum Problems  134

11.9 Lagrange Multipliers  144

12 Multiple Integrals 156

12.1 Double Integrals over Rectangular Regions 156

12.2 Integration over More General Regions  160

12.3 Calculation of Volumes of Solids  171

12.4 Polar Coordinates  179

12.5 Integrating in Polar Coordinates  188

12.6 Triple Integrals  200

12.7 Physical Applications  209

12.8 Other Coordinate Systems  215

13 Vector Calculus 222

13.1 Vector Fields  222

13.2 Line Integrals  228

13.3 Conservative Vector Fields and Path-Independence 236

13.4 Divergence, Gradient, and Curl  241

13.5 Green's Theorem  245

13.6 Surface Integrals  253

13.7 Stokes's Theorem  262

13.8 Flux and the Divergence Theorem  277

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