Multivariable Calculus

9. VECTORS AND THE GEOMETRY OF SPACE.
Three-Dimensional Coordinate Systems. Vectors. The Dot Product. The Cross Product. Discovery Project: The Geometry of a Tetrahedron. Equations of Lines and Planes. Laboratory Project: Putting 3D in Perspective. Functions and Surfaces. Cylindrical and Spherical Coordinates. Laboratory Project: Families of Surfaces. Review. Focus on Problem Solving.
10. VECTOR FUNCTIONS. Vector Functions and Space Curves. Derivatives and Integrals of Vector Functions. Arc Length and Curvature. Motion in Space: Velocity and Acceleration. Applied Project: Kepler's Laws. Parametric Surfaces. Review. Focus on Problem Solving.
11. PARTIAL DERIVATIVES. Functions of Several Variables. Limits and Continuity. Partial Derivatives. Tangent Planes and Linear Approximations. The Chain Rule. Directional Derivatives and the Gradient Vector. Maximum and Minimum Values. Applied Project: Designing a Dumpster. Discovery Project: Quadratic Approximations and Critical Points. Lagrange Multipliers. Applied Project: Rocket Science. Applied Project: Hydro-Turbine Optimization. Review. Focus on Problem Solving.
12. MULTIPLE INTEGRALS. Double Integrals over Rectangles. Iterated Integrals. Double Integrals over General Regions. Double Integrals in Polar Coordinates. Applications of Double Integrals. Surface Area. Triple Integrals. Discovery Project: Volumes of Hyperspheres. Triple Integrals in Cylindrical and Spherical Coordinates. Applied Project: Roller Derby. Discovery Project: The Intersection of Three Cylinders. Change of Variables in Multiple Integrals. Review. Focus on Problem Solving.
13. VECTOR CALCULUS. Vector Fields. Line Integrals. The Fundamental Theorem for Line Integrals. Green's Theorem. Curl and Divergence. Surface Integrals. Stokes' Theorem. Writing Project: Three Men and Two Theorems. The Divergence Theorem. Summary. Review. Focus on Problem Solving.
APPENDIXES.
A. Intervals, Inequalities, and Absolute Values.
B. Coordinate Geometry.
C. Trigonometry.
D. Precise Definitions of Limits.
E. A Few Proofs.
F. Sigma Notation.
G. Integration of Rational Functions by Partial Fractions.
H. Polar Coordinates.
I. Complex Numbers.
J. Answers to Odd-Numbered Exercises.