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. Functions and Surfaces. Cylindrical and Spherical Coordinates. Laboratory Project: Families and Surfaces. 10. Vector functions: Vector Functions and Space Curves. Derivatives and Integrals of Vector Functions. Arc Length and Curvature. Motion in Space. Applied Project: Kepler's Laws. Parametric Surfaces. 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 Optimization. Project: Hydro-Turbine. 12. Multiple integrals: Double Integrals over Rectangles. Integrated Integrals. Double Integrals over General Regions. Double Integrals in Polar Coordinates. Applications of Double Integrals. Surface Area. Triple Integrals. Applied Project: Roller Derby. Discovery Project: Volumes of Hyperspheres. Triple Integrals in Cylindrical and Spherical Coordinates. Discovery Project: The Intersection of Three Cylinders. Change of Variables in Multiple Integrals. 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: Green, Thomson, and Stokes. The Divergence Theorem. (Part contents).