This manual gives the solutions, with work, to the odd-numbered exercises in James Stewart’s Multivariable Calculus (fourth edition). This coincides with chapters 11-18 of his Calculus (fourth edition) and chapters 10-17 of his Calculus: Early Transcendentals.

The Fourth Edition has the mathematical precision, accuracy, clarity of exposition and outstanding examples and problem sets that have characterized the first four editions. Stewart retains the focus on problem solving and the pedagogical system that has made the book a favorite of students and instructors in a wide variety of colleges and universities throughout the world.

These refinements ensure that students and instructors have the best materials available. The number of pages in the book, however, remains unchanged from the 3rd edition support for students and instructors is now available through a vast array of supplementary material.

**10. PARAMETRIC EQUATIONS AND POLAR COORDINATES.**

Curves Defined by Parametric Equations. Laboratory Project: Families of Hypocycloids. Calculus with Parametric Curves. Laboratory Project: Bezier Curves. Polar Coordinates. Laboratory Project: Families of Polar Curves. Areas and Lengths in Polar Coordinates. Conic Sections. Conic Sections in Polar Coordinates. Review. Problems Plus.

**11. INFINITE SEQUENCES AND SERIES. **

Sequences. Laboratory Project: Logistic Sequences. Series. The Integral Test and Estimates of Sums. The Comparison Tests. Alternating Series. Absolute Convergence and the Ratio and Root Tests. Strategy for Testing Series. Power Series. Representations of Functions as Power Series. Taylor and Maclaurin Series . Laboratory Project: An Elusive Limit. Writing Project: How Newton Discovered the Binomial Series. Applications of Taylor Polynomials. Applied Project: Radiation from the Stars. Review. Problems Plus.

**12. 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. Cylinders and Quadric Surfaces. Review. Problems Plus.

**13. 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. Review. Problems Plus.

**14. PARTIAL DERIVATIVES. **

Functions of Several Variables. Limits and Continuity. Partial Derivatives. Tangent Planes and Linear Approximation. 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. Problems Plus.

**15. MULTIPLE INTEGRALS. **

Double Integrals over Rectangles. Iterated Integrals. Double Integrals over General Regions. Double Integrals in Polar Coordinates. Applications of Double Integrals. Triple Integrals. Discovery Project: Volumes of Hyperspheres. Triple Integrals in Cylindrical Coordinates. Discovery Project: The Intersection of Three Cylinders . Triple Integrals in Spherical Coordinates. Applied Project: Roller Derby. Change of Variables in Multiple Integrals. Review. Problems Plus.

**16. ECTOR CALCULUS. **

Vector Fields. Line Integrals. The Fundamental Theorem for Line Integrals. Green’s Theorem. Curl and Divergence. Parametric Surfaces and Their Areas. Surface Integrals. Stokes’ Theorem. Writing Project: Three Men and Two Theorems. The Divergence Theorem. Summary. Review. Problems Plus.

**17. SECOND-ORDER DIFFERENTIAL EQUATIONS. **

Second-Order Linear Equations. Nonhomogeneous Linear Equations. Applications of Second-Order Differential Equations. Series Solutions. Review. Problems Plus.

**APPENDIXES. **

F. Proofs of Theorems.

G. Graphing Calculators and Computers.

H. Complex Numbers.

I. Answers to Odd-Numbered Exercises.

REVIEW