Multivariable calculus

For a one-semester 2nd year course in multivariable calculus, for Engineering, Mathematics, or Science students.

Preface.

1. Vectors and Curves.

Vectors in the Plane.
Lines and Curves in the Plane. Acceleration.
The Dot Product. Coordinates and Vectors in Space.
Lines in Space.
The Dot Product of Space Vectors.
The Cross Product.
Planes. Curves in Space.
Normals of Curves.

2. Functions and Differentiation.

Functions of Two Variables.
Functions of Three Variables.
Limits and Continuity.
Partial Derivatives.
Linear Approximation.
The Chain Rule.
Directional Derivatives: Functions of Two Variables.
Directional Derivatives: Functions of Three Variables.
Higher Order Approximations. Local (or Relative) Extrema.
Constrained Extrema.
Global (or Absolute) Extrema.

3. Integration.

Double Integrals I - Description and Properties.
Double Integrals II - Cartesian Coordinates.
Double Integrals in Polar Coordinates.
Triple Integrals.
Triple Integrals in Cylindrical Coordinates.
Triple Integrals in Spherical Coordinates.

4. Vector Fields.

Vector Fields in the Plane.
Flow Lines of a Vector Field in the Plane.
Vector Fields in Space.
The Symbol .

5. Line Integrals.

Force Fields and Work.
Work and Line Integrals.
Conservative Vector Fields.
The Curl-Test.

6. Surface Integrals.

Parametrized Planes.
Parametrized Surfaces.
Velocity Fields and Flux.
Flux and Surface Integrals.

7. The Theorems of Green, Gauss, and Stokes.

The Boundary of Solids, Surfaces, and Curves.
Gauss Divergence Theorem.
Stokes Theorem.
Answers to Odd Problems.

• Students are engaged through formulas and geometric reasoning - in addition to calculating accurately, students are asked to draw accurately in both two and three dimensions, reason geometrically from figures, make estimates based on ruler-and pencil-constructions, and present their results verbally, helps students learn conceptual reasoning and reinforces learning by asking students to work the material in two different modes.
• This is a spiral bound text, lays flat so students can draw in blank diagrams while reading the text.
• A multitude of exercises are interwoven within the flow of the text - they appear in addition to the exercise sections.
• Encourages students to read the text actively and check their understanding of the material frequently. Habituates students to