CC by SA Calculus Text Books: Active Calculus Multivariable

Active Calculus - Multivariable, Second Edition, by Nicholas Long and Mitchel T. Keller, is an open-licensed textbook for learning calculus in several variables. Its contents move from the three-dimensional geometry and vector algebra needed for the subject to differentiation, integration, and vector calculus. The front matter includes the authors’ goals for the book, guidance on using it, and information about graphics and technology, while the chapters are divided into short sections with exercises.

The text begins with vectors, lines and planes, three-dimensional surfaces, and polar, cylindrical, and spherical coordinates. It then develops vector-valued functions and the calculus of curves, including arc length, curvature, and acceleration. The chapter on multivariable derivatives covers limits, partial derivatives, tangent planes, the chain rule, gradients, directional derivatives, and unconstrained and constrained optimization. Multiple integrals follow, with applications to area, mass, probability, and center of mass, as well as changes of coordinates. The final chapter treats vector fields and line and surface integrals, building toward Green’s, Stokes’, and the Divergence theorems. A supplementary section on conic sections rounds out the material.