Advanced Calculus using Mathematica

Advanced Calculus using Mathematica: NoteBook Edition is a complete text on calculus of several variables written in Mathematica NoteBooks.

“Just in time”

Another theme in these materials is a “just in time” approach to some topics.  For example, rather than developing all the linear material in the beginning, we study implicit linear equations just before implicit nonlinear equations.  In the more elementary course we teach with these materials, many of the “just in time” topics such as discussion of eigenvalues are skipped.  In the seond advanced course, students take a matrix algebra couse in the interim and gain some familiarity with eigenvalues in particular.  This allows us to take advantage of Mathematica in some higher dimensional MAX-min problems where hand computation of eigenvalues is burdensome.

The general multidimensional product and chain rules of calculus are also postponed in the elementary course, but not skipped.  For functions defined by formulas, basic partial derivatives can be computed with the one variable rules, and the real importance of the higher dimensional rules is abstract such as in understanding curvature or conservative vector fields.  The abstract rules are introduced when they are needed, but are treated in general in a section of Chapter 3 that can be skipped in a beginning course in multivariable calculus, but referred to in the second course.  Here are some examples of why the general abstract rules are important even though they are NOT needed to compute partial derivatives of specific formulas given in terms of sine, cosine, etc.:

Concrete Applications of Abstract use of the Product Rules and Chain Rules

Perpendicularity of the Curvature Vector

Tangential and Perpendicular Components of Acceleration

Conservation of Energy under F = m A

Planets Orbit in a Plane