John Zweck

Course Objectives and Advice for
Teachers and Students of Math 251 (Multivariable Calculus)

How does art come into being?
Out of volumes, motion, spaces,
Carved out within the surrounding space, the universe.
Out of directional line -
Vectors representing motion, velocity, acceleration, energy, etc.
- Lines which form significant angles and directions,
Making up one or several totalities.


Alexander Calder, "Comment realiser l'art", 1932.

I saw this quote as part of the wall text at the Calder-Miro exhibit at the Phillips Collection in Washington, D.C. in December 2004. Although the quote is about art, to me it captured in poetry what I try to teach the students in my Multivariable Calculus class. As Calder explains, much of 20th Century art was about exploring ways to depict and understand space and the objects in it. This is exactly what we do in Multivariable Calculus, or as I prefer to call it, Vector Calculus. The connotation behind the term "Multivariable Calculus" is that we just take Single Variable Calculus and add some more variables. Nothing could be further from the truth. In Vector Calculus, the primary goal is to study the geometry of change in two and three dimensional space. For this, rather than emphasizing (x,y,z)-coordinates, we work with vectors, which are geometric quantities with a magnitude and direction. In particular, we use vectors to mathematically describe curves and surfaces in space, and to study the derivatives (rates of change) and integrals (average properties) of functions and vector fields that are defined on curves and surfaces. Just as depictions of space and form in 20th dentury art can be beautiful to behold, so are the geometric and algebraic concepts we learn about in Vector Calculus. The unity between geometry and algebra is most succinctly expressed in the four versions of the Fundamental Theorem of Calculus we study in Math 251: The FTC for vector fields on curves, Green's Theorem, Stokes' Theorem and The Divergence Theorem.

I will now describe some of the overarching principles that underlie my approach to Vector Calculus. These principles double as learning goals for the course.

Last modified Jan 20, 2008

Valid HTML 4.01!