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.
- I teach people how to compute, but I am just as concerned that they
have a geometrical understanding of what their computations mean.
In the following I will focus on the meanings of computations rather than
their mechanics, though I spend a lot of time in class on the mechanics
as well.
- The course begins with a discussion of vectors and vector algebra.
Geometrically vectors are represented as quantities with a direction and
magnitude. We emphasize the geometric interpretations of the dot and cross
products, since these will be very important when we come to discuss
the equation of a plane,
the geometrical meaning of the gradient of a function and when we define
integrals of vector fields over functions later in the course.
With vector algebra, we do algebraic manipulations on vectors themselves,
rather than manipulating the individual x,y, and z coordinates of the vectors.
By working with the vectors themselves, we can better appreciate the
geometrical significance of the algebraic manipulations we are doing.
- A simple example of the relationship between vector algebra and geometry
arises when we discuss
parametrizations of lines and planes. Parametrizations of curves and surfaces
form the second main theme in the course. We emphasize parametrizations of lines
and planes early on in the course, since lines and planes are the
simplest examples of curves and surfaces, and because they are
tangents to curves and surfaces. Later in the course we
learn how to mathematically describe curves and surfaces in terms
of parametrizations, using both vector algebra and geometry.
The concept of a parametrization of a surface is
much more general and useful than that of a surface as the graph of a
function. To establish the connection between graphs and parametrizartions,
we emphasize that the graph of a function of 1 variable can be used to define
a parametrization of a curve and that the graph of a function of 2
variables can be used to define a parametrization of a surface.
The basic reasons for describing surfaces using parametrizations
rather than using graphs of functions, are that (1) the analogies between curves
and surfaces can be more readily studied; (2) many surfaces
are more readily described using a parametrization; (3) it is simpler
to develop all the theory for parametrizations and then
discuss the special case of the graph of a function than to have to
develop the theory twice, first for graphs and second for parametrizations;
(4) the relationship between Green's Theorem and Stokes' Theorem is easy
to establish; (5) the relationship between the Change of Variables Theorem
and the definition of the integral of a function over a surface can
be made clear.
- In the chapter on partial derivatives, we focus attention on the
geometrical meaning of the gradient of a function. To that end, we
formulate the chain rule for functions on curves as a dot product
involving the gradient of a function, and use the geometrical
interpretation of the dot product to derive the geometrical
interpretation of the gradient of a function
as the direction of steepest ascent for the graph of the function.
We also use the chain rule for functions on curves to establish the
orthogonality between tangent lines to the contour curves of the
graph of a function and the gradient of that function.
- In the chapter on multiple integrals, we emphasize how integrals
over curves and surfaces are defined using parametrizations in terms
of integrals over subsets of the real line and the plane.
- The four versions of the FTC form the pinnacle of the course.
In particular, we use Stokes' Theorem to provide a geometrical interpretation
of the curl of a vector field, and the Divergence Theorem to give a
geometrical interpretation of the divergence of a vector field.
- In his discussion of velocity and acceleration, Calder alluded to
Newton's Laws of Motion. Newton invented the Calculus to study planetary
motion, and since planets travel along curves in space, I imagine he really had
to invent a form of Vector Calculus right from the get-go.
Although Newton's and Kepler's Laws are discussed briefly in the course,
I prefer to conclude the course with
a discussion of Maxwell's equations for coupled electric and magnetic
vector fields, i.e., for light. These equations come in two flavours:
Integral and Differential. The wonderful thing is that the FTC can be used
to explain the relationship between these two flavors of Maxwell's equations.
Last modified Jan 20, 2008