Block 2: Functions
In Calculus Revisited Part 1, Professor Gross discussed the
calculus of a single real variable in which the domain of a function was a subset of the real numbers. Geometrically
speaking, the domain of a function was a subset of the x-axis. In this block he generalizes the domain as being a subset of
either the two-dimensional xy-plane and/or the three-dimensional xyz-space. In the language of vectors, in this block a function
maps 2 and 3 dimensional vectors into the set of real numbers. He then uses these functions to show how we compute the velocity
and acceleration of an object moving in space.
Block 3: Real Valued Functions of Several Real
In this block
Professor Gross uses the "game of mathematics" concept to develop an analytical way to extend the domain of a function
to beyond 3 dimensions. In particular, he shows how by using vector arithmetic, the rules of arithmetic that were used in
developing the calculus of a single variable turn out to be the same that we use to develop the calculus of several variables.
This leads to a discussion of how we replace the concept of slope in the 2 and 3-dimensional calculus by such concepts as
the directional derivative when dealing with more than 3 dimensions.
Block 4: Matrix Algebra
Block 4 extends the concept of inverse functions to
the case where y = f(x) with y = (y1, y2, ..., yn) and x = (x1, x2, ..., xn). In more user-friendly
terms this block asks us to determine when and how the system of equations that expresses y1, y2, ... and yn as
functions of x1, x2, and xn can be "inverted" to express
x1, x2, ... and xn as functions of y1,
y2, ... and yn. This motivates the study of matrix algebra since the process of inverting an n x n square
matrix is used to show how we decide whether a function f(x1,
x2, ..., xn) has an inverse and how we find the inverse function if it exists.