1.1 Definition

Euclidean 3-space R3 is the set of all ordered triples of real numbers. Such a triple p = (p1, p2, p3) is called a point of R3.

In linear algebra, it is shown that R3 is, in a natural way, a vector space over the real numbers. In fact, if p = (p1, p2, p3) and q = (q1, q2, q3) are points of R3, their sum is the point

The scalar multiple of a point p = (p1, p2, p3) by a number a is the point

It is easy to check that these two operations satisfy the axioms for a vector space. The point 0 = (0, 0, 0) is called the origin of R3.

Differential calculus deals with another aspect of R3 starting with the notion of differentiable real-valued functions on R3. We recall some fundamentals.

1.2 Definition

Let x, y, and z be the real-valued functions on R3 such that for each point p = (p1, p2, p3)

These functions x, y, z are called the natural coordinate functions of R3. We shall also use index notation for these functions, writing

Thus the value of the function xi on a point p is the number pi, and so we have the identity p = (p1, p2, p3) = (x1(p), x2(p), x3(p)) for each point p of R3. Elementary calculus does not always make a sharp distinction between the numbers p1, p2, p3 and the functions x1, x2, x3. Indeed the analogous distinction on the real line may seem pedantic, but for higher-dimensional spaces such as R3, its absence leads to serious ambiguities. (Essentially the same distinction is being made when we denote a function on R3 by a single letter f, reserving f(p) for its value at the point p.)

We assume that the reader is familiar with partial differentiation and its basic properties, in particular the chain rule for differentiation of a composite function. We shall work mostly with first-order partial derivatives ∂f/∂x, ∂f/∂y, ∂f/∂z and second-order partial derivatives ∂2f/∂x2, ∂2f/∂x∂y, … In a few situations, third- and even fourth-order derivatives may occur, but to avoid worrying about exactly how many derivatives we can take in any given context, we establish the following definition.

1.3 Definition

A real-valued function f on R3 is differentiable (or infinitely differentiable, or smooth, or of class C∞) provided all partial derivatives of f, of all orders, exist and are continuous.

Differentiable real-valued functions f and g may be added and multiplied in a familiar way to yield functions that are again differentiable and real-valued. We simply add and multiply their values at each point—the formulas read

The phrase “differentiable real-valued function” is unpleasantly long. Hence we make the convention that unless the context indicates otherwise, “function” shall mean “real-valued function,” and (unless the issue is explicitly raised) the functions we deal with will be assumed to be differentiable. We do not intend to overwork this convention; for the sake of emphasis the words “differentiable” and “real-valued” will still appear fairly frequently.

Differentiation is always a local operation: To compute the value of the function ∂f/∂x at a point p of R3, it is sufficient to know the values of f at all points q of R3 that are sufficiently near p. Thus, Definition 1.3 is unduly restrictive; the domain of f need not be the whole of R3, but need only be an open set of R3. By an open set of R3 we mean a subset of R3 such that if a point p is in , then so is every other point of R3 that is sufficiently near p. (A more precise definition is given in Chapter 2.) For example, the set of all points p = (p1, p2, p3) in R3 such that p1 > 0 is an open set, and the function yz log x defined on this set is certainly differentiable, even though its domain is not the whole of R3. Generally speaking, the results in this chapter remain valid if R3 is replaced by an arbitrary open set of R3.

We are dealing with three-dimensional Euclidean space only because this is the dimension we use most often in later work. It would be just as easy to work with Euclidean n-space Rn, for which the points are n-tuples p = (p1, …, pn ) and which has n natural coordinate functions x1, …, xn . All the results in this chapter are valid for Euclidean spaces of arbitrary dimensions, although we shall rarely take advantage of this except in the case of the Euclidean plane R2. In particular, the results are valid for the real line R1 = R. Many of the concepts introduced are designed to deal with higher dimensions, however, and are thus apt to be overelaborate when reduced to dimension 1.

1. Let f = x2y and g = y sin z be functions on R3. Express the following functions in terms of x, y, z:

2. Find the value of the function f = x2y – y2z at each point:

3. Express ∂f/∂x in terms of x, y, and z if

4. If g1, g2, g3, and h are real-valued functions on R3, then

is the function such that

†Express ∂f/∂x in terms of x, y, and z, if h = x2 – yz and

2.1 Definition‡

A tangent vector v p to R3 consists of two points of...