Modern Multidimensional Calculus

CHAPTER 1

FUNCTIONS AND VARIABLES

1–1 Elementary functions. A function is defined as a set of ordered pairs of numbers no two of which have the same first entry. The set of all first entries in the ordered pairs is called the domain of the function; the set of all second entries is called its range. If a function is displayed in a two-column table, the ordered pairs are read across. The domain appears as the first column of the table and the range as the second column.

If f is a function and (a, b) is one of its ordered pairs, the common practice is to indicate this fact by writing

The range entry b that corresponds to the domain entry a is called the value of f at a. The symbol f(a) is frequently read, "f of a, or in better modern usage, f at a. " A convenient way of describing a specific function f is to give a formula which yields f(a) in terms of a for each a in the domain. For example,

describes a simple polynomial function f.

Addition, subtraction, multiplication, and division of functions are described by the general rule, Pair off equal domain entries and perform the indicated operation on the corresponding range entries. More specifically, these operations are defined by the following formulas:

It is also convenient to define a real number times a function by the formula

Note that if f and g are functions and c is a number, then f + g, f — g, fg, f/g, and cf are functions.

With regard to this last point, multiplication by a number coupled with division can lead to some confusion. Suppose that

then one is tempted to write

This, however, equates a function and a number and cannot be correct. A correct deduction from (1) is that

The function f/f is a unit constant function. Each of its range entries is equal to 1. Let

(f to the zero power) be defined as the set of all ordered pairs (a, 1) where (a, b) is a member of f. Strictly speaking f⁰ and f/f may be different in that the domain of f⁰ may contain numbers a for which f(a) = 0 while that of f/f does not. It is a useful convention to agree that given (1),

The inverse of a function is obtained by reversing each of its ordered pairs. Since this may produce duplicate first entries, the inverse of a function may not be a function. The usual practice is to delete duplicate domain entries in some systematic way to obtain a principal inverse which is a function. For commonly used functions there are established conventions for constructing principal inverses. For example, the principal inverse of the square function is the nonnegative square root function. There are well-known conventions for defining principal inverses of the trigonometric and hyperbolic functions.

Given functions f and g, the composite function

(read f circle g) is defined as the set of all ordered pairs (a, c) such that for some b, (a, b) is in g and (b, c) is in f. The composite is an iterated mapping process shown schematically in Fig. 1–1.

In terms of a formula for function values, f ˚ g may be defined as follows:

FIGURE 1–1

Composition is associative; that is,

because each of these has for its value at a, f{g[φ(a)]}. It is not commutative; as a rule

Composition is distributive over the algebraic operations from the right:

It is not distributive from the left; in general

Because of the associative law, it is unambiguous to write

To find a value of this function one applies φ, then g, then f. Note, however, to differentiate it, one proceeds from left to right. The chain rule for differentiating f ˚ g ˚ φ may be written

The identity function, I, is defined by saying that

for every real number a; that is, I pairs each real number with itself.

It is natural to introduce positive integers as exponents on function symbols in terms of multiplication; that is, f² = ff; f³ = fff; and in general fn = fn–1f. This notation and the notation for the identity function furnish self-explanatory symbols for all polynomial functions. For example, the function f defined by

may be written

Fractional powers are defined by saying that I¹/n is the principal inverse of I.n

Calculus generally deals with the elementary functions. These are the functions generated by the identity function, the sine function, and the natural logarithm function through the operations of addition, subtraction, multiplication, division, multiplication by numbers, inversion, and composition.

Roughly speaking, elementary functions are those functions for which there are standard symbols. As noted above, the symbol I furnishes polynomial symbols. The trigonometric and hyperbolic functions (sin, cos, sinh, cosh, etc.) and their inverses (arcsin, arcsinh, etc.) have well-known symbols. The usual symbol for the natural logarithm is In. The exponential and absolute-value functions are familiar ones, but symbols for them are not commonly used. Let exponential (exp) and absolute (abs) be defined by

and

respectively.

Derivative formulas are specifically designed to fit the elementary functions. The standard formulas for derivatives of combinations are:

These, together with the three specific formulas,

will yield by direct substitution all the standard derivative formulas.

Of more basic importance than the organization of formulas is the fact that the structure of the class of elementary functions leads to a concise description of the continuity and differentiability properties of these functions.

A function f is continuous at a provided that

It is differentiable at a provided that

exists. Many results in calculus depend heavily on the continuity and differentiability properties of the functions involved. To apply these results with confidence, one must be able to verify that the required conditions hold, but for most practical purposes, this does not require an exhaustive study of the limit concept. Since specific applications generally involve elementary functions, it usually suffices to know about them. The necessary information is easily outlined in terms of the three basic functions and seven operations that generate the elementary functions.

The functions I and sin are continuous everywhere, and In is continuous except at 0. Addition, subtraction, multiplication, composition, and multiplication by numbers all preserve continuity, and for functions defined on intervals, inversion preserves continuity. Division preserves continuity except where the denominator is zero.

A similar set of statements describes the differentiability properties of the elementary functions. For I and sin, the derivative exists everywhere; for In, it exists everywhere except at 0. Addition, subtraction, multiplication, composition, and multiplication by numbers all preserve differentiability. If f and g are inverses and f is differentiable, then g is also differentiable, except where f’ ˚ g is zero. Division preserves differentiability except where the denominator is zero.

In a nutshell, elementary functions are continuous except where the formula for function values indicates division by 0 or ln 0, and they are differentiable, except where the derivative formula indicates division by 0 or ln 0.

This brief summary may be a little deceptive. For example, tan is discontinuous at π/2, but the division by zero appears only if one writes it as sin/cos. However, the complete statements preceding the summary give precise criteria for determining continuity and differentiability properties of specific elementary functions.

EXAMPLES

1. Let f = I² and g = Is + I⁰. Find f(0), f(—1), g(0), g(—1). As noted above, this is done by substitution; thus

2. Let f = I² — I¹/². In terms of an arbitrary positive number a, find f(a²), f(a¹/²), [f(a)]¹/², [f(a)]². Again, substitution is called for; hence

3. In words describe the rule for computing values of the function f in Example 2. Take the square of a number, then take its square root, and finally subtract the second of these two results from the first.

4. Write

using ˚ to denote composition. Answer.

5. Write

in a form without the ˚ symbol. Answer.

6. Locate the discontinuities of

This is an elementary function and has two discontinuities. It is discontinuous at —1 because division by zero is indicated there, and it is discontinuous at 1 because In 0 is indicated there.

7. Locate the discontinuities of

Division by zero seems to be indicated at 2, but note that the domain of the function is only the interval [—1, 1]. This function is continuous on its entire domain. At — 1 and 1 the best that can be said is that the appropriate one-sided limit is equal to the function value. However, at an end point of a domain this is regarded as continuity.

EXERCISES

1. Let f and g be as follows:

Find each of the following:

(a) f + g

(b) f — g

(c) fg

(d)

(e)

(f) f(0)

(g) g(0)

(h)

(i)

(j) f⁰

(k) g⁰

2. In each of the following take the function described and find the function. values listed.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

3. For each of the following verbal descriptions of an operation, describe the indicated function in symbols and give the domain.

(a) Square a number and add 1 to the result.

(b) Add 1 to a number and square the result.

(c) Square a number and add 1, then cube the given number and subtract 1; finally, divide the first of these results by the second.

(d) Square a number, add 1, and take the square root of the result.

(e) Add 1 to a number, take the square root of the result, add 1 to this; finally, square this last result.

(f) Add 1 to the reciprocal of a number.

(g) Add a number and its reciprocal.

(h) Add 1 to a number and take the reciprocal of the result.

(i) Add 1 to the square root of a number and take the reciprocal of the result.

(j) Add 1 to the reciprocal of a number and take the square root of the result.

4. Given functions f and g as follows:

Find each of the following:

(a) f ˚ g

(b) g ˚ f

(c) f ˚ g ˚ f

(d) g ˚ f ˚ g

(e) f ˚ f

(f) g ˚ g

(g) ( f ˚ g )(0)

(h) ( g ˚ f )(0)

(i) f[g(0)]

(j) g[f(0)]

5. In each of the following take the functions described and find the function values listed.

6. Write each of the following without the symbol ˚.

(a) I ¹/² ˚ (I ² — I ⁰ )

(b) I ³ ˚ (I + I ¹/² )

(c) I ² ˚ ( I ⁰ + I ¹/² ) ˚ ( I ² + I ⁰ )

(d) I ⁴ ˚ ( I ⁰ — I ) ˚ (3 I ⁰ + I ¹/² )

(e) 3 I ⁵ ˚ ( I ² — 2 I )

(f) I ¹/² ˚ ( I ³ + 3 I ² )

(g) I ¹/² ˚ ( I ⁰ — I ) ˚ I ¹/² ˚ ( I ⁰ — I )

(h) I ³ ˚ ( I ⁰ — 2 I ¹/² ) ˚ (3 I ⁰ + I )

7. Rewrite each of the following, using ˚ to denote composition.

(a) ( I — I ¹/² ) ⁵

(b) (I — I ¹/² ) ³

(c) (I — I ³ ) ¹/²

(d)

(e)

(f)

(g)

(h)

8. Functions are often informally described as machines; that is, one takes a set of numbers, feeds it into the machine, and gets another set of numbers out at the other end. In their simplest uses, modern computers are just such machines. One gives a computer instructions for processing data, then feeds in numbers and gets others out. For the computing-machine model, use the words input, instructions, and output to describe each of the following:

(a) function, as defined in this section,

(b) rule for computing function values,

(c) domain,

(d) range,

(e) composition of functions.

9. Let f be a function whose domain is the entire real number system. Discuss the relations among

10. A standard book of numerical tables will contain tables of logarithms, logarithms of trigonometric functions, and natural trigonometric functions. These tables display functions f, g, and φ such that

Which function comes from which table? Obtain a book of tables and check your answer numerically.

11. In general f ˚ g and g ˚ f are quite different functions; so the phrase "composite of f and g" is ambiguous. Commonly used terminology is as follows: f ˚ g is referred to as "the composite function consisting of g followed by f." Discuss this terminology in light of (a) the definition, and (b) the notation.

12. Locate the discontinuities for each of the following functions.

(a) ln ˚ abs ˚ sin

(b)

(c) cot ˚ 2 I

(d)

(e)

(f) 2 I + tan

(g) tanh

(h)

(i) ln ˚ ln ˚ abs

(j)

(k)

(l) ln ˚ abs ˚ ( I ⁰ — ln)

(m)

(n) ln ˚ abs ˚ tan

(o)

(p) I ( I ⁰ — I ) ¹/³

(q)

(r)

(s)

(t) exp ˚ ln ˚ abs

13. The function abs is not differentiable at 0. (a) Form the difference quotient and show that it has different one-sided limits, (b) Deduce this from the master rule on differentiability of elementary functions. [Hint: abs = I¹/² ˚ I².]

14. How many derivatives does Im/n have at 0?

1–2 Coordinate variables. A function is often referred to as a mapping of its domain onto its range. Here the word mapping will be used to denote a generalization of the notion of function. Define a mapping as any set of ordered pairs no two of which have the same first entry. A function is then a mapping with domain and range each a set of numbers.

The foundations of analytic geometry involve certain important mappings that are not functions by this definition. Specifically, the coordinate variables x and y are mappings. If p is a point of the plane, then x(p) is its abscissa, and the symbol x, itself, stands for the set of all ordered pairs (point, abscissa). Similarly, y is a mapping that pairs each point with its ordinate.

A variable is thus defined as a mapping whose domain is a set of geometric or physical entities and whose range is a set of numbers. The abscissa and ordinate variables are the prime examples, although many others will appear as physical and geometric applications of calculus are discussed.

If x is the abscissa variable and f is a function, the composite mapping f ˚ x is defined, and a basic notion in analytic geometry is that of the locus of

This is the set of all points p such that

Here a word about notation is in order. Equation (2) is usually written y = f(x) with no precise explanation of what x and y stand for. Presumably, f(x) means the value of f at the number x. However, parentheses are also used to denote composition, so f(x) could mean the composite denoted here by f ˚ x. In adapting the modern theory of the differential to elementary calculus it is essential that the coordinate variables be recognized as mappings. Thus, in the present context whether one writes f ˚ x or f(x), it must be understood that the relation between f and x is that of composition. To emphasize this fact, the unambiguous circle notation for composition will be used throughout this introductory chapter. Later, parentheses for composition will be introduced, and the notation will assume a more familiar form, although the meanings attached to standard symbols here must be retained.

The question of notation in (2) is very superficial. A much more basic question is the following. Is equation (2) to be taken by itself and regarded as an assertion that y and f ˚ x are two symbols for the same mapping? Unfortunately, the answer is no in analytic geometry and yes in calculus.

In analytic geometry each of the variables x and y has the entire plane for its domain, and the basic idea is that of the "locus of y = f ˚ x." The entire phrase in quotes is defined immediately following equation (2), and as long as the phrase in quotes is considered as a unit, there is no trouble.

Suppose the locus of y = f ˚ x is found to be a curve C. In calculus one wants to redefine the symbols x and y so that they still map points into their abscissas and ordinates, respectively, but in such a way that each mapping has only the curve C for its domain. Once this operation is performed, it becomes quite literally true that y = f ˚ x. Now, in calculus one wants equations to be simple assertions of fact like this because one of the principal manipulative tools in calculus is substitution, and substitution of one symbol for another leaves meanings unchanged only if the two symbols stand for the same thing.

Calculus equations generally involve variables, and the most polished way to introduce them is as it was done in the preceding paragraph. That is, define the variables carefully (including specification of domain) in advance, then make a valid statement about them in the form of an equation. In practice this becomes too involved. Each change of domain introduces a new variable and should call for a new symbol. Clearly, this is not practical. What one can do is to write statements of the form

Agree that x and y always mean abscissa and ordinate, respectively; then (3) means that, if x and y have domain C, y = f ˚ x is a statement of fact and may be used for purposes of substitution.

Abscissas and ordinates are not the only useful coordinate variables on the plane. Polar coordinates must certainly be considered, and as the study of multidimensional analysis progresses, other pairs will be introduced. The polar-coordinate variables r and θ are mappings just as x and y are. For each point p in the plane, r(p) is the distance from the origin to p, and θ(p) is the angle from the positive x-direction to that of the radial line through p. It is readily verified that the equations

hold over the entire plane. Therefore, substitutions from them are valid on any figure in the plane.

It is well to consider two different coordinate systems in the plane by pointing out that, in the description of a locus, the variables involved play just as important a role as the functions. For example,

is the equation of a parabola, while

is the equation of a spiral. These equations can be written

and

respectively. The function involved is the same, although the loci are quite different.

EXAMPLES

1. Rewrite

in a form that displays the structure of the mapping. Answer.

2. Rewrite

in standard notation for variables. Answer.

EXERCISES

1. Rewrite each of the following in a form that displays the structure of the mapping.

(a)

(b)

(c) 3 + (2 — x ) ⁴

(d)

(e)

(f)

(g)

(h)

(i)

(j)

2. Rewrite each of the following in standard notation. (See Example 2 for an illustration of standard notation.)

(a) I ¹/² ˚ ( I ⁰ + I ) ˚ x

(b) I s ˚ ( I ¹/² — I ² ) ˚ x

(c) I ² ˚ (2 I — I ² ) ˚ ( I ⁰ — I ) ˚ x

(d) I ¹/² ˚ ( I ⁰ + I ¹/² ) ˚ x

(e)

(f) ( I — I ³ ) ˚ ( I — I ¹/² ) ˚ x

(g)

(h)

(i)

(j)

3. Let x and y be the usual coordinate variables; let f and g be functions; let p and q be points of the plane; let a and b be numbers. Assume that each of the functions f and g has the entire real number system for both domain and range. Each of the following symbols falls in exactly one of four categories: a number, a function, a variable, meaningless. Classify each of them.

4. Write each of the following expressions in symbols.

(a) the sum of the abscissa and ordinate variables;

(b) the product of the abscissa and ordinate variables applied to the point p;

(c) the product of the abscissa of p and the ordinate of p;

(d) the composite consisting of the abscissa variable followed by the cube of the identity function; [Note: See Exercise 11, Section 1–1.]

(e) the value at the number a of the sum of the cube of the identity function and the square of the identity function;

(f) the composite consisting of the sum of the abscissa and ordinate variables followed by the square of the identity function;

(g) the sum of the two composites consisting of the abscissa and ordinate variables, each followed by the square of the identity function;

(h) the square root of the sum of the unit constant function and the identity function;

(i) the sum of the unit constant function and the square root of the identity function.

5. Each of the following lists a pair of variables (with explanations of the symbols following). In each case name a function that maps the first variable into the second. Note that the three answers are the same. One function may appear in many different connections.

(a) s, A ; s measures the length of a side and A the area, both applied to squares.

(b) r, A/π; r measures the radius and A the area, both applied to circles.

(c) t , 2 s/g; t measures elapsed time and s measures distance traveled, both applied to freely falling bodies which start from rest; g is the constant of the acceleration due to gravity.

1–3 Derivatives and integrals. If f is a function defined on an interval containing a, the derivative of f at a is defined by

whenever this limit exists. The derivative of f is another function f’ consisting of all ordered pairs [a, f’(a)] for which f’(a) is defined. As indicated in Section 1–1, a systematic set of formulas can be compiled to yield the derivative of any elementary function.

All these considerations belong to what might be called the calculus of functions. That is, the notion of derivative and the entire battery of formulas can be studied without any mention of the notion of variable as defined in Section 1–2.

The notion of integral may also be defined strictly within the framework of the calculus of functions. Suppose f is continuous on [a, b]. Take numbers ci such that

It can be shown that there is a unique limit,

where the limit is taken over any sequence of sets {ci} for which the maximum difference, ci — ci–1, tends to zero. This limit depends on a, b, and f. It is denoted by and called the integral from a to b of f.

The fundamental theorem of calculus may also be embedded in the calculus of functions. Essentially, the theorem says that if f’ is continuous, then

Because of this theorem, derivative formulas (which are fairly easy to come by) yield a useful list of integral formulas.

If the functions involved are used in conjunction with variables as indicated in Section 1–2, a calculus of variables begins to emerge. First let us consider derivatives. Let u and v be variables, and let f be a function. Suppose that on some domain

Given this situation, the derivative of v with respect to u is denoted by Duv and defined by

The symbol Du stands for what is called a derivative operator. This is a mapping; it maps variables into variables. Specifically, for example,

hence Du maps un into nun–1.

From this point of view, successive differentiations mean successive applications of the operator Du. So, the second derivative of v with respect to u is

and this is commonly written

The nth derivative of v with respect to u is then

With regard to integrals, the situation is similar but the explanation is a little more involved. The general notion of integration of a variable will be presented in Section 2–5. Consider for the present an important special case. Let x be the abscissa variable with the horizontal axis only as a domain. Let f be a function, and think of f ˚ x as measuring vertical distances from the domain of x to the locus of y = f ˚ x. That is, for each point p on the x-axis, x(p) is the abscissa of p, and f[x(p)] is the vertical distance from p to the curve.

For each p in the domain of x there is defined a variable denoted by dxp and called the differential of x at p. A comprehensive definition of differentials will appear later (Section 2–2), but for the specific model being considered here, the following is an accurate description of dxp. Its domain is the horizontal axis, and for each q in this domain, dxv(q) is the abscissa of q with respect to p; in other words, dxp(q) is the directed distance from p to q.

In general, differentials measure distances like this, and different differentials measure them to different scales. Roughly speaking, the theory developed in Section 2–2 shows how a variable u determines the scale to which dup measures distances from p. Suffice it to say for the present that for the abscissa variable on a horizontal line, the differential measures distances to the same scale as the original variable.

Now, take the interval on the x-axis from x = a to x = b, and partition it by points p0, p1, p2, . . . , pn with x(p0) = a and x(pn) = b. Form the sum

The limit of this sum, as the number of partition points tends to infinity and the maximum distance between adjacent ones tends to zero, is denoted

and is called the integral from a to b of f ˚ x with respect