Advanced Calculus: Second Edition

1

Partial

Differentiation

§1.Introduction

We shall be dealing in this chapter with real functions of several real variables, such as u = f(x, y), u = f(x, y, z), etc. In these examples the variables x, y, z, … are called the independent variables or arguments of the function, u is the dependent variable or the value of the function. Unless otherwise stated, functions will be assumed single-valued; that is, the value is uniquely determined by the arguments. Multiple-valued functions may be studied as combinations of single-valued ones. For example, the equation

defines two single-valued functions,

A function of two variables clearly represents a surface in the space of the rectangular coordinates x, y, u. In the study of functions of more than two variables, geometrical language is often retained for purposes of analogy, even though geometric intuition then fails.

1.1PARTIAL DERIVATIVES

A partial derivative of a function of several variables is the ordinary derivative with respect to one of the variables when all the rest are held constant. Various notations are used. The partial derivatives of u = f(x, y, z) are

An important advantage of the subscript notation is that it indicates an operation on the function that is independent of the particular letters employed for the arguments. Thus, if f(x, y, z) = xzy, we have

It shares this advantage with the familiar f′(x) for the derivative of a function of one variable. The notations for the value of a derivative at a point are illustrated by

For example,

1.2IMPLICIT FUNCTIONS

The example of §1 serves to illustrate how a function may be defined implicitly. Thus, equation (1) defines the two functions (2) and (3), which are said to be defined implicitly by (1) or explicitly by (2) and (3). In other cases, a function may be defined implicitly even though it is impossible to give it explicit form. For example, the equation

defines one single-valued function u of x and y. Given any real values of the arguments, the equation could be solved by approximation methods for u. Yet u cannot be given in terms of x and y by use of a finite number of the elementary functions.

The partial derivatives of a function defined implicitly may be obtained without using an explicit expression for the function. One has only to differentiate both sides of the defining equation with respect to the independent variable in question, remembering that the dependent variable is really a function of the independent ones. For example, differentiating equation (1) gives

These results can be checked directly by use of equation (2) or of equation (3). From equation (4) we would have

The method applies equally well if several functions are defined by simultaneous equations.

One could also solve for . To obtain the derivatives with respect to y, one has only to differentiate the defining equations with respect to that variable.

1.3HIGHER ORDER DERIVATIVES

Partial derivatives of higher order are obtained by successive application of the operation of differentiation defined above. The notations employed will be sufficiently illustrated by the following examples. If u = f(x, y, z),

A function of two variables has two derivatives of order one, four of order two, and 2n of order n. A function of m independent variables will have mn derivatives of order n. Later we shall see that many of the derivatives of a given order will be equal under very general conditions. In fact, the number of distinct derivatives of order n is then the same as the number of terms in a homogeneous polynomial in m variables of degree n:

EXERCISES(1)

1.Find .

2.Find log (r² + s).

3.Find f1(x, y) and f2(1, 2) if f(x, y) = tan² (x² − y²).

4.If z² = x² − 2xy − 1, find when x = 1, y = −2, z = −2 first by the explicit and then by the implicit method.

5.If

find .

6.If f(x, y) = x tan−1 (x² + y), find f1(1, 0), f2(x, y).

7.If f(x, y, z) = x log y² + yez, find f1(1, −1, 0), f2(x, xy, y + z).

8.If u = xyz, find .

9.If u = xu + uy, find .

10.If

find .

11.If u = xy, show that

12.Prove the statement in the text about the number of terms

in a homogeneous polynomial.

13.In Example C find .

14.If

find u(x, y) explicitly and then u1(π/2, π). Find the same derivative by the implicit function method.

15.If

find u1(x, y) when x = 1, y = −2, u = 1, v = 0, by the two methods of the previous problem.

§2.Functions of One Variable

We recall here certain notions about functions of one variable, which the student is assumed to have met before, perhaps in a less precise form. We shall also introduce certain abbreviating notations that will facilitate the statement of theorems.

2.1LIMITS AND CONTINUITY

A function f(x) approaches a limit A as x approaches a if, and only if, for each positive ∈ number e there is another, δ, such that whenever 0 < | x − a | < δ we have | f(x) − A | < ∈. That is, when x is near a (within a distance δ from it), f(x) is near A (within a distance ∈ from it). In symbols we write

EXAMPLE A.

For, in this example, we may choose δ equal to the given ∈. We have

If 0 < | x − 1 | < δ = ∈, we obtain

Here f(x) has no limit as x approaches zero. Since f(x) takes on the values −1 and +1 infinitely often in every neighborhood of the origin, it is certainly not within a distance less than 1 from any number throughout any neighborhood of the origin.

If, in the definition of limit, the first inequalities are replaced by 0 < x − a < δ (0 < a − x < δ), we say that lim f(x) = A as x approaches a from above (below) and write

It is now easy to formulate what is meant by a continuous function. Let vis first introduce the following symbols:

Definition 1. f(x) ∈ C at

This may be read, "f(x) belongs to the class of functions continuous at x = a (or f(x) is continuous at x = a) if, and only if, the limit off(x) is f(a) as x approaches a."

In Example A, the function is continuous at x = 1, since also . Observe that the last equality in Definition 1 is equivalent to

For a function to be continuous at x = a, it certainly must be defined there. Thus, f(x) = (sin x)/x is not continuous at x = 0 in the first instance, since division by zero is undefined. However, if f(0) is defined as 1, f(x) becomes continuous at x = 0. In Example B, f(x) is discontinuous at x = 0 on two counts: f(0) is undefined, and the limit involved does not exist. No choice of definition for f(0) could make f(x) continuous at x = 0.

If in Definition 1 "x → a is replaced by x → ("x → a−"), f(x) is said to be continuous on the right (left) at x = a. Thus, in Example C, f(x) is continuous on the right at x = 0 if f(0) = 0. We say that

f(x) ∈ C,a < x < b,⇔f(x) ∈ C

at each x of the interval a < x < b. Further,

f(x) ∈ C,a x b,⇔f(x) ∈ C,a < x < b

and

In Example C, with f(0) = 0,

2.2DERIVATIVES

We now introduce further classes of functions, those which have derivatives of certain orders.

These three numbers are called, respectively, the derivative, the derivative on the right, and the derivative on the left off(x) at x = a. For example, if f(x) = | x |, then f′(0) does not exist, but f′+(0) = 1 and f′−(0) = − 1. Distinguish between f′+(a) and f′(a+) = .

The latter limit clearly does not exist.

Higher derivatives are defined in the obvious way by successive application of Definition 2.

It is easy to see that when f′(x) exists then f(x) ∈ C. Hence, if f(x) ∈ Cn, we also have

f(x) ∈ Ck for k = 0, 1, 2, …, n − 1 (C⁰ = C)

These examples show how to construct a function f(x) ∈ Cn for which f(x) ∉ Cn + ¹. Note the difference between Example E and the first case under Example F. These two functions fail to belong to C¹ for different reasons. The first has a derivative at every point but this derivative is not continuous at x = 0, the second has no derivative at x = 0. This suggests that it would be profitable to define a class of functions between C and C¹. This is, in fact, the case (see §3.6). In the interests of simplicity, we shall not use a special letter for the class.

2.3ROLLE’S THEOREM

CASE I. f(x) ≡ 0. Then f′(x) = 0 for all x.

CASE II. f(x) 0. Then there is a number c, a < c < b, where f(c) ≠ 0. If f(c) > 0 (< 0), then f(x) has an absolute maximum* (minimum) at a point ξ, a < ξ < b. Hence,

Allowing Δx to approach zero, we see by hypothesis 1 that both quotients approach f′(ξ), which must therefore be nonnegative and nonpositive. Hence f′(ξ) = 0. Observe that hypotheses 1 and 2 are satisfied if f(x) ∈ C¹ in a x b. The latter weaker hypothesis is often sufficient for applications.

2.4LAW OF THE MEAN

The function

satisfies all hypotheses of Theorem 1. The conclusion φ′(ξ) = 0 leads at once to the desired result. One can easily see the origin of the function φ(x) by observing that it gives the length of the line segment AB in Figure 1.

If we set a = c, b = c + h or if we set b = c, a = c + h (h < 0), the law becomes in either case

Fig. 1.

EXERCISES(2)

1.Evaluate the following limits if they exist:

2.If [x] means "the largest integer x," is the function [x] continuous on the right or continuous on the left at x = 2? Does

3.The function [x] is continuous in which of the following intervals:

0 < x < 1,0 x 1,0 x < 1,0 < x 1?

4.Does the function f(x) = | x | ³ belong to C²? Compute f′″+0 and f′″−0.

5.Define f(x) of Example C as 0 at x = 0 and compute f′+(0). You may assume that te−t → 0 as t → +∞.

6.Find ξ in the law of the mean if

f(x) = tan−1 x, a = 1, b = ,

and show that it lies in the required interval.

7.Find ξ in Rolle’s theorem for f(x) = x³(1 − x)⁵, and show that it lies in the required interval.

8.Find θ in the law of the mean for f(x) = Ax² + Bx + C, (A ≠ 0), and show that 0 < θ < 1.

9.If f(x) = , show that f(x) satisfies the hypotheses of Rolle’s theorem for a = 0, b = 1. Find ξ. Show that f(x) ∉ C¹ in 0 x 1.

10.If f(x) = x sin (1/x), x ≠ 0, f(0) = 0, show that f(x) satisfies the hypotheses of Rolle’s theorem for a = 0, b = 1/π. How many values of ξ exist? Hint: Apply Rolle’s theorem on smaller intervals.

11.In the previous exercise does f(x) ∈ C¹ in 0 x 1/π?

12.Formulate exact definitions for the following:

13.Construct f(x) such that f(x) ∈ Cn, f(x) ∉ Cn + ¹.

14.Prove by the method employed for Example H that

15.Prove from the definition of limit that a function cannot have two different limits as its independent variable approaches a limit.

16.Prove that the existence of f′(a) implies the continuity of f(x) at x = a.

17.If f(x) = [x] as in Exercise 2, show that f′(2−) = 0 and that f′−(2) does not exist.

18.If f(x) ∈ C¹ in a x b, show that f′+(a) = f′(a+). Hint: Use the law of the mean.

19.If [x] is defined as in Exercise 2, is [1 − [x]] continuous on the right (left)? Answer the question for each x.

§3.Functions of Several Variables

We now proceed with a systematic treatment of partial differentiation. We develop first the method of differentiating composite functions analogous to

for functions of one variable.

3.1LIMITS AND CONTINUITY

We begin by defining the limit of a function of two variables. A function f(x, y) approaches a limit A as x approaches a and y approaches b,

if, and only if, for each positive number ∈ there is another, δ, such that whenever | x − a | < δ, | y − b | < δ, 0 < (x − a)² + (y − b)² we have | f(x, y) − A | < ∈. That is, when (x, y) is at any point inside a certain square with center at (a, b) and width 2δ (except at the center), f(x, y) differs from A by less than ∈.

EXAMPLE A.f(x, y) = x² + y²

Given ∈, we may choose δ = . For, the inequalities | x | < , | y | < imply (x² + y²) < ∈. Hence,

then f(x, y) approaches no limit as (x, y) approaches the origin. For f(x, y) is as large as we like at points near the line x = −y. On the other hand, observe that

Any collection of points (x, y) is called a point set. The set of points | x − a | < δ, | y − b | < δ is known as an open square or two-dimensional interval or a δ-neighborhood of the point (a, b). A point (a, b) is a limit point of a set S if every δ-neighborhood of (a, b) contains points of S different from (a, b). A set S is closed if it contains all its limit points. A point is an interior point of S if it is the centre of a δ-neighborhood composed entirely of points of S. A set is open if it is composed entirely of interior points. For example, if S is the set of points (x, y) for which x² + y² < a², S is open. Limit points of this set not in it are those for which x² + y² = a². The boundary of a set is the set of all limit points not interior points.

A domain is an open set, any two of whose points can be joined by a broken line having a finite number of segments, all of whose points belong to the set. A region is either a domain or a domain plus some or all of its boundary. If it contains all of its boundary, it is a closed region.

We say that f(x, y) ∈ C in a domain D if, and only if, f(x, y) ∈ C at each point of D. Also f(x, y) ∈ C at a boundary point (a, b) of a region R where f(x, y) is defined if, and only if,

That is, the point (x, y) approaches (a, b) only through points of R. This corre sponds to one-sided approach for functions of one variable. Then f(x, y) ∈ C in R if f(x, y) ∈ C at each point of R.

3.2DERIVATIVES

We now define the classes Cn for functions of several variables. We first give limit definitions of the partial derivatives described in §1. We use the letter R to indicate a region.

Definition 6.

It can be shown that if f(x, y) satisfies the condition of this definition, then f(x, y) ∈ Ck (k = 0, 1, 2, …, n − 1), just as for functions of a single variable.

3.3A BASIC MEAN-VALUE THEOREM

We are now able to establish a result of fundamental importance in the theory of partial differentiation. It may be considered analogous to Theorem 2, the law of the mean for functions of a single variable. We shall the letter D to indicate a domain.

Set

and rewrite it as follows:

Δf = [f(a + Δx, b) − f(a, b)] + [f(a + Δx, b + Δy) − f(a + Δx, b)].

Here we have added and subtracted f(a + Δx, b) on the right-hand side of equation (1). Now apply Theorem 2 to the function f(x, b) of the single variable x. Its derivative is f1(x, b). We thus obtain for the first bracket above

Fig. 2.

Next apply the same theorem to the function f(a + Δx, y). We thus obtain

There is no reason to suppose that θ1 = θ2, and in general these two numbers will be different. A more symmetric form of the law of the mean, a form involving a single θ, will appear in §9, equation (4).

Observe the force of hypothesis 2. If we were to replace it by the hypothesis that (a, b) and (a + Δx, b + Δy) are both points of D, equation (2) might not be true. A glance at Figure 2 will show why.

We can determine for this particular example the exact values of θ1 and θ2, Δx > 0,

3.4COMPOSITE FUNCTIONS

We use the result of Theorem 3 to differentiate a function of functions, one case of which is stated in the following theorem.

Theorem 4.1.f(x, y), g(r, s), h(r, s) ∈ C¹

The regions in which the given function ∈ C¹ are not stated, in the interests of simplicity. It is understood, of course, that the region for (r, s) and the one for (x, y) must be such that the functions g, h can be substituted in f(x, y) to form

f(g(r, s), h(r, s))

From the definition of a partial derivative, we have

Δf = f(g(r0 + Δr, s0), h(r0 + Δr, s0)) − f(g(r0, s0), h(r0, s0)).

Now apply Theorem 3, setting

By the continuity of g and h, we see that Δx and Δy tend to zero with Δr. We have

Now let Δr approach zero and make use of Definition 5 and Definition 6 to obtain

Replacing x0, y0 by their values and dropping subscripts, we have equation (3). Equation (4) is obtained in a similar way. The results are easily remembered by putting them in the following form, analogous to the second equation of this section:

EXAMPLE D.f(x, y) = xy,f1 = y,f2 = x

Thus the rule for differentiation of a product is the same whether the factors are functions of one or of two variables, a fact which is also evident from the definition of a partial derivative.

3.5FURTHER CASES

The following cases are proved in a manner analogous to that used for the proof of Theorem 4:

CASE I.

CASE II.

CASE III.

To prove these and analogous results, one must use Theorem 3 and suitable modifications thereof (Theorem 2 or Exercise 6 of the present section). Distinguish carefully between total and partial derivatives.

3.6DIFFERENTIABLE FUNCTIONS

For a more delicate analysis it is useful to introduce a class of functions lying between C and C¹, the class of differentiable functions. The name might suggest that f(x, y) is differentiable if f1 and f2 exist. Actually it is desirable to require somewhat more. We define f(x, y) to be differentiable at (a, b) ⇔ f1(a, b), f2(a, b) exist and if

where φ(Δx, Δy) and ψ(Δx, Δy) → 0 as (Δx, Δy) → (0, 0). For example, if

f(x, y) = 2 − y + 2x² − x²y

f is differentiable at every point. In particular at (0, 0), f = 2, f1 = 0, f2 = −1, and

f(Δx, Δy) − f(0, 0) = f2(0, 0) Δy + (2Δx − Δx Δy) Δx.

Hence we may take φ(Δx, Δy) = 2 Δx − Δx Δy and ψ(Δx, Δy) = 0. Or we could choose φ = 2 Δx, ψ = −(Δx)², and in either case φ and ψ have the desired limit zero.

Let us observe first that if f(x, y) ∈ C¹ in D then f(x, y) is differentiable at every point of D. For, if we set

equation (5) becomes equation (2), and from the assumed continuity of f1 and f2 the functions φ and ψ tend to zero as (Δx, Δy) → (0, 0).

EXAMPLE G.f(x, y) = | x | (1 + y) ∈ C at (0, 0) but is not differentiable there. For f1(0, 0) does not exist.

EXAMPLE H.f(x, y) = x when | y | < | x |, f(x, y) −x when | y | | x |. Here f1(0, 0) = 1, f2(0, 0) = 0. Now if f(x, y) were differentiable at (0, 0), equation (5) would become when Δy = Δx

But this is a contradiction, as one sees by canceling Δx and letting Δx → 0. Hence f(x, y) is not differentiable at (0, 0). It is continuous there.

This function is differentiable at (0, 0) (see Exercise 20). It does not belong to C¹ there. To prove this, it is enough to show that f1(x, x) has no limit as x → 0+. But

and we saw in Example E, §2, that g′(0+) does not exist.

Let us summarize the results just obtained.

I.f ∈ C¹ ⇒ f is differentiable (Theorem 3).

II.f is differentiable ⇒ f ∈ C (Exercise 17).

III.There exist continuous functions not differentiable (Example G).

IV.There exist nondifferentiable functions having partial derivatives (Example H).

V.There exist differentiable functions not belonging to C¹ (Example I).

EXERCISES(3)

1.If

u = x² + y², x = r cos θ, y = r sin θ

compute and first by Theorem 4 and then by eliminating x and y.

2.Same problem if u = xy.

3.Compute the derivative of u with respect to t if

u = xyz, x = t, y = t², z = t³

4.Find the numbers θ1 and θ2 of Theorem 3, if

f(x, y) = x² + 3xy + y², a = b = 0, Δx = 1, Δy = −1

5.Same problem if

f = exy, a = b = Δx = Δy = 1

6.Prove a theorem analogous to Theorem 3 for a function of three variables.

7.Prove Case I.

8.Prove Case II.

9.Prove Case III.

10.If

u = f(x, y), x = g(r, s), y = h(t)k(r)

find

11.If u = f(x, y), x = r cos θ, y = r sin θ, show that

Explain the exact meaning of the equation, dissolving the mystery of a function of (x, y) equated to a function of (r, θ).

12.

13.

14.In Example C, compute the limit of θ1 as Δx → 0.

15.If f(x, y) ∈ C at (a, b), show that f(x, b) is continuous at x = a and f(a, y) is continuous at y = b.

16.If f(x, y) = 1 when xy ≠ 0 and = 0 when xy = 0, show that f(x, 0) ∈ C for all x and f(0, y) ∈ C for all y but that f(x, y) ∉ C at (0, 0). Show that

17.Show that if f(x, y) is differentiable at a point it is continuous there. Hint: Let (Δx, Δy) → (0, 0) in (5).

18.If

show that f(x, y) is differentiable at (0, 0).

19.If , show thatf(x, y) is not differentiable at (0, 0).

20.In Example I show that f1(0, 0) = f2(0, 0) = 0. Show that f(x, y) is differentiable at (0, 0). Hint:

φ = Δx sin (Δx² + Δy²)−1/2, ψ = Δy sin (Δx² + Δy²)−1/2

21.Theorem 4 is still true if f is only differentiable and if g and h possess partial derivatives. Prove.

§4.Homogeneous Functions. Higher Derivatives

A polynomial in x and y is said to be homogeneous if all its terms are of the same degree. For example,

f(x, y) = x² − 2xy + 3y²

is homogeneous. It is easy to generalize the property so that functions not polynomials can have it. Observe, in the above example, that

f(λx, λy) = λ²f(x, y)

for any positive number λ. We use this characteristic of homogeneous polynomials to make the generalization. The definition is stated for a function of two variables, but it is easily altered to apply to a function of any number of variables.

4.1DEFINITION OF HOMOGENEOUS FUNCTIONS

Definition 7. A function f(x, y) is homogeneous of degree n in a region R if, and only if, for (x, y) in R and for every positive value of λ

The number n is positive, negative, or zero and need not be an integer. In the above example n = 2 and R is the whole xy-plane. The region R must be such that (λx, λy) is a point of it for all positive λ whenever (x, y) is a point of it. For example, R may be an angular region between two infinite rays emanating from the origin or the whole plane.

EXAMPLE A.f(x, y) = x¹/³y−4/3 tan−1 (y/x)

Here n = − 1; R is any quadrant without the axes.

EXAMPLE B.f(x, y) = 3 + log (y/x)

This function is homogeneous of degree 0; R is the first or third quadrant without the axes.

EXAMPLE C. f(x, y) =

Here n = 3; R is the whole plane. Observe that if λ is a negative number, equation (1) is not satisfied for this function. For,

f(λx, λy) = | λ | ³ f(x, y)

EXAMPLE D.f(x, y) = x¹/³ y−2/3 + x²/³ y−1/3

This function is not homogeneous.

4.2EULER’S THEOREM

To prove this, differentiate equation (1) partially with respect to λ,

xf1(λx, λy) + yf2(λx, λy) = nλn − ¹ f(x, y)

Finally, set λ = 1.

We point out in passing that certain authors* define homogeneity in a different way, demanding that equation (1) should hold for all real values of λ. With this definition the function of Example C is not homogeneous. But this definition would have the disadvantage that the converse of Euler’s theorem would be false, whereas we shall now prove that the converse is valid under Definition 7.

It is to be understood in this theorem that R is the type of region described under Definition 7. Choose (x0, y0) an arbitrary point of R, and form the function

φ(λ) = f(λx0, λy0)

defined for all positive values of λ. Then by hypothesis 2

Now differentiate φ(λ)λ−n with respect to λ, and obtain

[φ(λ)λ−n]′ = φ′(λ)λ−n − nφ(λ)λ− n − ¹

The right-hand side of this equation is zero by virtue of the previous equation. Hence,

φ(λ)λ− n = C

where C is a constant which may be determined by setting λ = 1,

Since (x0, y0) was an arbitrary point of R, the theorem is proved.

4.3HIGHER DERIVATIVES

Higher derivatives of composite functions may be computed by the principles already at our disposal. As an example, let us compute the three derivatives of order two for the function u = f(φ(r, s), ψ(r, s)). We assume that the three functions involved belong to C².

Differentiating again, remember that f1 and f2 are themselves composite

We have omitted the arguments in these functions to save space. In each φ or ψ with any subscript, they are (r, s); in each f they are (φ(r, s), ψ(r, s)). If we admit that f12 = f21, φ12 = ψ21, φ12 = ψ21, facts that we shall prove later, we see that . This will also be evident later without computation.

EXAMPLE E.

From the formulas above, we have, for example,

This result can be checked directly by eliminating x and y before differentiating.

This result could also be obtained from equation (4) by replacing φ(r, s) by g(t) φ(r, s) by h(t) etc.

EXERCISES(4)

1.Verify Theorem 5 for Examples A and B by computing both sides of Euler’s equation directly.

2. Which of the following functions are homogeneous?

(a)

(b)log y − log x

(c)(x³ + y³)²/³