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.
(iv) If f (x) ’ a and g(x) ’ b, with respect to the direction , then,
f (x) + g(x) ’ a + b.
(v) If f (x) ’ a and g(x) ’ b, with respect to the direction , then
f (x)g(x) ’ ab.
(vi) Suppose that f (x) ’ a, with respect to the direction . If f (x) = 0
for each x ∈ X and a = 0, then f (x)’1 ’ a’1 .
(vii) If f (x) ¤ A for each x ∈ X and f (x) ’ a, with respect to the
direction , then a ¤ A. If g(x) ≥ B for each x ∈ X and g(x) ’ b, with
respect to the direction , then b ≥ B.

As one might expect, we can recover Lemma 1.2.2 from Exercise D.8.

Exercise D.9. (i) If N+ is the set of strictly positive integers, show that >
(with its ordinary meaning) is a direction on N+ . Show further that, if f is
a function from N+ to F (an ordered ¬eld) and a ∈ F, then f (n) ’ a, with
respect to the direction >, if and only if f (n) ’ a as n ’ ∞ in the sense of
De¬nition 1.2.1.
(ii) Deduce Lemma 1.2.2 from Exercise D.8.
(iii) Show that (i) remains true if we replace > by ≥. Show that (i)
m if n ≥ 10m + 4. Thus
remains true if we replace > by with n
di¬erent succession relations can produce the same notion of limit.

The real economy of this approach appears when we extend it.

Exercise D.10. (i) Let a, t and b be real with a < t < b. Show that, if we
on (a, b) \ {t} by x y if |x ’ t| < |y ’ t|, then
de¬ne the relation
is a direction. Suppose f : (a, b) ’ R is a function and c ∈ R. Show that
f (x) ’ c with respect to the direction , if and only if f (x) ’ c as x ’ t,
in the traditional sense of De¬nition D.3.
398 A COMPANION TO ANALYSIS

(ii) Deduce the properties of the traditional limit of De¬nition D.3 from
Exercise D.8.
(iii) Give a treatment of the classical ˜limit from above™ de¬ned in De¬-
nition D.4 along the lines laid out in parts (i) and (ii).

Exercise D.11. Obtain a multidimensional analogue of De¬nition D.7 along
the lines of De¬nition 4.1.8 and prove a multidimensional version of Exer-
cise D.8 along the lines of Lemma 4.1.9.

A little thought shows how to bring De¬nition D.2 into this circle of ideas.

De¬nition D.12. If X is a direction on a non-empty set X, Y is a di-
rection on a non-empty set Y and f is a function from X to Y , we say that
f (x) ’ —Y as x ’ —X if, given y ∈ Y , we can ¬nd x0 (y) ∈ X such that
f (x) Y y for all x x0 (y).

Exercise D.13. (i) Show that if we take X = N+ , X to be the usual rela-
tion > on X, Y = R and Y to be the usual relation > on Y , then saying that
a function f : X ’ Y has the property f (x) ’ —Y as x ’ —X is equivalent
to the classical statement f (n) ’ ∞ as n ’ ∞.
(ii) Give a similar treatment for the classical statement that a function
f : R ’ R satis¬es f (x) ’ ’∞ as x ’ ∞.
(iii) Give a similar treatment for the classical statement that a function
f : R ’ R satis¬es f (x) ’ a as x ’ ∞.

In all the examples so far, we have only used rather simple examples of
direction. Here is a more complicated one.

Exercise D.14. This exercise assumes a knowledge of Section 8.2 where we
de¬ned the Riemann integral. It uses the notation of that section. Consider
the set X of ordered pairs (D, E) where D is the dissection

D = {x0 , x1 , . . . , xn } with a = x0 ¤ x1 ¤ x2 ¤ · · · ¤ xn = b,

and

E = {t0 , t1 , . . . , tn } with xj’1 ¤ tj ¤ xj .

If f : [a, b] ’ R is a bounded function, we write
n
σ(f, D, E) = f (tj )(xj ’ xj’1 ).
j=1
399
Please send corrections however trivial to twk@dpmms.cam.ac.uk

Show that, if we write (D , E ) (D, E) when D ⊇ D, then is a
direction on X. (Note that we place no conditions on E and E .) Show that
not all dissections are comparable.
Using the results of Section 8.2, show that f is Riemann integrable, in
the sense of Section 8.2, with integral I if and only if
σ(f, D, E) ’ I
with respect to the direction .
Here is another example, this time depending on the discussion of metric
spaces in Section 10.3. If we wish to de¬ne the notion of a limit for a func-
tion between two metric spaces the natural classical procedure is to produce
something along the lines of De¬nition 10.3.22
De¬nition D.15. Let (X, d) and (Z, ρ) be metric spaces and f be a map
from X to Z. Suppose that x ∈ X and z ∈ Z. We say that f (y) ’ z as
y ’ x if, given > 0, we can ¬nd a δ( , x) > 0 such that, if y ∈ X and
d(x, y) < δ( , x), we have
ρ(f (y), z) < .
Here is an alternative treatment using direction.
Exercise D.16. Let (X, d) and (Z, ρ) be metric spaces and x ∈ X and z ∈
Z. We take X to be the collection of open sets in X which contain x and Z
to be the collection of open sets in Z which contain z.
Show that if we de¬ne a relation on X by U X V if V ⊇ U , then X is
a direction on X . Is it always true that two elements of X are comparable?
Let Z be de¬ned similarly. If f is a map from X to Z, show that
f (y) ’ z as y ’ x in the sense of De¬nition D.15 if and only if f (y) ’ —Z
as x ’ —X in the sense of De¬nition D.12.
The advantage of the approach given in Exercise D.16 is that it makes
no reference to the metric and raises the possibility of of doing analysis on
more general objects than metric spaces.
We close with a couple of interesting observations (it will be more conve-
nient to use De¬nition D.7 than our more general De¬nition D.12).
Exercise D.17. Suppose that is a direction on a non-empty set X and f
is a function from X to R.
Suppose that there exists an M ∈ R such that f (x) ¤ M for all x ∈ X and
suppose that f is ˜increasing™ in the sense that x y implies f (x) ¤ f (y).
Show that there exists an a ∈ F such that that f (x) ’ a with respect to the
direction .
[Hint. Think about the supremum.]
400 A COMPANION TO ANALYSIS

If the reader thinks about the matter she may recall points in the book
where such a result would have been useful.

Exercise D.18. Prove Lemma 9.2.2 using using Lemma D.17.

Exercise D.19. The result of Lemma D.17 is the generalisation of the state-
ment that every bounded increasing sequence in R has a limit. Find a similar
generalisation of the general principle of convergence. Use it to do Exer-
cise 9.2.10.

If the reader wishes to see more, I refer her to the elegant and e¬cient
treatment of analysis in [2].
Appendix E

Traditional partial derivatives

One of the most troublesome culture clashes between pure mathematics and
applied is that to an applied mathematician variables like x and t have mean-
ings such as position and time whereas to a pure mathematician all variables
are ˜dummy variables™ or ˜place-holders™ to be interchanged at will. To a pure
mathematician, v is an arbitrary function de¬ned by its e¬ect on a variable
so that v(t) = At3 means precisely the same thing as v(x) = Ax3 whereas,
to an applied mathematician who thinks of v as a velocity, the statements
v = At3 and v = Ax3 mean very di¬erent (indeed incompatible) things.
dv δv
The applied mathematician thinks of as representing ˜when every-
dt δt
thing is so small that second order quantities can be neglected™. Since
δv δt δv
= ,
δt δx δx
it is obvious to the applied mathematician that
dv dv dx
= , (A)
dt dx dt
but the more rigid notational conventions of the pure mathematicians prevent
them from thinking of v as two di¬erent functions (one of t and one of x) in
the same formula. The closest a pure mathematician can get to equation (A)
is the chain rule
d
v(x(t)) = v (x(t))x (t).
dt
Now consider a particle moving along the x-axis so its position is x at
time t. Since
δx δt
= 1,
δt δx
401
402 A COMPANION TO ANALYSIS

it is obvious to the applied mathematician that
dx dt
= 1,
dt dx
and so
dt dx
=1 (B)
dx dt

dx
What does equation (B) mean? In the expression we treat x as a function
dt
dt
of t, which corresponds to common sense, but in the expression we treat t
dx
as a function of x, which seems a little odd (how can the position of a particle
in¬‚uence time?). However, if the particle occupies each particular position
at a unique time, we can read o¬ time from position and so, in this sense, t
is indeed a function of x. A pure mathematician would say that the function
x : R ’ R is invertible and replace equation (B) by the inverse function
formula
d ’1 1
x (t) = .
x (x’1 (t))
dt

One of the reasons why this formula seems more complicated than equa-
dx dt
tion (B) is that the information on where to evaluate and has been
dt dx
suppressed in equation (B), which ought to read something like

dt dx
(x0 ) = 1 (t0 ) ,
dx dt
or
dt dx
=1 ,
dx dt
x=x0 t=t0

where the particle is at x0 at time t0 .
The clash between the two cultures is still more marked when it comes
to partial derivatives. Consider a gas at temperature T , held at a pressure
P in a container of volume V and isolated from the outside world. The
applied mathematician knows that P depends on T and V and so writes P =
P (T, V ) or P = P (V, T ). (To see the di¬erence in conventions between pure
and applied mathematics, observe that, to a pure mathematician, functions
f : R2 ’ R such that f (x, y) = f (y, x) form a very restricted class!) Suppose
403
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that, initially, T = T0 , P = P0 , V = V0 . If we change T0 to T0 + δT whilst
keeping V ¬xed, then P changes to P0 + δP and the applied mathematician
‚P δP
thinks of as representing ˜when everything is so small that
‚T V =V0 ,T =T0 δT
‚P
second order quantities can be neglected™. In other words, is
‚T V =V0 ,T =T0
the rate of change of P when T varies but V is kept ¬xed. Often applied
mathematicians write
‚P ‚P
=
‚T ‚T V =V0 ,T =T0

but you should note this condensed notation suppresses the information that
V (rather than, say, V + T ) should be kept constant when T is varied.
There is good reason to suppose that there is a well behaved function
g : R3 ’ R such that, if a particular gas is at temperature T , held a pressure
P in a container of volume V , then the temperature, pressure, volume triple
(T, P, V ) satis¬es

g(T, P, V ) = 0

(at least for a wide range of values of T , P and V ). We may link the pure
mathematician™s partial derivatives with those of the applied mathematician
by observing that, if (T0 , P0 , V0 ) and (T0 + δT, P0 + δP, V0 ) are possible
temperature, pressure, volume triples, then, to ¬rst order,

0 = g(T0 + δT, P0 + δP, V0 )
= g(T0 , P0 , V0 ) + g,1 (T0 , P0 , V0 )δT + g,2 g(T0 , P0 , V0 )δP
= g,1 (T0 , P0 , V0 )δT + g,2 (T0 , P0 , V0 )δP

and so, to ¬rst order,
δP g,1 (T0 , P0 , V0 )
=’ .
δT g,2 (T0 , P0 , V0 )
It follows that
‚P g,1 (T0 , P0 , V0 )
=’ .
‚T g,2 (T0 , P0 , V0 )
V =V0 ,T =T0

Essentially identical calculations show that
‚T g,3 (T0 , P0 , V0 )
=’
‚V g,1 (T0 , P0 , V0 )
P =P0 ,V =V0
404 A COMPANION TO ANALYSIS

and
‚V g,2 (T0 , P0 , V0 )
=’ .
‚P g,3 (T0 , P0 , V0 )
T =T0 ,P =P0

Putting the last three equations together, we obtain
‚P ‚T ‚V
= ’1.
‚T ‚V ‚P
V =V0 ,T =T0 P =P0 ,V =V0 T =T0 ,P =P0

This is a very beautiful equation. It can be made much more mysterious by
leaving implicit what we have made explicit and writing
‚P ‚T ‚V
= ’1. (C)
‚T ‚V ‚P
If we further neglect to mention that T , P and V are restricted to the surface
g(T, P, V ) = 0, we get ˜an amazing result that common sense could not
possibly have predicted . . . It is perhaps the ¬rst time in our careers, for
most of us, that we do not understand 3-dimensional space™ ([34], page 65).
Exercise E.1. Obtain the result corresponding to equation (C) for four vari-
ables. Without going into excessive details, indicate the generalisation to n
variables with n ≥ 2. (Check that if n = 2 you get a result corresponding to
equation (B).)
The di¬erence between the ˜pure™ and ˜applied™ treatment is re¬‚ected in
a di¬erence in language. The pure mathematician speaks of ˜di¬erentiation
of functions on many dimensional spaces™ and the applied mathematician of
˜di¬erentiation of functions of many variables™. Which approach is better de-
pends on the problem at hand. Although (T, P, V ) is a triple of real numbers,
it is not a vector, since pressure, volume and temperature are quantities of
di¬erent types. Treating (T, P, V ) as a vector is like adding apples to pears.
The ˜geometric™ pure approach will yield no insight, since there is no ge-
ometry to consider. On the other hand theories, like electromagnetism and
relativity with a strong geometric component, will bene¬t from a treatment
which does not disguise the geometry.
I have tried to show that the two approaches run in parallel but that, al-
though statements in one language can be translated ˜sentence by sentence™
into the other, there is no word for word dictionary between them. A good
mathematician can look at a problem in more than one way. In particular a
good mathematician will ˜think like a pure mathematician when doing pure
mathematics and like an applied mathematician when doing applied math-
ematics™. (Great mathematicians think like themselves when doing mathe-
matics.)
405
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Exercise E.2. (In this exercise you should assume that everything is well
behaved.) Rewrite the following statement in pure mathematics notation and
prove it using whichever notation (pure, applied or mixed) you prefer.
Suppose that the equations

f (x, y, z, w) = 0
g(x, y, z, w) = 0

can be solved to give z and w as functions of (x, y). Then

‚f ‚g ‚f ‚g

‚z ‚x ‚w ‚w ‚x
=’ ‚f ‚g ‚f ‚g
‚x ’
‚z ‚w ‚w ‚z
‚f ‚g ‚f ‚g

‚z ‚y ‚w ‚w ‚y
=’ .
‚f ‚g ‚f ‚g
‚y ’
‚z ‚w ‚w ‚z


Exercise E.3. Each of the four variables p, V , T and S can be regarded as a
well behaved function of any two of the others. In addition we have a variable
U which may be regarded as a function of any two of the four variables above
and satis¬es

‚U ‚U
= ’p.
= T,
‚S ‚V
V S

(i) Show that

‚V ‚T
= .
‚S ‚p
p S


‚2U
(ii) By ¬nding two expressions for , or otherwise, show that
‚p‚V

‚S ‚T ‚S ‚T
’ = 1.
‚V ‚p ‚p ‚V
p V V p


Exercise E.4. You should only do this exercise if you have met the change
of variable formula for multiple integrals. Recall that, if (u, v) : R 2 ’ R2 is
a well behaved bijective map, we write
‚u ‚u
‚(u, v) ‚x ‚y
J= = det ‚v ‚v
‚(x, y) ‚x ‚y
406 A COMPANION TO ANALYSIS

and call J the Jacobian determinant1 . Recall the the useful formula

‚(u, v) ‚(x, y)
= 1.
‚(x, y) ‚(u, v)

Restate it in terms of Df and Df ’1 evaluated at the correct points. (If you
can not see what is going on, look at the inverse function theorem (Theo-
rem 13.1.13 (ii)) and at our discussion of equation (B) in this appendix.)
Restate and prove the formula

‚(u, v) ‚(s, t) ‚(u, v)
=
‚(s, t) ‚(x, y) ‚(x, y)

in the same way.




1
J is often just called the Jacobian but we distinguish between the Jacobian matrix
(see Exercise 6.1.9) and its determinant.
Appendix F

Another approach to the
inverse function theorem

The object of this appendix is to outline a variant on the approach to the
inverse function theorem given in section 13.1.
We begin with a variation on Lemma 13.1.2.

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