Dirichlet, Peter Gustav Lejeune 139

which can be rewritten as the

DOT PRODUCT

of two

vectors:

Dvf= f· v

where is the

GRADIENT

of f. This pro-

vides the easiest method for computing the directional

derivative of a function.

Note that f· v = |f| · |v| cos(θ), where θis the

angle between the two vectors. Since the cosine func-

tion has maximal value for θ= 0°, this shows that the

direction vof steepest slope for a graph at a point P

occurs in the direction v= f. This proves:

The vector fpoints in the direction in which f

increases most rapidly.

Similarly, the cosine function has minimal value for θ=

180°, which shows that the steepest decline occurs in

precisely the opposite direction:

The vector –fpoints in the direction in which

fdecreases most rapidly.

These ideas extend to functions of more than just two

variables.

direct proof Most claims made in mathematics are

statements of the form:

If the premise Ais true, then the conclusion B

is true.

A direct proof of such a statement attempts to establish

the validity of the claim by assuming that the premise A

is true and showing that the conclusion Bfollows from

a series of logical inferences based on Aand other pre-

viously established known facts. Typically, a direct

proof has the form:

1. Assume Ais true.

2. Show that Aimplies B.

3. Conclude that Bis true.

The main part of the proof is the demonstration that A

implies B.

As a simple example, we prove: if a natural num-

ber n is even, then n2is a multiple of 4. We will base its

proof on the known fact that any even number is a

multiple of two (as well as the standard algebraic

manipulations).

Proof: Assume that nis even.

Then ncan be written in the form n= 2k, for

some number k.

Consequently, n2= (2k)2= 4k2, and so is a

multiple of four.

This completes the proof.

An

INDIRECT PROOF

or a

PROOF BY CONTRADICTION

attempts to establish that the conclusion Bmust be true

by showing that it cannot be false.

See also

DEDUCTIVE

/

INDUCTIVE REASONING

;

CON

-

TRAPOSITIVE

;

LAWS OF THOUGHT

;

PROOF

;

QED

;

THEOREM

.

Dirichlet, Peter Gustav Lejeune (1805–1859) Ger-

man Analysis, Number theory Born on February 13,

1805, near Liège, now in Belgium (although he consid-

ered himself German), scholar Lejeune Dirichlet is

remembered for his significant contributions to the

field of

ANALYTIC NUMBER THEORY

and to the study of

F

OURIER SERIES

. In particular, he is noted for proving

that any

ARITHMETIC SEQUENCE

a, a+d, a+2d, a+3d, …

must contain an infinite number of primes, provided

the starting number aand the difference dare

RELA

-

TIVELY PRIME

. (This shows, for instance, that there are

infinitely many prime numbers that are 7 greater than a

multiple of 13.) Dirichlet was the first to provide the

modern definition of a

FUNCTION

we use today and, in

the study of trigonometric series, was the first to pro-

vide conditions that ensure that a given Fourier series

will converge. For this reason, despite the work of

J

EAN

-B

APTISTE

J

OSEPH

F

OURIER

(1768–1830), Dirichlet

is often referred to as the founder of the theory of

Fourier series.

Dirichlet graduated from the gymnasium (high

school) in Bonn at the age of 16 and went to Paris to

study mathematics. He never formally completed an aca-

demic program there and consequently never obtained a

university degree. In 1825, at the age of 20, Dirichlet

received instant fame as a worthy mathematician by

publishing a proof that there can be no positive-integer

solutions to the fifth-degree equation x5+ y5= z5. This is

a special case of F

ERMAT

’

S LAST THEOREM

, and Dirich-

let’s work on it represented the first significant step