144 div

k

(a+ b) ×c= c×(a+ b)

= c×a+ c×b

= a×c+ b×c

In

SET THEORY

, “intersection” distributes over “union”

both from the left and from the right:

A∩(B∪C) = (A∩B)∪(A∩C)

(A∪B)∩C= (A∩C)∪(B∩C)

As a mnemonic device, it is helpful to think of the

phrase “distributes over” as synonymous with “sprin-

kles over.” We have: multiplication “sprinkles over”

additions and intersection “sprinkles over” union. In

any

RING

, the distributive property is the single axiom

that combines the two defining operations.

See also

ASSOCIATIVE

;

COMMUTATIVE PROPERTY

.

div A

VECTOR FIELD

assigns to every point (x,y,z) in

space a vector F= <f1(x,y,z),f2(x,y,z),f3(x,y,z)>. The

divergence of F, denoted div F, is the quantity:

given as a sum of

PARTIAL DERIVATIVE

s. Physicists have

shown that this quantity represents the amount of flux

leaving an element of volume in space. For example, if

Frepresents the velocity field of a turbulent fluid, and ρ

is the density of the fluid, then ρdivF, calculated at a

point P, is the rate at which mass is lost from an

(infinitely small) box drawn around P. (At any instant,

fluid is flowing into and out of this box.)

The divergence operator is often written as though

it is a

DOT PRODUCT

of two vectors:

divF= · F

where is the del operator. The

CROSS

PRODUCT

of with Fis called the curl of F:

If Fis the velocity field of a turbulent liquid, then

physicists have shown that 1/2 curl F, calculated at a

point P, is the angular velocity of an element of fluid

located at P, that is, it is a measure of the amount of

turning it undergoes.

See also

GRADIENT

.

divergent This term simply means “does not con-

verge.” For example, an infinite

SEQUENCE

is said to

diverge if it has no

LIMIT

, and an infinite

SERIES

diverges if the sequence of partial sums diverges.

A divergent sequence is said to be “properly diver-

gent” if it tends to infinity. For example, the sequence

1,2,3, … is properly divergent (but the sequences 1,

–1,1, –1,1, … and 1, –2,3, –4,5, … are not). One also

describes a series as properly divergent if the corre-

sponding sequence of partial sums has this property.

For example, the series 1 + 2 + 4 + 8 + 16 +… is prop-

erly divergent.

An

INFINITE PRODUCT

is divergent if it has value

zero or does not converge. An

IMPROPER INTEGRAL

diverges if the limit defining it does not exist.

See also

CONVERGENT SEQUENCE

.

divisibility rules A number is said to be divisible by

nif, working solely within the integers, the number

leaves a remainder of zero when divided by n. For

example, 37 leaves a remainder of 1 when divided by

3, and so is not divisible by 3. On the other hand, 39,

leaving a remainder of zero, is divisible by 3.

There are a number of rules to quickly test the

divisibility of numbers by small integers. We present

divisibility rules for the first 12 integers.

Divisibility by 1

All numbers are divisible by 1.

Divisibility by 2

As all multiples of 10 are divisible by 2, it suffices to

check whether or not the final digit of a number is divisi-

ble by 2. For example, 576 = 57 ×10 + 6. That 6 is a

multiple of 2 ensures that 576 is too. We have the rule:

A number is divisible by 2 only if its final digit

is 0, 2, 4, 6 or 8.

Divisibility by 3

That 10, 100, 1000,… all leave a remainder of 1 when

divided by 3 allows us to quickly determine the remain-