(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 “sprinkles 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 =. The
divergence of F, denoted div F, is the quantity:
given as a sum of PARTIAL DERIVATIVEs. Physicists have
shown that this quantity represents the amount of flux
leaving an element of volume in space. For example, if
F represents 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 F is called the curl of F:
If F is 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 converge.” 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 divergent” 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 corresponding sequence of partial sums has this property.
For example, the series 1 + 2 + 4 + 8 + 16 +… is properly 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
n if, 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 divisible 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 remaincurl
f
y
f
z
f
z
f
x
f
x
f
y
F
F
= ∇× =
∂
∂
−
∂
∂
+
∂
∂
−
∂
∂
∂
∂
−
∂
∂
3
2
1
3
2
1
i
j
k
+
∇ =
∂
∂
∂
∂
∂
∂
x y z
, ,
div F
f
x
f
y
f
z
=
∂
∂
+
∂
∂
+
∂
∂
1
2
3
144 div
= 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 “sprinkles 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 =
divergence of F, denoted div F, is the quantity:
given as a sum of PARTIAL DERIVATIVEs. Physicists have
shown that this quantity represents the amount of flux
leaving an element of volume in space. For example, if
F represents 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 F is called the curl of F:
If F is 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 converge.” 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 divergent” 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 corresponding sequence of partial sums has this property.
For example, the series 1 + 2 + 4 + 8 + 16 +… is properly 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
n if, 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 divisible 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 remaincurl
f
y
f
z
f
z
f
x
f
x
f
y
F
F
= ∇× =
∂
∂
−
∂
∂
+
∂
∂
−
∂
∂
∂
∂
−
∂
∂
3
2
1
3
2
1
i
j
k
+
∇ =
∂
∂
∂
∂
∂
∂
x y z
, ,
div F
f
x
f
y
f
z
=
∂
∂
+
∂
∂
+
∂
∂
1
2
3
144 div