148 double integral

between these two vectors is 90°, their dot product i· j

will be zero (i· j= 1.0 + 0.1 + 0.0 = 0). In general:

Two vectors aand bare at right angles if,

and only if, their dot product a· bis zero.

The dot product has the following commutative and

distributive properties:

a· b= b· a

a· (b+ c) = a· b+ a· c

See also

CROSS PRODUCT

;

NORMAL TO A PLANE

;

ORTHOGONAL

;

TRIPLE VECTOR PRODUCT

;

VECTOR

EQUATION OF A PLANE

.

double integral The volume under a graph z= f(x,y)

of two variables (which is drawn as a surface sitting in

three-dimensional space) above a region Rin the xy-

plane is computed via a double integral, denoted:

One approximates this volume by subdividing the

region Rinto small rectangular pieces, drawing a rect-

angular cuboid above each rectangle with height reach-

ing the surface, and summing the volumes of each of

these cuboids. As one takes finer and finer approxima-

tions, this process produces better and better approxi-

mations to the true volume under the graph. The limit

of this process is the double integral:

where dAkdenotes the area of the kth rectangular

region used to approximate R.

G

OTTFRIED

W

ILHELM

L

EIBNIZ

(1646–1716) showed

that if the region Ris itself a rectangle, say, given by

a≤x≤band c≤y≤d, then the double integral can be

computed as either of the two iterated integrals:

(In an iterated integral, one integrates one variable at a

time, regarding the second variable as a constant.) This

result holds true for other shaped regions Ras well, as

long as they are not too complicated.

For example, the volume under the graph z= xy

above the rectangle R= [1,2] ×[2,3] is:

Notice that the integration is performed from the

inside out.

A triple integral of a function of

three variables f(x,y,z),computed over a volume Vin

space, can often be computed as a triple iterated inte-

gral, integrating each variable in turn. Again, the order

of the integration, typically, does not matter.

See also G

EORGE

G

REEN

.

double point A location on a curve where the curve

either crosses itself, or is tangential to itself, is called a

double point. In the first case, the point of intersection

is called a node, and the curve has two distinct tangents

at that point. In the second case, the point of contact is

called a tacnode or an osculation. The two tangents to

the curve coincide at this point.

See also

ISOLATED POINT

;

SINGULAR POINT

;

TANGENT

.

double root See

ROOT

.

dummy variable A variable appearing in a mathe-

matical expression is a dummy variable if it is assigned

no specific meaning and if the letter being used for it

could equally well be replaced by another letter. An

index of

SUMMATION

, for instance, is a dummy variable:

the sum of denoting 13+ 23+ 33+ 43, for exam-

ple, could equally well be represented as or

, say. The variable used for the integrand of a

DEFINITE INTEGRAL

is a dummy variable. The two

expressions ∫1

0x2dx and ∫1

0t2dt, for instance, represent