Physical Chemistry Third Edition

B Some Useful Mathematics 1243

The scalar product of a vector with itself is the square of the magnitude of the vector:

A·A|A|^2 A^2 (B-32)

The scalar products of the unit vectors are

i·ji·kj·k 0 (B-33)

i·ij·jk·k 1 (B-34)

If the vectorsAandBare represented by Cartesian components as in Eq. (B-28),

Eq. (B-33) implies that six of the nine terms in the productA·Bvanish, leaving

A·BAxBx+AyBy+AzBz (B-35)

The scalar product of a vector with itself is the square of the magnitude of the vector:

A·A|A|^2 A^2 x+A^2 y+A^2 z (B-36)

|A|A

√

A^2 x+A^2 y+A^2 z (B-37)

Thecross productorvector productof two vectors is a vector quantity that is

perpendicular to the plane containing the two vectors with magnitude equal to the

product of the magnitudes of the two vectors times the sine of the angle between them:

|A×B||A||B|sin(α) (B-38)

The direction of the product vector is the direction in which an ordinary (right-handed)

screw moves if it is rotated in the direction which the vector on the left must be rotated

to coincide with the vector on the right, rotating through an angle less than or equal to

180 ◦. The cross product is not commutative:

A×B−B×A (B-39)

The cross products of the unit vectors are

i×i0, j×j0, k×k 0 (B-40)

i×jk, i×k−j, j×ki (B-41)

In terms of Cartesian components, we can deduce from Eqs. (B-40) and (B-41) that

A×Bi[AyBz−AzBy]+j[AzBx−AxBz]+k[AxBy−AyBx] (B-42)

Vector Derivatives Thegradientis a vector derivative of a scalar function. Iffis a

function ofx,y, andz, its gradient is given by

∇fi

∂f

∂x

+j

∂f

∂y

+k

∂f

∂z

(B-43)

The symbol for the gradient operator,∇, is called “del.” At a given point in space, the

gradient points in the direction of most rapid increase of the function, and its magnitude

is equal to the rate of change of the function in that direction. The gradient of a vector

function is also defined, and the gradient of each component is as defined in Eq. (B-43).

The gradient of a vector quantity therefore has nine components, and is called adyadic