Physical Chemistry Third Edition

B Some Useful Mathematics 1241

I

∫∞

0

∫π

0

∫ 2 π

0

f(r,θ,φ)r^2 sin(θ)dφdθdr (B-27)

Since the limits are constants, this integral is carried out in the same way as that of

Eq. (B-24), with theφintegration done first and theθintegration done next.

For other coordinate systems, a factor analogous to the factorr^2 sin(θ) must be used.

This factor is called aJacobian. For example, for cylindrical polar coordinates, where

the coordinates arez,φ(the same angle as in spherical polar coordinates, andρ(the

projection ofrinto thexyplane), the Jacobian is the factorρ, so that the element of

volume isρdρdzdφ. We use the symbold^3 rfor the three-dimensional volume element

in any coordinate system, so thatdxdydz,r^2 sin(θ)dφdθdr,ρdφdρdz, and so on are all

denoted byd^3 r.

B.3 Vectors

A vector is a quantity with both magnitude and direction. The vectorAcan be repre-

sented by its Cartesian components,Ax,Ay, andAz:

AiAx+jAy+kAz (B-28)

wherei,j, andkare unit vectors that point in thex,y, andzdirections, respectively,

andAx,Ay, andAzare the Cartesian components of the vector. The product of the unit

vectoriand the scalarAxis a vector of lengthAxpointing in the positivexdirection if

Axis positive and in the negativexdirection ifAxis negative. The other two products

in Eq. (B-28) are analogous. The sum of two vectors can be obtained by moving the

second vector so that its tail coincides with the head of the first vector without rotating

it or changing its length. The sum vector is then drawn from the tail of the first vector

to the head of the second. Figure B.5 indicates how the three vectors in thex,y, andz

directions add in the case that all three components are positive. The sum ofiAxand

jAylies in thexyplane and thekAzvector is added to it by bringing its tail to the head

of the first sum and constructing the final sum as shown.

The sum of two arbitrary vectors is obtained in the same way. IfA+BC, then

Cis obtained by moving the tail ofBto the head ofAand drawingCfrom the tail

ofAto the head ofB. The sum vector is then drawn from the tail of the first vector

to the head of the second. Figure B.6 shows the sumA+BC. Vector addition is

commutative, so thatA+BB+A, as shown in the figure. The vector sum is also

obtained by adding the components. IfA+BC, then

CxAx+Bx

CyAy+By

CzAz+Bz

(B-29a)

Thedot productorscalar productof two vectors is a scalar quantity equal to the

product of the magnitudes of the two vectors times the cosine of the angle between

them:

A·B|A||B|cos(α)ABcos(α) (B-30)

whereαis the angle between the vectors and where the magnitude of the vectorsAand

Bare denoted either by|A|and|B|or byAandB. The scalar product is commutative:

A·BB·A (B-31)