Begin2.DVI

which by the divergence theorem can be expressed in the form

∫∫

S

q ·dS
=

∫∫∫

V

div 
q dV. (9 .97)

Substituting the heat flow given by equation (2.46) into equation (2.49) produces

the relation ∫∫

S

q·dS
=−

∫∫∫

V

kdiv (grad T)dV, (9 .98)

which depicts the total amount of heat leaving the arbitrary volume Venclosed by S.

From equation (2.47), one can calculate the rate of change of decreasing heat within

the volume. Such a change is given by

−∂H

∂t

=−

∫∫

V

cρ ∂T

∂t

dV (9 .99)

and must equal the change given by the flux integral (2.50). Equating these quan-

tities produces the relation

∫∫∫

V

[

cρ ∂T

∂t

−kdiv (grad T)

]

dV = 0 (9 .100)

which must hold for any arbitrary volume Vwithin the material. Since the volume

is arbitrary, it is required that the integrand be identically zero and write

cρ ∂T

∂t

−kdiv (grad T) = 0 = ⇒ cρ

k

∂T

∂t

=∂

(^2) T
∂x^2
+∂
(^2) T
∂y^2
+∂
(^2) T
∂z^2
=⇒ cρ
k
∂T
∂T
=∇^2 T (9 .101)

This result is known as the heat equation.

For steady-state temperature distributions, write ∂T∂t = 0,and consequently equa-

tion (9.101) reduces to Laplace’s equation.

In the study of heat flow the level curves T(x, y, z ) = c are called isothermal

surfaces, and the field lines associated with the heat flow
qwithin the material are

called heat flow lines.

Two-Body Problem

Newton’s law of gravitation states that two masses m and M, are attracted

toward each other with a force of magnitude GmMr 2 ,where Gis a constant and ris the

distant between the masses. Let M represent the mass of the Sun and mrepresent

the mass of a planet and assume that the motion of one mass with respect to the