1.3 Fundamentals of the Theory of Field
27
If a scalar function ϕ in the closed surface S has the continuous first partial
derivative, and V is the volume surrounded by S, then there is the following gradient
formula
S
ϕdS =
S
ϕndS =
S
nϕdS =
V
grad ϕdV =
V
∇ϕdV (1.3.75)
Equation (1.3.75) is called the Gauss formula of the gradient form or gradient
theorem.
Proof Let a = ϕC, where, C is an arbitrary constant vector, taking the divergence
of the expression and using the property (2) of gradient, we obtain
∇ · a = ∇ · (ϕC) = C · ∇ϕ
Substituting the above expression into the Gauss formula, we obtain
S
ϕC · dS =
V
C · ∇ϕdV
Since C is an arbitrary constant vector, so it can be extracted from the integral
sign, we obtain
C ·
S
ϕdS = C ·
V
∇ϕdV
Since C is an arbitrary constant vector, so that
S
ϕdS =
V
grad ϕdV =
V
∇ϕdV
Quod erat demonstrandum.
By the way it is pointed out, according to Eq. (1.3.75), another definition of
gradient expression can be written as
grad ϕ = ∇ϕ = lim
V →0
1
V
S
ϕdS = lim
V →0
1
V
S
nϕdS
(1.3.76)
Equation (1.3.76) can be obtained by the integral mean value theorem. This is a
gradient expression defined in integral form.
Example 1.3.4 Let v( pu) = u( pv), verify
∇ · [v∇( pu) − u∇( pv) + pv∇u − pu∇v] = 0