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1 Preliminaries
1.3.3 The Gauss Theorem and Green’s Formulae
Theorem 1.3.2 Suppose that a vector a within a closed surface S has a continuous first order partial derivative, then the flux of a on S equals the integral of the
divergence of the vector with respect to the volume V surrounded by S, that is
S
a · dS =
S
a · ndS =
V
div adV =
V
∇ · adV
(1.3.45)
where, n is the unit outward normal vector on S; ∇ is the Hamiltonian operator.
Equation (1.3.45) is called the Gauss-Ostrogradsky theorem, it is called the
Gauss theorem for short, it is also called the Gauss formula in the form of divergence
or divergence theorem. It has established the relationship between the triple integral
of a continuous function in the spatial domain V and the surface integral on the
boundary surface S. The Gauss formula in the theoretical research and practical
work has been widely used, it is a powerful mathematical tool, many formulae are
derived from the Gauss formula.
Proof The volume V surrounded by the closed surface S is divided into n volume
element V 1 , V 2 , …, V n surrounded by closed surface S 1 , S 2 , …, S n . taking
out the kth surface S k and the volume V k surrounded by it, by the definition of
divergence, there is the following relation
div a =
S k a · dS
V k
+ ε k
namely
V k div a =
S k
a · dS + ε k V k
where, the divergence is the value at some point M in the elemental volume; ε k is
small enough, and when V k → 0, ε k → 0. Summing the above expression to k
from 1 to n, we obtain
n
k=1
V k div a =
n
k=1
S k
a · dS +
n
k=1
ε k V k
In the interior of the closed surface S, the two outward normal directions are
opposite on the public surface of the adjacent two elemental volumes, the two integrals cancel each other out, in this way, on the right side of the above expression
there is only the integral of the closed surface S. Meanwhile, let n → ∞, such that
V k → 0, then the limit on the left side of the above expression is the volume
integral
V div adV. Moreover for the vector a, it is thought when n is sufficiently
1 Preliminaries
1.3.3 The Gauss Theorem and Green’s Formulae
Theorem 1.3.2 Suppose that a vector a within a closed surface S has a continuous first order partial derivative, then the flux of a on S equals the integral of the
divergence of the vector with respect to the volume V surrounded by S, that is
S
a · dS =
S
a · ndS =
V
div adV =
V
∇ · adV
(1.3.45)
where, n is the unit outward normal vector on S; ∇ is the Hamiltonian operator.
Equation (1.3.45) is called the Gauss-Ostrogradsky theorem, it is called the
Gauss theorem for short, it is also called the Gauss formula in the form of divergence
or divergence theorem. It has established the relationship between the triple integral
of a continuous function in the spatial domain V and the surface integral on the
boundary surface S. The Gauss formula in the theoretical research and practical
work has been widely used, it is a powerful mathematical tool, many formulae are
derived from the Gauss formula.
Proof The volume V surrounded by the closed surface S is divided into n volume
element V 1 , V 2 , …, V n surrounded by closed surface S 1 , S 2 , …, S n . taking
out the kth surface S k and the volume V k surrounded by it, by the definition of
divergence, there is the following relation
div a =
S k a · dS
V k
+ ε k
namely
V k div a =
S k
a · dS + ε k V k
where, the divergence is the value at some point M in the elemental volume; ε k is
small enough, and when V k → 0, ε k → 0. Summing the above expression to k
from 1 to n, we obtain
n
k=1
V k div a =
n
k=1
S k
a · dS +
n
k=1
ε k V k
In the interior of the closed surface S, the two outward normal directions are
opposite on the public surface of the adjacent two elemental volumes, the two integrals cancel each other out, in this way, on the right side of the above expression
there is only the integral of the closed surface S. Meanwhile, let n → ∞, such that
V k → 0, then the limit on the left side of the above expression is the volume
integral
V div adV. Moreover for the vector a, it is thought when n is sufficiently