1.3 Fundamentals of the Theory of Field
17
It is obvious that the divergence of point M has nothing to do with the shape of
a closed surface it is a quantity that does not depend on a selected coordinate
system, so it is a scalar. Equation (1.3.36) shows that the divergence is the rate of
change of flux to volume. The vector field in which the divergence vanishes is called
a field without source, source-free field, zero-divergence field, solenoidal field or
tubular field.
Although the divergence of a vector field is a scalar that is not dependent on the
choice of coordinate system, in the concrete when calculating the divergence of a
vector field, it often needs to use the components of the vector to represent the vector,
and the components of the vector have to do with the choice of coordinate. So it is
necessary to discuss the relationship between vector divergence and its components.
Let the vector a = a x i + a y j + a z k in a rectangular coordinate system, the
elemental volume in Eq. (1.3.36) is an elemental hexahedron, its edge length
is and respectively, as shown in Fig. 1.1, and suppose that the center
coordinates of micro hexahedron are (x, y, z). As a first order approximation, there
is the following relationship in x-direction
a x | x+
2
= a x | x +
∂a x
∂ x
2
, a x | x−
2
= a x | x −
∂a x
∂ x
2
The net flux flowing out of the elemental hexahedron in x-direction is
a x | x+
2
− a x | x−
2
=
∂a x
∂ x
O
x
x
z
z
y
y
Fig. 1.1 Elemental volume
17
It is obvious that the divergence of point M has nothing to do with the shape of
a closed surface it is a quantity that does not depend on a selected coordinate
system, so it is a scalar. Equation (1.3.36) shows that the divergence is the rate of
change of flux to volume. The vector field in which the divergence vanishes is called
a field without source, source-free field, zero-divergence field, solenoidal field or
tubular field.
Although the divergence of a vector field is a scalar that is not dependent on the
choice of coordinate system, in the concrete when calculating the divergence of a
vector field, it often needs to use the components of the vector to represent the vector,
and the components of the vector have to do with the choice of coordinate. So it is
necessary to discuss the relationship between vector divergence and its components.
Let the vector a = a x i + a y j + a z k in a rectangular coordinate system, the
elemental volume in Eq. (1.3.36) is an elemental hexahedron, its edge length
is and respectively, as shown in Fig. 1.1, and suppose that the center
coordinates of micro hexahedron are (x, y, z). As a first order approximation, there
is the following relationship in x-direction
a x | x+
2
= a x | x +
∂a x
∂ x
2
, a x | x−
2
= a x | x −
∂a x
∂ x
2
The net flux flowing out of the elemental hexahedron in x-direction is
a x | x+
2
− a x | x−
2
=
∂a x
∂ x
O
x
x
z
z
y
y
Fig. 1.1 Elemental volume