8
1 Preliminaries
Obviously, when both α(y) and β(y) are constants, α
(y) = β
(y) = 0,
Theorem 1.2.4 is converted into Theorem 1.2.3, namely Theorem 1.2.3 is a special
case of Theorem 1.2.4.
1.3 Fundamentals of the Theory of Field
A field is a form of relationship between physical quantities in the real world and
space and time, it is a form of material existence. If at each point in a space region, it
corresponds to a certain value of a physical quantity, then this space region is called
the field of the physical quantity existing in it. A physical quantity distribution in
the field can be expressed as a function of spatial position, this function is called the
point function of the physical quantity. Of course, a physical quantity in the field
may also change with time, therefore a point function can also be related to time.
If a physical quantity has the property of the quantity, then the field formed by the
physical quantity is called the scalar field. If a physical quantity has the property of
the vector, then the field formed by the physical quantity is called the vector field. If a
physical quantity has the property of the tensor, then the field formed by the physical
quantity is called the tensor field. In the field of physical quantity, a function that its
value is a quantity is called the scalar function, a function that its value is a vector
is called the vector function, a function that its value is a tensor is called the tensor
function. The point function, scalar function, vector function and tensor function
can all be called the function for short.
1.3.1 Directional Derivative and Gradient
A quantity with magnitude and direction is called the vector. The magnitude of the
vector is called the length of vector or modulus of vector. The modulus of vector
a is expressed by |a|. A vector that its modulus equals 1 is called the unit vector or
vector of unit length. A vector that its modulus equals 0 is called the zero vector,
it is written as 0.
The first partial derivative
∂ϕ
∂ x
,
∂ϕ
∂ y
and
∂ϕ
∂z
of the function ϕ = ϕ(M) = ϕ(x, y, z)
denote respectively the rate of change on three specific directions along the x, y and
z axis at point M. However, in many problems, the rate of changed of the function
ϕ = ϕ(x, y, z) along the other direction also has practical significance, so it is
necessary to study its derivative in the other direction.
Suppose that M 0 is a determined point of the function ϕ(M), a straight line L is
drawn through the point, a moving point M near M 0 is chosen on the straight line,
the distance from point M 0 to point M is M 0 M, when M → M 0 , if the limit of ratio
ϕ(M) − ϕ(M 0 )
M 0 M
1 Preliminaries
Obviously, when both α(y) and β(y) are constants, α
(y) = β
(y) = 0,
Theorem 1.2.4 is converted into Theorem 1.2.3, namely Theorem 1.2.3 is a special
case of Theorem 1.2.4.
1.3 Fundamentals of the Theory of Field
A field is a form of relationship between physical quantities in the real world and
space and time, it is a form of material existence. If at each point in a space region, it
corresponds to a certain value of a physical quantity, then this space region is called
the field of the physical quantity existing in it. A physical quantity distribution in
the field can be expressed as a function of spatial position, this function is called the
point function of the physical quantity. Of course, a physical quantity in the field
may also change with time, therefore a point function can also be related to time.
If a physical quantity has the property of the quantity, then the field formed by the
physical quantity is called the scalar field. If a physical quantity has the property of
the vector, then the field formed by the physical quantity is called the vector field. If a
physical quantity has the property of the tensor, then the field formed by the physical
quantity is called the tensor field. In the field of physical quantity, a function that its
value is a quantity is called the scalar function, a function that its value is a vector
is called the vector function, a function that its value is a tensor is called the tensor
function. The point function, scalar function, vector function and tensor function
can all be called the function for short.
1.3.1 Directional Derivative and Gradient
A quantity with magnitude and direction is called the vector. The magnitude of the
vector is called the length of vector or modulus of vector. The modulus of vector
a is expressed by |a|. A vector that its modulus equals 1 is called the unit vector or
vector of unit length. A vector that its modulus equals 0 is called the zero vector,
it is written as 0.
The first partial derivative
∂ϕ
∂ x
,
∂ϕ
∂ y
and
∂ϕ
∂z
of the function ϕ = ϕ(M) = ϕ(x, y, z)
denote respectively the rate of change on three specific directions along the x, y and
z axis at point M. However, in many problems, the rate of changed of the function
ϕ = ϕ(x, y, z) along the other direction also has practical significance, so it is
necessary to study its derivative in the other direction.
Suppose that M 0 is a determined point of the function ϕ(M), a straight line L is
drawn through the point, a moving point M near M 0 is chosen on the straight line,
the distance from point M 0 to point M is M 0 M, when M → M 0 , if the limit of ratio
ϕ(M) − ϕ(M 0 )
M 0 M