1.3 Fundamentals of the Theory of Field
9
exists, then it is called the directional derivative of the function ϕ(M) along direction
L at point M 0 , and it is written as
∂ϕ(M 0 )
∂ L
= lim
M 0 M→0
ϕ(M) − ϕ(M 0 )
M 0 M
(1.3.1)
Thus it can be seen that the directional derivative is the rate of change of the
function ϕ(M) to the distance along a certain direction at a given point. If
∂ϕ
∂ L
> 0,
the value of the function ϕ increases along the direction L; If
∂ϕ
∂ L
< 0, the value of the
function ϕ the decreases L; If
∂ϕ
∂ L
= 0, the value of the function ϕ does not change
along the direction L.
Infinitely multiple directions can be chosen through point M 0 , every direction
corresponds with the directional derivative. In the rectangular coordinate system, the
given formula by the following theorem can calculate the direction derivative.
Theorem 1.3.1 If a scalar field ϕ = ϕ(x, y, z) is differentiable at point
M 0 (x 0 , y 0 , z 0 ), cos α, cos β and cos γ are the direction cosines in the direction L,
then the directional derivative of ϕ along the direction L at point M 0 is bound to
exist, and it is given by the following formula
∂ϕ
∂ L
=
∂ϕ
∂ x
cos α +
∂ϕ
∂ y
cos β +
∂ϕ
∂z
cos γ
(1.3.2)
where,
∂ϕ
∂ x
,
∂ϕ
∂ y
and
∂ϕ
∂z
are the various partial derivatives of the function ϕ at point M 0 .
Proof Suppose that the coordinate of the moving point is M(x +x, y+y, z+z),
the distance between point M 0 and M is ρ =
((x) 2 + ((y) 2 + ((z) 2 . Since ϕ is
differentiable at point M 0 , so the increment of ϕ can be expressed as
ϕ = ϕ(M) − ϕ(M 0 ) =
∂ϕ
∂ x
x +
∂ϕ
∂ y
y +
∂ϕ
∂z
z + ω · ρ
where, ω tends to zero when ρ → 0, both ends of the above expression is divided
by ρ and to take limit, we obtain
∂ϕ
∂ L
= lim
ρ→0
ϕ
ρ
=
∂ϕ
∂ x
cos α +
∂ϕ
∂ y
cos β +
∂ϕ
∂z
cos γ
Thus the directional derivative exists and it is Eq. (1.3.2). Quod erat demonstrandum.
In Eq. (1.3.2), the directional derivative
∂ϕ
∂ L
can be expressed as the scalar product
of two vectors, namely
∂ϕ
∂ L
=
∂ϕ
∂ x
i +
∂ϕ
∂ y
j +
∂ϕ
∂z
k
· (cos αi + cos β j + cos γ k)
(1.3.3)
9
exists, then it is called the directional derivative of the function ϕ(M) along direction
L at point M 0 , and it is written as
∂ϕ(M 0 )
∂ L
= lim
M 0 M→0
ϕ(M) − ϕ(M 0 )
M 0 M
(1.3.1)
Thus it can be seen that the directional derivative is the rate of change of the
function ϕ(M) to the distance along a certain direction at a given point. If
∂ϕ
∂ L
> 0,
the value of the function ϕ increases along the direction L; If
∂ϕ
∂ L
< 0, the value of the
function ϕ the decreases L; If
∂ϕ
∂ L
= 0, the value of the function ϕ does not change
along the direction L.
Infinitely multiple directions can be chosen through point M 0 , every direction
corresponds with the directional derivative. In the rectangular coordinate system, the
given formula by the following theorem can calculate the direction derivative.
Theorem 1.3.1 If a scalar field ϕ = ϕ(x, y, z) is differentiable at point
M 0 (x 0 , y 0 , z 0 ), cos α, cos β and cos γ are the direction cosines in the direction L,
then the directional derivative of ϕ along the direction L at point M 0 is bound to
exist, and it is given by the following formula
∂ϕ
∂ L
=
∂ϕ
∂ x
cos α +
∂ϕ
∂ y
cos β +
∂ϕ
∂z
cos γ
(1.3.2)
where,
∂ϕ
∂ x
,
∂ϕ
∂ y
and
∂ϕ
∂z
are the various partial derivatives of the function ϕ at point M 0 .
Proof Suppose that the coordinate of the moving point is M(x +x, y+y, z+z),
the distance between point M 0 and M is ρ =
((x) 2 + ((y) 2 + ((z) 2 . Since ϕ is
differentiable at point M 0 , so the increment of ϕ can be expressed as
ϕ = ϕ(M) − ϕ(M 0 ) =
∂ϕ
∂ x
x +
∂ϕ
∂ y
y +
∂ϕ
∂z
z + ω · ρ
where, ω tends to zero when ρ → 0, both ends of the above expression is divided
by ρ and to take limit, we obtain
∂ϕ
∂ L
= lim
ρ→0
ϕ
ρ
=
∂ϕ
∂ x
cos α +
∂ϕ
∂ y
cos β +
∂ϕ
∂z
cos γ
Thus the directional derivative exists and it is Eq. (1.3.2). Quod erat demonstrandum.
In Eq. (1.3.2), the directional derivative
∂ϕ
∂ L
can be expressed as the scalar product
of two vectors, namely
∂ϕ
∂ L
=
∂ϕ
∂ x
i +
∂ϕ
∂ y
j +
∂ϕ
∂z
k
· (cos αi + cos β j + cos γ k)
(1.3.3)