in terms of the arc length parameter s, where sis measured from some fixed point
on the curve. In general, the scalar field φ=φ(x, y, z )varies with position and has
different values when evaluated at different points in space. Let us evaluate φat
points along the curve r to determine how φchanges with position along the curve.
The rate of change of φwith respect to arc length along the curve is given by
dφ
ds
=∂φ
∂x
dx
ds
+∂φ
∂y
dy
ds
+∂φ
∂z
dz
ds
dφ
ds
=
(
∂φ
∂x
ˆe 1 +∂φ
∂y
eˆ 2 +∂φ
∂z
ˆe 3
)
·
(
dx
ds
ˆe 1 +dy
ds
ˆe 2 +dz
ds
ˆe 3
)
dφ
ds
= grad φ·dr
ds
=∇φ·ˆet,
where the right-hand side is to be evaluated at a point P on the arbitrary curve r (s)
in R. The right-hand side of this equation is the dot product of the gradient vector
with the unit tangent vector to the curve at the point P and physically represents
the projection of the vector grad φin the direction of this tangent vector. Note that
the curve r (s)represents an arbitrary curve through the point P, and hence, the unit
tangent vector represents an arbitrary direction. Therefore, one may interpret the
derivative dφds = grad φ·e as representing the rate of change of φas one moves in
the direction e. Here the derivative equals the projection of the vector grad φin the
direction e. Such derivatives are called directional derivatives.
(Directional derivative) The component of the gradient φ = φ(x,y,z ) in
the direction of a unit vector ˆe = cos αˆe 1 + cos βˆe 2 + cos γˆe 3 is equal to the
projection ∇φ·ˆe and is called the directional derivative of φin the direction ˆe.
The directional derivative is written as
dφ
ds
=grad φ·e =∇φ·ˆe
=
(
∂φ
∂x
ˆe 1 +
∂φ
∂y
ˆe 2 +
∂φ
∂z
ˆe 3
)
·(cos αˆe 1 + cos βˆe 2 + cos γˆe 3 )
(7 .71)
where sdenotes distance in the direction e. If ˆe =ˆe nis a unit normal vector to
a surface, the notation
∂φ
∂n
= grad φ·ˆe n is used to denote a normal derivative
to the surface.
The directional derivative is a measure of how the scalar field φchanges as you move
in a certain direction. Since the maximum projection of a vector is the magnitude
of the vector itself, the gradient of φis a vector which points in the direction of
