7Directional derivative
IA Differential Equations
7.1 Gradient vector
Consider a function
f
(
x, y
) and a displacement ds = (d
x,
d
y
). The change in
f(x, y) during that displacement is
df =
∂f
∂x
dx +
∂f
∂y
dy
We can also write this as
df = (dx, dy) ·
∂f
∂x
,
∂f
∂y
= ds · ∇f
where
∇f
=
gradf
=
∂f
∂x
,
∂f
∂y
are the Cartesian components of the gradient of
f.
We write ds =
ˆ
s ds, where |
ˆ
s| = 1. Then
Definition (Directional derivative). The directional derivative of
f
in the
direction of
ˆ
s is
df
ds
=
ˆ
s · ∇f.
Definition (Gradient vector). The gradient vector
∇f
is defined as the vector
that satisfies
df
ds
= ˆs · ∇f.
Officially, we take this to be the definition of
∇f
. Then
∇f
=
∂f
∂x
,
∂f
∂y
is a
theorem that can be proved from this definition.
We know that the directional derivative is given by
df
ds
= ˆs · ∇f = |∇f|cos θ
where
θ
is the angle between the displacement and
∇f
. Then when
cos θ
is
maximized,
df
ds
= |∇f|. So we know that
(i) ∇f
has magnitude equal to the maximum rate of change of
f
(
x, y
) in the
xy plane.
(ii) It has direction in which f increases most rapidly.
(iii)
If ds is a displacement along a contour of
f
(i.e. along a line in which
f
is
constant), then
df
ds
= 0.
So ˆs · ∇f = 0, i.e. ∇f is orthogonal to the contour.