3Partial differentiation
IA Differential Equations
3.2 Chain rule
Consider an arbitrary displacement in any direction (
x, y
)
→
(
x
+
δx, y
+
δy
).
We have
δf = f(x + δx, y + δy) − f(x, y)
= f(x + δx, y + δy) − f(x + δx, y) + f(x + δx, y) −f(x, y)
= f
y
(x + δx, y)δy + o(δy) + f
x
(x, y)δx + o(δx)
= (f
y
(x, y) + o(1))δy + o(δy) + f
x
(x, y)δx + o(δx)
δf =
∂f
∂x
δx +
∂f
∂y
δy + o(δx, δy)
Take the limit as δx, δy → 0, we have
Theorem (Chain rule for partial derivatives).
df =
∂f
∂x
dx +
∂f
∂y
dy.
Given this form, we can integrate the differentials to obtain the integral form:
Z
df =
Z
∂f
∂x
dx +
Z
∂f
∂y
dy,
or divide by another small quantity. e.g. to find the slope along the path
(x(t), y(t)), we can divide by dt to obtain
df
dt
=
∂f
∂x
dx
dt
+
∂f
∂y
dy
dt
.
If we pick the parameter t to be the arclength s, we have
df
ds
=
∂f
∂x
dx
ds
+
∂f
∂y
dy
ds
=
dx
ds
,
dy
ds
·
∂f
∂x
,
∂f
∂y
= ˆs · ∇f,
which is known as the directional derivative (cf. Chapter 7).
Alternatively, the path may also be given by
y
=
y
(
x
). So
f
=
f
(
x, y
(
x
)).
Then the slope along the path is
df
dx
=
∂f
∂x
y
+
∂f
∂y
dy
dx
.
The chain rule can also be used for the change of independent variables, e.g.
change to polar coordinates x = x(r, θ), y = y(r, θ). Then
∂f
∂θ
r
=
∂f
∂x
y
∂x
∂θ
r
+
∂f
∂y
x
∂y
∂θ
r
.