4Surfaces and surface integrals
IA Vector Calculus
4.4 Change of variables in R
2
and R
3
revisited
In this section, we derive our change of variable formulae in a slightly different
way.
Change of variable formula in R
2
We first derive the 2D change of variable formula from the 3D surface integral
formula.
Consider a subset
S
of the plane
R
2
parametrized by
r
(
x
(
u, v
)
, y
(
u, v
)). We
can embed it to R
3
as r(x(u, v), y(u, v), 0). Then
∂r
∂u
×
∂r
∂v
= (0, 0, J),
with J being the Jacobian. Therefore
Z
S
f(r) dS =
Z
D
f(r(u, v))
∂r
∂u
×
∂r
∂v
du dv =
Z
D
f(r(u, v))|J| du dv,
and we recover the formula for changing variables in R
2
.
Change of variable formula in R
3
In R
3
, suppose we have a volume parametrised by r(u, v, w). Then
δr =
∂r
∂u
δu +
∂r
∂v
δv +
∂r
∂w
δw + o(δu, δv, δw).
Then the cuboid
δu, δv, δw
in
u, v, w
space is mapped to a parallelopiped of
volume
δV =
∂r
∂u
δu ·
∂r
∂v
δv ×
∂r
∂w
δw
= |J| δu δv δw.
So dV = |J| du dv dw.