represents a small change in r. One can think of the differential dr as the diagonal
of a parallelepiped having the vector sides
∂r
∂u
du, ∂r
∂v
dv, and ∂r
∂w
dw.
The volume of this parallelepiped produces the volume element dV of the curvilinear
coordinate system and this volume element is given by the formula
dV =
∣∣
∣∣∂r
∂u ·
(
∂r
∂v ×
∂r
∂w
)∣∣
∣∣du dv dw.
This result can be expressed in the alternate form
dV =
∣∣
∣∣J
(
x, y, z
u, v, w
)∣∣
∣∣du dv dw,
where one can make use of the property of representing scalar triple products in
terms of determinants to obtain
J
(
x, y, z
u, v, w
)
=
∣∣
∣∣
∣∣
∂x
∂u
∂x
∂v
∂x
∂y ∂w
∂u
∂y
∂v
∂y
∂z ∂w
∂u
∂z
∂v
∂z
∂w
∣∣
∣∣
∣∣
The quantity J
(
x, y, z
u, v, w
)
is called the Jacobian of the transformation from x, y, z co-
ordinates to u, v, w coordinates. The absolute value signs are to insure the element
of volume is positive.
As an example, the volume element dV =dx dy dz under the change of variable
to cylindrical coordinates (r, θ, z), with coordinate transformation
x=x(r, θ, z ) = rcos θ, y =y(r, θ, z) = rsin θ, z =z(r, θ, z ) = z
has the Jacobian determinant
∣∣
∣∣J
(
x, y, z
r, θ, z
)∣∣
∣∣=
∣∣
∣∣
∣∣
cos θ −rsin θ 0
sin θ r cos θ 0
0 0 1
∣∣
∣∣
∣∣=rwhich gives the
new volume element dV =r dr dθ dz.
As another example, the volume element dV =dx dy dz under the change of vari-
able to spherical coordinates (ρ, θ, φ ), where
x=x(ρ, θ, φ ) = ρsin θcos φ, y =y(ρ, θ, φ ) = ρsin θsin φ, z =z(ρ, θ, φ ) = ρcos θ
one finds the Jacobian
∣∣
∣∣J
(
x, y, z
ρ, θ, φ
)∣∣
∣∣=
∣∣
∣∣
∣∣
sin θcos φ ρ cos θcos φ −ρsin θsin φ
sin θsin φ ρ cos θsin φ ρ sin θcos φ
cos θ −ρsin θ 0
∣∣
∣∣
∣∣=ρ
(^2) sin θ