Begin2.DVI

represents a small change in
r. One can think of the differential d
r 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

giving the new volume element dV =ρ^2 sinθ dρ dφ dθ.

Verification of the above results is left as an exercise.