Jacobians

Jacobians are very useful to determine the elements of integrals when transforming them from a system coordinates to another.

1. 1D Jacobian

In 1D problems we are used to a simple change of variables, e.g. from x to u

∫ f(x) dx (a → b ) = ∫ f(x(u)) (dx/du) du ( α → β)

dx/du is one-dimension Jacobian

2. 2D Jacobian

Transformation from (x,y) to (u,v):
∫∫(R) f(x,y) dxdy = ∫∫(R') f(x(u,v),y(u,v)) (∂(x,y)/∂(u,v) dudv
∂(x,y)/∂(u,v) is the 2D Jacobian

∂(x,y)/∂(u,v) =
|∂x/∂u ∂x/∂v|
|∂y/∂u ∂y/∂v|
=
|x_u x_v |
|y_u y_v |
= x_uy_v - x_vy_u

The Jacobian is the determinant of the Jacobian Matrix

Example:
Area of circle of radius R:

x = r cos θ
y = r sin θ

dxdy → dr dθ

A = ∫∫(S) dxdy = ∫∫(S') (Jacobian)drdθ

Jacobian =
|x_r x_θ |
|y_r y_θ |
=
|cos θ -r sinθ |
|sin θ r cos θ |
= r

Thereore:

A = ∫∫ (Jacobian)drdθ = ∫∫ r drdθ = (1/2)R² 2π = πR²

3. 3D Jacobian

x = x(u, v, w), y = y(u, v, w), and z = z(u, v, w)
Transforms vlume dxdydz to dudvdw.
∫∫∫ (V) f(x,y,z) dxdydz =
∫∫∫ (V')F(u,v,w) ∂(x,y,z)/∂(u,v,w) dudvdw

The Jacobian ∂(x,y,z)/∂(u,v,w) is equal to:

|x_u x_v x_w|
|y_u y_v y_w|
|z_u z_v z_w|

Example:

Transformation of volume elements between Cartesian and spherical polar coordinate systems: dxdydz → dudvdw = drdθdφ
x = r sinθcosφ
y = r sinθsinφ
x = r cosθ

x_u = ∂x/∂r = sinθcosφ
x_θ = ∂x/∂θ = r cosθcosφ
x_φ = ∂x/∂φ = - r sinθsinφ

....

The Jacobian is:

∂(x,y,z)/∂(u,v,w) =
|sinθcosφ r cosθcosφ - r sinθsinφ|
|sinθsinφ r cosθsinφ r sinθcosφ|
|cosθ - r sinθ 0| =
r² sin θ

Therefore:
The volume element is:
dV = dxdydz = (Jacobian) drdθdφ = r² sin θ drdθdφ