2D Jacobian
2D Jacobian
Let , be a transformation of the - plane into the - plane. Let be a small rectangle with sides parallel to the and axes with side lengths and . The image of this rectangle in the - plane is a curved rectangle . The coordinates of these points are:
x=x(u,v)
y=y(u,v)
u
v
x
y
A
1
A
2
A
3
A
4
u
v
du
dv
x
y
B
1
B
2
B
3
B
4
B
1
B
2
B
3
B
4
These points can be approximated by the points of the parallelogram
C
1
C
2
∂x
∂u
∂y
∂u
C
3
∂x
∂u
∂x
∂v
∂y
∂u
∂y
∂v
C
4
∂x
∂v
∂y
∂v
The area of this parallelogram is the absolute value of
det
=det
dudv
∂x ∂u | ∂y ∂u |
∂x ∂v | ∂y ∂v |
∂x ∂u | ∂x ∂v |
∂y ∂u | ∂y ∂v |
The last determinant is called the Jacobian and is usually denoted by .
∂(x,y)
∂(u,v)