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2D Jacobian

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