14.8 Jacobians
For more interesting illustrations of this topic, visit
http://www.math.umn.edu/~rogness/multivar/nonlineartransformation.html
Definition of the Jacobian
Example 1
Find the Jacobian for the change of variables
x = r cosө and
y = r sinө
Example 1 Solution
Find the Jacobian for the change of
variables
x = r cosө and
y = r sinө
Why would we change variables?
Example 2
Let R be the region bounded by the lines
x - 2y = 0, x – 2y = -4, x + y =4 and x + y = 1
Find a transformation T from region R to
region S such that S is a rectangular region.
Example: 2 Solution
Example 2 Solution
We can convert individual points between coordinate systems
Similarly, we could use these formulas
to convert in the other direction
Change of variables
Example 3 use a change of variables
to simplify a region
Let R be the region bounded by the lines
x - 2y = 0, x – 2y = -4, x + y =4 and x + y = 1
as shown below. Evaluate the double integral.
Example 3 Solution slide 1
Example 3 Solution slide 2
Example 4
Let R be the region bounded by vertices (0,1),(1,2)
(2,1), (1,0)
a) Sketch the transformed region
b) Evaluate the integral
Example 4 a
Let u = x + y
Let v = x- y
Example 4
solution
Let u = x + y
Let v = x- y
Wisdom from
Singapore:
Explaining a
joke is like
dissecting a
frog.
You learn
more about it
but you kill it
in the process.
-Niel Chong