For problems 1 4 compute the Jacobian of each transformation.
1.
2.
3.
4.
5. If R is the
region inside determine the region we would get applying the
transformation ,
to R.
6. If R is the
triangle with vertices ,
and determine the region we would get applying the
transformation ,
to R.
7. If R is the
parallelogram with vertices ,
,
and determine the region we would get applying the
transformation ,
to R.
8. If R is the
square defined by and determine the region we would get applying the
transformation ,
to R.
9. If R is the
parallelogram with vertices ,
,
and determine the region we would get applying the
transformation ,
to R.
10. If R is the
region bounded by ,
,
and determine the region we would get applying the
transformation ,
to R.
11. Evaluate where R
is the region bounded by ,
,
and using the transformation ,
.
12. Evaluate where R
is the triangle with vertices ,
and using the transformation ,
to R.
13. Evaluate where R
is the parallelogram with vertices ,
,
and using the transformation ,
to R.
14. Evaluate where R
is the region bounded by ,
,
and using the transformation ,
.
15. Evaluate where R
is the parallelogram with vertices ,
,
and using the transformation ,
to R.
16. Derive a transformation that will transform the ellipse into a unit circle.
17. Derive the transformation used in problem 12.
18. Derive the transformation used in problem 13.
19. Derive a transformation that will convert the
parallelogram with vertices ,
,
and into a rectangle in the uv system.
20. Derive a transformation that will convert the
parallelogram with vertices ,
,
and into a rectangle with one
corner occurring at the origin of the uv
system.