You are Here: Home >< Maths

# Jacobian help plz! watch

1. Hi, was wondering if anyone could help me with the following question:

By changing varaibles u = xy and v = y/x, show that the integral (can be seen in my solution sheet) = 15/8 where R is the region in the first quadrant bounded by xy = 1, xy = 2, y = x and y = 2x.

This is my solution so far:

My Solution here

I'm trying to follow the examples in my notes but I still don't understand the use of Jacobian? Can anyone help me with the question, thanks!
2. (Original post by rockiee)
Hi, was wondering if anyone could help me with the following question:

By changing varaibles u = xy and v = y/x, show that the integral (can be seen in my solution sheet) = 15/8 where R is the region in the first quadrant bounded by xy = 1, xy = 2, y = x and y = 2x.

I'm trying to follow the examples in my notes but I still don't understand the use of Jacobian? Can anyone help me with the question, thanks!
That region of the plane is given by 1 < u < 2 and 1 < v < 2. It's, from the uv-point of view, a rectangle but is in the xy-plane a stretched rectangle. The Jacobian (or rather its modulus) is a measure at each point of how much the area's getting stretched, which is why it goes into the double integral formula.

Specifically you need to:
write your integrand in terms of u and v
change the limits to 1,2 and 1,2
recall that dx dy = |J| du dv for the Jacobian you calculated.
3. Oh thanks!

Hmm that's the part which I don't understand. Changing the integrand in terms of u and v? The example I have in my notes is so straightforward because u = the term in the integrand so I just substitute it in. However in this one I'm not sure how to approach the conversion of the integrand Can offer some help?

so far I've got the integrand:

∫∫ (x²+2y²) | ∂(x,y)/∂(u,v) du dv
4. x = rt(u/v), y = rt(uv)
5. thanks!

hmm I don't think I've integrated it properly....I got logs in my integration..is that right? and I didnt get 15/8 either
6. Your Jacobian is wrong - you've not got v_x correct
7. ohh its suppose to be -y/x² not y right? so the Jacobian is...1/(2y/x) = x/2y?
8. (Original post by rockiee)
ohh its suppose to be -y/x² not y right? so the Jacobian is...1/(2y/x) = x/2y?
which is 1/(2v)

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: December 4, 2005
Today on TSR

### Don't know what to study?

These A-levels give you the most options

### University open days

• University of Lincoln
Mini Open Day at the Brayford Campus Undergraduate
Wed, 19 Dec '18
• University of East Anglia
Fri, 4 Jan '19
• Bournemouth University
Wed, 9 Jan '19
Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams