Question

not sure where am i going wrong here...

Answer 1

I see wolfram alpha's series here
What's giving you trouble?

Answer 2

you see what ?

i wanted to prove

\(∂(u,v)/∂(x,y) =6r^3 sin 2θ\)

but i got
\(∂(u,v)/∂(x,y) =2r^2 sin 2θ\)
instead...

Answer 3

yes it is using double angle identity

Answer 4

Looking at it now, Indeed the differentiations are all correct...

Yes, it's a bit confusing myself why they have an extra 3r
Do we have the right definition for partial (u,v) / partial(x,y)
the jacobian right?

Answer 5

yes, thats correct

Answer 6

Hm, I cannot see why it's not correct unless they are looking for another transformation

Because partial (x,y) / partial (r, theta) = r

Answer 7

So maybe they want partial (u, v) / partial (r, theta)
not sure? Can you check?

Answer 8

checked...they want partial (u,v)/ partial (x,y) only

Answer 9

There's no opportunity for it to be anything other than quadratic...
given the transformations

Answer 10

maybe they have mis-typed either of the function u or v

Answer 11

this jacobian problem is pretty straightforward computation

Yes, or they really want partial (u,v) / partial (r, theta) or something like this
I think that's what they really want

Answer 12

partial (u,v) / partial (r, theta) gives 6r^3 sin 2 theta ?

Answer 13

checking if it does

Answer 14

it gives you 2r^3 sin 2\theta

uv-->xy = 2r^2 sin2\theta
xy-->r,theta = r

Answer 15

oh, jacobians can be multiplied that way ?

Answer 16

jacobian is just an exchange rate for areas between two coordinate systems
so you can safely multiply them

Answer 17

To be fully sure, we can calculate the jacobian for uv-->r,theta directly and see
but it looks like a painful computation

Answer 18

http://prntscr.com/4mpjzr

Answer 19

Okay... so that confirms I am not insane
You just stack the jacobians... we could have saved ourselves a lot of trouble and just multiplied by r form the area element for polar coordinates r dr d theta
jacobian is r

Answer 20

so the mysterious factor of 3 I don't know where they got it from
but that's why my suspicion is it's some kind of typo
maybe both in the jacobian they want and also the transformations

Answer 21

thanks Mira, rational! :)
I will mention that there is a typing error in the question ans it should be 2r^3 instead of 6r^3

Answer 22

yw :)