Find the following relations for x and​ y, and compute the Jacobian​ J(u,v)?

Question

Solve the following relations for x and​ y, and compute the Jacobian​ J(u,v).

u=x+3y

v=2x-y

what is x and y

Answer 1

Hi Samir! I hope you're doing well and that this answer isn't too late.

A Jacobian is used when you are converting an area (or volume) into a different coordinate plane. Basically, we take some region in the XY-plane and convert it into a UV-plane in order to make integration of that region easier. Doing this requires us to multiple the integral by a "Jacobian".

First, let's get our u and v equations into x and y ones. This just takes some algebra:

u=x+3y --> x=3y-u Just rearrange the equations.

v=2x-y --> y=2x-v

y=2(3y-u)-v Now, we use substitution to solve for y and x.

y=6y-2u-v

-5y=-2u-v

y=2/5u+1/5v Here's our y equation! Now, let's just substitute this in to find our x equation.

x=3(2/5u+1/5v)-u

x=6/5u+3/5v-u

x=1/5u+3/5v Here's our x equation.

Now, we can compute the Jacobian. Here's the equation for computing a Jacobian: ∂(x,y)/∂(u,v). We can compute this using the determinant of the matrix formed by ∂(x,y)/∂(u,v):

(∂x/∂u)(∂y/∂v) - (∂x/∂v)(∂y/∂u)

We do end up taking the absolute value of this determinant when we put it in our integral, though we aren't actually asked to integrate anything in this problem. Here's our matrix:

|1/5 3/5|

|2/5 1/5|

Here's our determinant: (1/5)(1/5)-(2/5)(3/5) = -1/5

And, that's our answer!