Homework Help: Derivative and Jacobian of a transformation

1. Feb 13, 2012

ElijahRockers

1. The problem statement, all variables and given/known data

For the transformation, draw the lattice lines, calculate the derivative, and calculate the Jacobian.

$x=rcos\theta$
$y=rsin\theta$

3. The attempt at a solution

I drew the lattice lines correctly. What I am confused about is the derivative. Since x and y are both functions of r and theta, what derivative are they talking about? Wouldn't I have to take the partial with respect to R or theta? This section is supposed to be on calculating determinants.

I understand that the Jacobian is the determinant of a particular matrix, but where does this matrix come from?

2. Feb 13, 2012

lanedance

the Jacobian matrix is the matrix of partial derivatives
$$\begin{pmatrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} \\ \end{pmatrix}$$

the Jacobian determinant is the determinant of that matrix and is probably what you're referring to with the shorthand "Jacobian"

3. Feb 13, 2012

lanedance

4. Feb 13, 2012

ElijahRockers

Alright thanks. Yea I looked at the Jacobian wikipedia earlier, before I posted this, but it all just seemed greek to me. I am about to try to finish the assignment.

Thanks again, I'll let you know how it goes.

Last edited: Feb 13, 2012
5. Feb 14, 2012

lanedance

note if you had a small displacement (dr,dtheta)^T, multiplying this by the jacobian would give you the corresponding (dx,dy), similar to the chain rule