Derivative and Jacobian of a transformation

[ElijahRockers]

Homework Statement

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

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

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?

 

[lanedance]

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

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

 

[lanedance]

see here for more
http://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant

 

[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.

 

[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