# Derivative and Jacobian of a transformation

#### ElijahRockers

Gold Member
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?

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#### lanedance

Homework Helper
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"

Homework Helper

#### ElijahRockers

Gold Member
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.

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#### lanedance

Homework Helper
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

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