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Derivative and Jacobian of a transformation

ElijahRockers

Gold Member
270
10
1. The problem statement, all variables and given/known data

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

[itex]x=rcos\theta[/itex]
[itex]y=rsin\theta[/itex]

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?
 

lanedance

Homework Helper
3,304
2
the Jacobian matrix is the matrix of partial derivatives
[tex]
\begin{pmatrix}
\frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \\
\frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} \\
\end{pmatrix}
[/tex]

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

ElijahRockers

Gold Member
270
10
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:

lanedance

Homework Helper
3,304
2
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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