How to know which variable comes first in the Jacobian?

[Rijad Hadzic]

Homework Statement

Find the Jacobian of the transformation:

$x = e^{-r}sinθ , y = e^rcosθ$

Homework Equations

The Attempt at a Solution

formula for Jacobian is absolute value of the determinant

$\begin{vmatrix} \frac {∂x}{∂u} & \frac {∂x}{∂v}\\ \frac {∂y}{∂u} & \frac {∂y}{∂v}\\ \end{vmatrix}$

But how am I suppose to know which one set u and v = to?

For example, if u = r and v = θ, my answer is (sinθ)^2 - (cosθ)^2 which is my books answer, which is different then when u = θ and v = r, where the answer is (cosθ)^2 - (sinθ)^2

 

[Dick]

Homework Statement

Find the Jacobian of the transformation:

$x = e^{-r}sinθ , y = e^rcosθ$

Homework Equations

The Attempt at a Solution

formula for Jacobian is absolute value of the determinant

$\begin{vmatrix} \frac {∂x}{∂u} & \frac {∂x}{∂v}\\ \frac {∂y}{∂u} & \frac {∂y}{∂v}\\ \end{vmatrix}$

But how am I suppose to know which one set u and v = to?

For example, if u = r and v = θ, my answer is (sinθ)^2 - (cosθ)^2 which is my books answer, which is different then when u = θ and v = r, where the answer is (cosθ)^2 - (sinθ)^2

Since you've defined the Jacobian as the absolute value of the determinant, it doesn't matter. Those answers have the same absolute value. Changing sign is what happens in general when you interchange rows or columns of a matrix.

 

[Ray Vickson]

Homework Statement

Find the Jacobian of the transformation:

$x = e^{-r}sinθ , y = e^rcosθ$

Homework Equations

The Attempt at a Solution

formula for Jacobian is absolute value of the determinant

$\begin{vmatrix} \frac {∂x}{∂u} & \frac {∂x}{∂v}\\ \frac {∂y}{∂u} & \frac {∂y}{∂v}\\ \end{vmatrix}$

But how am I suppose to know which one set u and v = to?

For example, if u = r and v = θ, my answer is (sinθ)^2 - (cosθ)^2 which is my books answer, which is different then when u = θ and v = r, where the answer is (cosθ)^2 - (sinθ)^2

You and the book are both mistaken. The absolute value of the Jacobian determinant is $|\sin^2 \theta - \cos^2 \theta|$, so the order does not matter.