Why is the Jacobian for polar coordinates sometimes negative?

In summary, the Jacobian for polar coordinates can be negative due to the orientation of the coordinate system. Specifically, when transforming from Cartesian to polar coordinates, the determinant of the Jacobian matrix can yield a negative value if the transformation involves a reflection or if the angle is measured in the opposite direction. This negative value indicates a change in orientation, highlighting that the area element in polar coordinates may be inverted in certain contexts.
  • #1
laser
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Homework Statement
description
Relevant Equations
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723883159279632384.png


Proving this geometrically [1] gives ##J = r.##

Why is the ##-r## one wrong? Why is ##(x, y) \rightarrow (\theta, r)## is different from ##(x, y) \rightarrow (r, \theta)##? Edit: In Paul's Notes [2] it seems like ##J## is always positive, but online says it can be negative...

[1] The first answer on https://math.stackexchange.com/questions/1656814/how-to-prove-dxdy-r-dr-d-theta
[2] https://tutorial.math.lamar.edu/Classes/CalcIII/ChangeOfVariables.aspx
 
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  • #2
I don't think I can delete this post - but I think I've figured out the issue. ##J## can be positive or negative but in the integration formula we take the absolute value of ##J##. My bad for reading it wrong!
 
  • #3
If you switch a row or column of a determinant, the sign of the determinant changes. Regarding the text in the image you posted, in both cases the transformations are from polar to Cartesian rather than from Cartesian to polar as written in that text. The relevant equations are ##x = r\cos(\theta)## and ##y = r\sin(\theta)##.

The equations for converting from Cartesian to polar are different.
 
  • #4
laser said:
##J## can be positive or negative but in the integration formula we take the absolute value of ##J##. My bad for reading it wrong!
How is your post related to integration? You didn't mention anything of the sort in your earlier post.
 
  • #5
Mark44 said:
How is your post related to integration? You didn't mention anything of the sort in your earlier post.

Aside from in the title, which contains "dA = rdrdtheta".

@laser: The change of varaible formula in area integrals is derived from [tex]
\mathbf{n}\,dS = \frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v}\,du\,dv[/tex] so that [tex]
dS = \left\| \frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v} \right\|\,du\,dv.[/tex] Swapping the order of [itex]u[/itex] and [itex]v[/itex] changes the direction of [itex]\mathbf{n}[/itex] but does not change [itex]dS[/itex]. We treat 2D as being in the plane [itex]z = 0[/itex], so that the cross-product reduces to [tex]
\left(\frac{\partial x}{\partial u} \frac{\partial y}{\partial v} - \frac{\partial x}{\partial v}
\frac{\partial y}{\partial u} \right)\mathbf{e}_z[/tex] and taking the norm of this gives [tex]
\left| \frac{\partial x}{\partial u} \frac{\partial y}{\partial v} - \frac{\partial x}{\partial v}
\frac{\partial y}{\partial u} \right|.[/tex]
 
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  • #6
Just to point out that this change of coordinates is only valid locally, i.e., there's no global change of coordinates in this case.
 

FAQ: Why is the Jacobian for polar coordinates sometimes negative?

1. What is the Jacobian in the context of polar coordinates?

The Jacobian in the context of polar coordinates refers to the determinant of the matrix that represents the transformation from Cartesian coordinates (x, y) to polar coordinates (r, θ). It describes how area scales when converting between these two coordinate systems.

2. How do you calculate the Jacobian for polar coordinates?

To calculate the Jacobian for polar coordinates, we start with the transformations: x = r cos(θ) and y = r sin(θ). The Jacobian matrix is then formed by taking the partial derivatives of x and y with respect to r and θ. The determinant of this matrix, which is |J| = r, gives the Jacobian for the transformation.

3. Why might the Jacobian be negative in polar coordinates?

The Jacobian itself is not negative; it is always positive (r ≥ 0). However, the sign of the Jacobian can be influenced by the orientation of the coordinate system. In certain contexts, particularly when considering the direction of integration or the orientation of the area element, the Jacobian can be treated as negative if the transformation reverses the orientation.

4. What does it mean for a Jacobian to be negative in a transformation?

A negative Jacobian in a transformation indicates that the transformation reverses the orientation of the space. This can happen in transformations that involve reflections or when the limits of integration are switched, leading to a change in the direction of integration.

5. How should one handle the sign of the Jacobian when performing integrals in polar coordinates?

When performing integrals in polar coordinates, it is crucial to consider the limits of integration and the orientation of the region being integrated over. If the transformation involves a reversal of orientation, the integral should account for this by potentially introducing a negative sign. However, since the Jacobian itself is positive, one typically does not need to worry about its sign unless the context explicitly involves orientation-reversing transformations.

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