Differential operators in 2D curvilinear coordinates

[Robin04]

Homework Statement

I’m studying orthogonal curvilinear coordinates and practice calculating differential operators.
However, I’ve run across an exercise where the coordinate system is only in 2D and I’m confused about how to proceed with the calculations.

Homework Equations

A point in the plane is given by $u$ and $v$ coordinates where $u=\frac{\sqrt{x^2+y^2}+y}{2}$ and $v=\frac{\sqrt{x^2+y^2}-y}{2}$
I’m using these formulas to calculate the operators: https://en.m.wikipedia.org/wiki/Orthogonal_coordinates#Differential_operators_in_three_dimensions

The Attempt at a Solution

I’ve calculated the Lamé coefficients (scale factors) and got the following: $h_u=\sqrt{\frac{v}{u}+1}$, $h_v=\sqrt{\frac{u}{v}+1}$

In case of the gradient and the divergence I simply just didn’t calculate the third term but I don’t see how to do this with the curl and Laplace.

 

[Ray Vickson]

Homework Statement

I’m studying orthogonal curvilinear coordinates and practice calculating differential operators.
However, I’ve run across an exercise where the coordinate system is only in 2D and I’m confused about how to proceed with the calculations.

Homework Equations

A point in the plane is given by $u$ and $v$ coordinates where $u=\frac{\sqrt{x^2+y^2}+y}{2}$ and $v=\frac{\sqrt{x^2+y^2}-y}{2}$
I’m using these formulas to calculate the operators: https://en.m.wikipedia.org/wiki/Orthogonal_coordinates#Differential_operators_in_three_dimensions

The Attempt at a Solution

I’ve calculated the Lamé coefficients (scale factors) and got the following: $h_u=\sqrt{\frac{v}{u}+1}$, $h_v=\sqrt{\frac{u}{v}+1}$

In case of the gradient and the divergence I simply just didn’t calculate the third term but I don’t see how to do this with the curl and Laplace.

For the curl, you need a third dimension, so you could use coordinates $(x,y,z)$ or $(u,v,z)$ and a vector field of the form $\mathbf{A} = A(x,y) \mathbf{i} + B(x,y) \mathbf{j} + 0 \mathbf{k}$.

 

[Robin04]

For the curl, you need a third dimension, so you could use coordinates $(x,y,z)$ or $(u,v,z)$ and a vector field of the form $\mathbf{A} = A(x,y) \mathbf{i} + B(x,y) \mathbf{j} + 0 \mathbf{k}$.

And what should I do with the scale factor? Is it 1 in this case?

 

[Ray Vickson]

And what should I do with the scale factor? Is it 1 in this case?

The link you cite provides all the information you need; just use the formulas given there.

 

[Robin04]

The link you cite provides all the information you need; just use the formulas given there.

I think I'm missing something here. So the curl only exists in three dimensions therefore I must introduce arbitrarily a third coordinate?

 

[Ray Vickson]

I think I'm missing something here. So the curl only exists in three dimensions therefore I must introduce arbitrarily a third coordinate?

I thought that is what I said in #2.

 

[Robin04]

I thought that is what I said in #2.

Aah, I see. And does it make sense that only the Z component of the curl is non zero? I’m still struggling to understand what these operators really mean.

 

[Ray Vickson]

Aah, I see. And does it make sense that only the Z component of the curl is non zero? I’m still struggling to understand what these operators really mean.

For any vector field $\mathbf{A}(x,y,z)$ the curl of $\mathbf{A}$ is pependicular to $\mathbf{A}$, so if the field lies in the $(x,y)-$plane the curl points along the $z$ direction.

 

[Robin04]

For any vector field $\mathbf{A}(x,y,z)$ the curl of $\mathbf{A}$ is pependicular to $\mathbf{A}$, so if the field lies in the $(x,y)-$plane the curl points along the $z$ direction.

Thank you very much for your help! :)