# Curl of a vector in a NON-orthogonal curvilinear coordinate system

• MasterD
So, for the given non-orthogonal curvilinear coordinate system in 3D, the curl of a vector can be defined as the Hodge dual of the exterior derivative of a covariant vector, where the exterior derivative is defined by (dv)_{ij} = \partial_i v_j - \partial_j v_i and the Hodge dual is defined by (\nabla \times v)^i = \sqrt{g} \epsilon^{ijk} (dv)_{jk}. There may be some factors missing and care must be taken with covariant and contravariant vectors, but the definition remains the same in all coordinate systems. Further information can be found on the Wikipedia article on curl.
MasterD
Hi,

I have a certain NON-orthogonal curvilinear coordinate system in 3D (in the metric only $$g_{13}=g_{23}=g_{31}=g_{32}=0$$) and I want to take the curl ($$\nabla\times\mathbf{v}$$) of a vector.

Any idea on how to do this? The only information I can find is about taking the curl of a vector in an orthogonal curvilinear coordinate system.

Very much thanks in advance for any insights.

Dirk

The thing that's more natural to define than curl (ie, it's defined in any dimension, where as curl is only defined in 3D) is what's called the exterior derivative of a (covariant) vector. It's defined by (in any coordinate system):

$$(dv)_{ij} = \partial_i v_j - \partial_j v_i$$

The curl is then the "Hodge dual" of this, defined by:

$$(\nabla \times v)^i = \sqrt{g} \epsilon^{ijk} (dv)_{jk}$$

where g is the determinant of the metric, and $\epsilon^{ijk}$ is the Levi-civita symbol. I might have some factors missing, but you can check this by computing simple cases.

Ok; thanks a lot; I will look further into this.

One more question: Shouldn't there be any scaling WITHIN the exterior derivative?

Something like: $$(dv)_{ij}=\frac{1}{h_1}\partial_i h_1 v_j - \frac{1}{h_2}\partial_j h_2 v_i$$ ?

No, the definition of the exterior derivative takes the same form in all coordinate systems. That's what's nice about it. But you need to be careful about covariant vs. contravariant vectors, and there are some constant factors associated with the Hodge dual (these depend partly on convention, but in this case you'll need to pick them to match up with the usual definition of curl). Look up the wikipedia article on curl, it has a more explicit version of the formula I mentioned.

$$(\nabla \times v)^i = \frac{1}{\sqrt{g}} \epsilon^{ijk} (dv)_{jk}$$

- in a coordinate basis.

## What is the curl of a vector in a non-orthogonal curvilinear coordinate system?

The curl of a vector in a non-orthogonal curvilinear coordinate system is a measure of the rotation of the vector field at a specific point in the coordinate system. It takes into account the change in direction and magnitude of the vector as it moves through the coordinate system.

## How is the curl of a vector calculated in a non-orthogonal curvilinear coordinate system?

The curl of a vector in a non-orthogonal curvilinear coordinate system is calculated using the general formula:$\bg_white&space;\textbf{curl}&space;\textbf{F}&space;=&space;\Big(&space;\frac{1}{h_1h_2h_3}&space;\frac{\partial&space;F_3}{\partial&space;x_2}&space;-&space;\frac{1}{h_1h_3h_2}&space;\frac{\partial&space;F_2}{\partial&space;x_3},&space;\frac{1}{h_1h_3h_1}&space;\frac{\partial&space;F_1}{\partial&space;x_3}&space;-&space;\frac{1}{h_1h_2h_1}&space;\frac{\partial&space;F_1}{\partial&space;x_2},&space;\frac{1}{h_2h_1h_2}&space;\frac{\partial&space;F_2}{\partial&space;x_1}&space;-&space;\frac{1}{h_2h_3h_2}&space;\frac{\partial&space;F_3}{\partial&space;x_1}&space;\Big)$

## What are the units of the curl of a vector in a non-orthogonal curvilinear coordinate system?

The units of the curl of a vector in a non-orthogonal curvilinear coordinate system depend on the units of the vector field and the coordinate system. Generally, it will have the same units as the vector field divided by the product of the coordinate system scaling factors.

## What is the significance of the curl of a vector in a non-orthogonal curvilinear coordinate system?

The curl of a vector in a non-orthogonal curvilinear coordinate system is an important quantity in fluid mechanics, electromagnetics, and other fields where vector fields are present. It can be used to determine the rotational behavior of a vector field and can be used in equations to model and predict physical phenomena.

## How does the curl of a vector in a non-orthogonal curvilinear coordinate system relate to the divergence of the vector?

The curl of a vector in a non-orthogonal curvilinear coordinate system and the divergence of the vector are related through the vector identity known as the curl-divergence theorem. This theorem states that the integral of the curl of a vector over a closed surface is equal to the integral of the divergence of the vector over the volume enclosed by that surface. This relationship is used in many mathematical and physical applications.

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