# Generalizing: Curl of a Function

• pbandjay
In summary, the curl of a vector field can be generalized to dimensions greater than three if the vector field is represented by a 1-form and its exterior derivative is a 2-form.
pbandjay
Hello. I have tried to search all day for the answer to my question but could not find anything.

Is there a way to generalize the curl of a vector field to dimensions greater than three? It seems pretty straightforward to find the gradient and divergence of higher dimensional vector fields, but my calculus book only offers the definition: curl F := ∇×F, which appears to only work if F has three components. Or am getting something wrong?

I am also only in my first real analysis course, so I have not come across this kind of topic in that class yet.

Not easily. The problem is that the best way to generalize the cross product itself is to use the "alternating tensor". That is the operator represented by $\epsilon_{i_1i_2\cdot\cdot\cdot i_n}$ equal to 1 if $i_1i_2\cdot\cdot\cdot i_n$ is an even permutation of $1 2 3 \cdot\cdot\cdot\ n$, -1 if it s an odd permutation, and 0 if neither of those is true.
For example, when n= 3, $\epsilon_{ijk}$ has 27 entries. $\epsilon_{123}= \epsilon_{231}= \epsilon_{312}= 1$, $\epsilon_{132}= \epsilon_{213}= \epsilon_{321}= -1$, and the other 21 entries are all 0.

With that $w_i= \sum_{j=1}^3\sum_{k=1}^3\epsilon_{ijk}u_jv_k$ gives $w_1= \epsilon_{123}u_{2}v_{3}+ \epsilon_{132}u_3v_2= u_2v_3- u_3v_2$
$w_2= \epsilon_{213}u_1v_3+ \epsilon_{231}u_3v_2= -u_2v_3+ u_3v_2$
$w_3= \epsilon_{312}u_1v_2+ \epsilon_{321}u_2v_1= u_1v_2- u_2v_3$
precisely the cross product.

But, in n dimensions, $\epsilon$ has n indices. In order to get a vector result, we would have to multiply n-1 vectors, not just two.

In a general space whose dimension is not necessarily three, the generalization of curl is a Hodge dual of an exterior derivative of the vector field. The vector field is a 1-form, its exterior derivative is a 2-form, and its Hodge dual is a (n-2)-form. Which is a scalar (a number) for n=2, a vector for n=3, a higher-rank tensor for n>=4.

Thank you for the help! Looks like I have a few things to learn about the topic.

## 1. What is the definition of curl?

The curl of a function is a vector operator that describes the rotation or circular movement of a vector field at a given point.

## 2. How is curl calculated?

Curl is calculated by taking the partial derivatives of the function with respect to each of the three coordinates (x, y, z) and then taking the cross product of these derivatives.

## 3. What does the curl of a function represent?

The curl of a function represents the local rotation or angular velocity of a vector field at a specific point.

## 4. What is the significance of the curl in physics?

The curl of a vector field is important in physics as it helps describe the rotational properties of electromagnetic fields and fluid flow. It is also used in the equations of motion for rotating bodies.

## 5. Can the curl of a function be generalized to higher dimensions?

Yes, the concept of curl can be generalized to higher dimensions, but it becomes more complex and is known as the "curl operator". In 4 or more dimensions, the curl operator is a tensor rather than a vector.

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