electromagnetism - When can I use $wedge$ instead of curl?

It seems in some circles the wedge product is used in preference to curl. I have a basic understanding of Green and Stokes' formula, I wish to use the $\wedge$ notation from now on.

Can someone tell me if this is commonly done, and if so what is the underlying assumption of the surface. If it is not too much to ask, can someone show me how to write say Maxwell's equations using $\wedge$ instead of Curl

Answer

The wedge product has its roots in exterior algebra. Exterior algebra lets you talk about objects like planes or volumes as algebraic elements of their own, separate from ordinary vectors, but still obeying the same notions of being "vectors" in their own vector spaces. The wedge product of two vectors is a bivector, and many concepts you may have been taught in vector calculus can be thought of in terms of bivectors instead. Normal vectors are just the unique vectors perpendicular to bivectors that are tangent to some surface instead. Rotations can be thought of as rotating within a bivector, instead of around a rotation axis (and incidentally, rotating within a bivector is a concept that still works in Minkowski space, unlike rotation axes).

Exterior algebra by itself is not enough, though: you need to have a way for vectors and bivectors to interact. There are a couple formalisms that give all the tools to do this:

1. 
   

   Differential forms. Forms are very common and extensively used in high-level mathematics and for advanced electromagnetism and general relativity. Differential forms contributes the machinery to do calculus on bivectors and the like, using the "exterior derivative" $d$. It also has the concept of Hodge duality, with the Hodge star operator $\star$ that turns bivectors in 3d to their normal vectors and vice versa. This ultimately gives you the power to use forms effectively in metrical contexts: like special and general relativity.

   
   
   


2. 
   

   Clifford algebra and geometric calculus. Clifford algebra is extensively used in quantum mechanics, through the so-called gamma or pauli matrices. The algebra of these objects, however, is worthy of study independent of the idea that these are matrices using matrix multiplication. In this mindset, you can use clifford algebra for run-of-the-mill 3d vector geometry and calculus. Clifford algebra introduces a "geometric product" of vectors instead, which incorporates both the metric and the exterior algebra into one nifty operation. Many of the core concepts are the same as with differential forms, but the notation is often a little closer to traditional vector calculus in look.

   
   

You should use wedge products anytime you're not in 3d space, as the cross product is only used in 3d or 7d and not an immediately generalizable concept.

A traditional differential forms writing of Maxwell's equations might be like this:

$$\star d (\star E) = \rho, \quad dE = -\partial_t B, \quad dB = 0, \quad \star d (\star B) = j + \partial_t E$$

This form takes $B$ as a 2-form (a bi-covector), but it's not particularly common, as forms are more often used in the context of spacetime, where $E$ is also a 2-form and the two fields come together in the Faraday 2-form $F$. Maxwell's equation in vacuum then take on the simpler form

$$d(\star F) = J, \quad dF = 0$$

In geometric calculus, the derivative operator $\nabla$ has the capability to do both divergences and curls in one operation, so the last equation is typically written

$$\nabla F = \nabla \cdot F + \nabla \wedge F = J + 0 = J$$

The relationship between wedges and the cross product, is as follows: In differential forms,

$$a \times b = \star (a \wedge b)$$

While in clifford algebra, we typically keep explicit the unit trivector $i$:

$$a \times b = i^{-1} (a \wedge b)$$