I am math enthusiast who is always trying to expand his knowledge.
Now I have one inquiry about vector calculus:
Is there any “better” or “more reputable” notation for divergence and curl? I have seen two notations:
- For curl, $ \text{curl}(\mathbf{E})$ and $ \nabla \times \mathbf{E}$
- For divergence, $ \text{div}({\mathbf{E}}) $ and $ \nabla \cdot \mathbf{E} $
Both used in various educational videos and textbooks. I have also seen both notations used in Maxwells’ equations. Sometime I have also seen the divergence and curl notations written without brackets, in the form $ \text{div}\mathbf{E} $ and $\text{curl}\mathbf{E} $.
Can we use them interchangeably? On my side, I prefer $ \text{div}(\mathbf{E}) $ and $ \text{curl}(\mathbf{E}) $, but if the dot and cross products are more widely accepted notations them I will use them from that point on.
For example, I have seen Gauss’ law for magnetism and Faraday’s law of induction in the two following ways:
- Gauss’ law for magnetism as $ \text{div}(\mathbf{B}) = 0 $ and as $ \nabla \cdot \mathbf{B} = 0 $
- Faraday’s law of induction as $ \text{curl}(\mathbf{E}) = -\frac{\partial\mathbf{B}}{\partial t} $ and as $\nabla \times \mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t} $
This is another case:
In the YouTube video by 3Blue1Brown about divergence and curl, $ \text{div}(\mathbf{E}) $ and $ \text{curl}(\mathbf{E}) $ are used, while the cross and dot products are used in the Wikipedia article about Maxwells’ equations.
Is it a thing to say that the text notation is used to make the conceptual idea clearer, while the dot and cross products are used in the more professional context?
Also, why is divergence represented as a dot product while curl is a cross product? And why is del (also called nabla) there with the dot and cross product?
Some references are given below:
- This video (Khan Academy – Divergence 1) says that text is the correct way to write it but that the dot product is a mnemonic.
- This article (Wikipedia Page – Divergence) uses them both.
A quotation from the Wikipedia about “Curl (mathematics)” is:
“ The notation curl F is more common in North America. In the rest of the world, particularly in 20th century scientific literature, the alternative notation rot F is traditionally used, which comes from the "rate of rotation" that it represents. To avoid confusion, modern authors tend to use the cross product notation with the del (nabla) operator, as in $ {\displaystyle \nabla \times \mathbf {F} } $ which also reveals the relation between curl (rotor), divergence, and gradient operators.”
—— Curl (mathematics) (5 Feb, 2024)
How is it that I have never heard of $\text{rot}(\mathbf{F})$ in my life despite not being in North America?
Edit: Looks like both are interchangeable, and the cross and dot products are supposed to remind of the fact that $\nabla \cdot \mathbf{F} = \text{a dot product with nabla which is scalar-valued}$, and that we can similarly think of the curl.
Best Answer
The short answer is, you can use whichever you prefer.
The text notation is perhaps clearer at a conceptual level.
The nabla ($\nabla$) notation serves also as a mnemonic for the formulas involved. In 3D Euclidean space, you can think of $\nabla$ as the "vector"*: $$\left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right)$$ Then the formula for the divergence can be thought of as dot product with $\nabla$, and the formula for curl can be thought of as cross product with $\nabla$, so that's where we get the notations $\nabla \cdot \mathbf{F}$ and $\nabla \times \mathbf{F}$ from.
(*) Of course, this is not a vector of numbers, it's just a mnemonic. But if you want you can even formalize it as a vector of linear operators acting on 3D vector fields.