From: 3blue1brown
Curl is a fundamental concept in vector calculus, often discussed alongside divergence. It describes the tendency of a vector field to rotate around a given point [00:04:36]. While the concept is best understood by imagining the vector field as representing fluid flow, it applies to various other physical phenomena [02:03:00].
Understanding Curl through Fluid Flow Analogy
When a vector field is conceptualized as describing the velocity of a fluid at each point in space [02:03:00], curl measures how much this imagined fluid tends to rotate around a particular point [00:04:36].
- Rotation Tendency: If a small object, like a twig, were dropped into the fluid and fixed at its center, curl indicates whether it would tend to spin [00:04:41].
- Direction of Rotation:
- Regions where the rotation is clockwise are said to have positive curl [00:04:49].
- Regions where the rotation is counterclockwise have negative curl [00:04:54].
- Net Rotational Influence: Non-zero curl can occur even if not all surrounding vectors point in a rotational direction. For example, if flow is slow at the bottom of a region but quick at the top, it can still result in a net clockwise influence, indicating non-zero curl [00:05:03].
Dimensionality of Curl
While the visual intuition for curl is often provided in two dimensions [01:17:00], true, proper curl is a three-dimensional idea [00:05:19]. In three dimensions, each point in space is associated with a new vector characterizing the rotation around that point, determined by a right-hand rule [00:05:22]. However, for a two-dimensional analysis, curl associates each point in 2D space with a single number [00:05:38].
Applications of Curl
Even though the intuition for curl is often rooted in fluid dynamics, it holds significant importance in other areas of physics and mathematics.
Maxwell’s Equations
Curl is a key component in Maxwell’s equations, which describe electricity and magnetism [00:05:55]. The last two of these four equations illustrate how the change in one field (electric or magnetic) depends on the curl of the other field [00:07:13]. This interrelationship is also what gives rise to light waves [00:07:34].
Dynamic Systems
Curl can also be useful in contexts that do not initially appear spatial [00:07:37]. For instance, in analyzing dynamic systems like predator-prey population models, the “state” of the system (e.g., population sizes) can be represented as a point in a multi-dimensional “phase space” [00:07:55]. The rates of change of these variables form a vector field in this phase space [00:08:42]. Operations like curl can help understand the system’s behavior, such as the presence of cyclic patterns or stable/unstable cycles [00:09:56].
Notation and Connection to Cross Product
The curl of a vector field function (F) is commonly written as a cross product with the “nabla” operator (∇), denoted ∇ x F [00:10:51]. This notation is more than just a mnemonic; there is a real connection between curl and the cross product [00:11:09].
The cross product measures how perpendicular two vectors are [00:12:52]. In the context of curl, the cross product of a small “step” vector with the resulting change in the vector field’s output tends to be positive in regions of positive curl [00:12:57]. Conceptually, curl can be thought of as an average of this “step vector difference vector cross product” [00:13:08]. If a step in a certain direction leads to a change in the vector perpendicular to that step, it corresponds to a tendency for flow rotation [00:13:13].