Divergence and curl: The language of Maxwell's equations, fluid flow, and more
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when someone asks for a visual, intuitive picture of divergence and curl, or why the del notation looks like a dot and cross product.
Recommended by 10 people across 9 different months, 2019 to 2024.
Why Physics Stack Exchange recommends it
Physics Stack Exchange users link it when someone asks for a physical or visual picture of curl and divergence, especially in electromagnetism and fluid contexts.
Recommended by 6 people across 6 different months, 2018 to 2022.
What it is
A 3Blue1Brown animated explainer on vector fields and the two core operations of vector calculus, divergence and curl. It builds fluid-flow intuition, connects it to Maxwell's equations, and ends by explaining the del notation with the dot and cross product.
Cited in 2 Wikipedia articles, including Curl (mathematics), Divergence.
What people said
Strong rec bluesky · 2021-02 · Mathematics Stack Exchange · on “How do I imagine why divergence of curl and curl of gradient is $0$?”
Imagining water flows or looking at Maxwells equations is definitely a good idea. A video for visualizing I can strongly recommend is this one: [this video] Lets say $\vec{F}$ is a vector field. First start by actually understanding what div( $\vec{F}$ ) really means. It represents the sources and sinks of the flow. In terms of electrodynamics these are the electrons and protons as they create the field. The curl on the other hand is a measure how much a field (flow) rotates around a certain point (for visuals watch the video). A rotation does not have a starting or end point or phrased differently it does not have a source or a sink. That is why the divergence of curl of $\vec{F}$ must be zero.
Recommends Toby Peterken · 2018-06 · Physics Stack Exchange · on “How to know whether electric field is rotating without divergence? (basic question)”
In the bottom field, it will rotate at every point, as the outside vectors are greater. In the top field it will move in a circle but not rotate as the outside vectors are smaller (however it will rotate in the centre). I just found this video that might be useful [this video]
Recommends e_s_ · 2019-05 · Mathematics Stack Exchange · on “Intuition of divergence and curl”
There's a nice video on divergence and curl by 3Blue1Brown that gives that gives some great visual intuition: [this video]
Recommends AXO · 2019-09 · Mathematics Stack Exchange · on “Could you a give a intutive interpretation of curl?”
The link in the accepted answer is broken so here are some videos that I found: Divergence and curl: The language of Maxwell's equations, fluid flow, and more [this video] (by 3Blue1Brown [LINK] ) A few more videos on Khan Academy > Math > Multivariable calculus > Derivatives of multivariable functions > Curl [LINK] including: 2d curl intuition [LINK] 2d curl formula [LINK] 2d curl example [LINK] 2d curl nuance [LINK] Describing rotation in 3d with a vector [LINK] 3d curl intuition, part 1 [LINK] 3d curl intuition, part 2 [LINK] 3d curl formula, part 1 [LINK] 3d curl formula, part 2 [LINK] 3d curl computation example [LINK]
Recommends Ajay Mohan · 2020-03 · Physics Stack Exchange · on “How does the physical meaning of curl is in agreement with these scenarios?”
In the link [LINK] that OP has cited, the author writes the following. The curl of a vector field [at a given point] measures the tendency for the vector field to swirl around [the given point]. Swirling is different from a mere curving of the vector field. If the sentence is misinterpreted, it would seem to imply that if a vector field merely curves at some point, then it definitely has a non-zero curl at that point. This misinterpretation is not true: e.g. this [LINK] Math.SE answer (and the examples that OP has mentioned). The characteristic of swirling is difficult to ascertain from a simple visual inspection of the field. Instead, a good intuitive interpretation is to imagine the vector
Recommends Davi Barreira · 2020-06 · Mathematics Stack Exchange · on “Divergence of cross product of two vectors, intuition”
After hustling with this identity for a bit, this is what I was able to come up with. First thing to pay attention to is that $\nabla \cdot (\vec A\times \vec B )$ is the divergence of the cross product vector field. The interpretation for the cross product vector field depends on the domain of the problem, but we can abstract that and just think that $\vec A\times \vec B$ generates a new vector field $\vec C$ . Hence, $\nabla \cdot \vec C$ can be understood using the common intuition for divergence, which is that it measures the amount of "fluid" being removed or added to the field (sink/source). For a better explanation on this, you can check out this videos by ThreeBrownOneBlue [this video] .
Recommends novawarrior77 · 2020-08 · Physics Stack Exchange · on “Relation between computation of curl and divergence and their formal definitions”
For some intuition on the $del$ operator formulas of divergence and curl, take a look at the last portion of this video (the whole thing is excellent): 10:18-13:28 [this video]
Recommends jng224 · 2021-06 · Mathematics Stack Exchange · on “What is divergence of a field?”
Re "What is divergence of a field?" – 3blue1brown has made a video [this video] which visually explains divergence and curl.
Recommends jng224 · 2021-08 · Physics Stack Exchange · on “Non-conservative electromagnetic fields $ \displaystyle \nabla \times \vec{E} = - \frac {\partial \vec{B} }{\partial t}. $”
Re. "How can I physically visualise curl?" – you might want to check out this video [this video] by 3b1b which explains divergence and curl conceptually.
Recommends Hans Lundmark · 2021-11 · Mathematics Stack Exchange · on “How do you understand the nature of curl?”
You'll need to take into account the speed of the flow. If the flow on one side of the ball is faster than on the other side, it will cause the ball to rotate, even if the velocity vectors are parallel. Have a look at this 3b1b video: [this video] (the part about the curl starts at 4:32).