On a tangent: I remember asking my Calculus 101 professor what the "intuitive meaning" of divergence and curl was, outside of the formal math and equations. He was shocked that one could ask to sully these perfect mathematical concepts with dirty intuitive reductions. A guide like this, or 3-Blue-1-Brown videos would have made my day, then.
<math term> is a <math term2> that <math term3s> a <math term4>'s <math term5>. It is an example of a <math term6> that cannot <math term7> a <math term8>.
Me: I learned nothing from that.
(Yes, which is partly my fault, but it's really not helpful for an intro to any math topic unless you already have mastered 99% of the topic and just want to resolve the last few things. The encyclopedia model really goes against what you need for the general population here. Arbital tried to do a more helpful model here but is defunct; Khan Academy is a generally better for this but requires more investment.)
In this community, we’ve witnessed the difficultly first-hand with the much-maligned monad tutorials. A mathematician would have no trouble understanding monads —- they’re just a matter of reading the definition and the laws. That’s what you do every day when you’re studying math in university. Read some definitions and some theorems, play with things a bit, try to prove stuff, then move on to the next topic.
That said, Wikipedia's mathy pages have proved repeatedly useful to me as a reference, e.g., whenever I remember a mathematical concept only vaguely or intuitively and just need to find a detailed formalization to implement it in code. My browser is always open, so Wikipedia is often the "reference of least resistance."
FWIW, there's Simple Wikipedia (https://simple.wikipedia.org), but it is, quite frankly, terrible for learning about anything of a mathematical nature beyond high-school algebra.
Perhaps Wikipedia should add a "Learn About This" section to math, physics, engineering, and hard-science pages?
Not that they should for some reason, but too often editing wikipedia is presented/thought of as something reserved to a happy few with deep social consequences, when it's in fact the simplest thing in the world.
Rigour is better left to computers these days. They are better at following rules. We can have better proofs with software like metamath. Surprisingly, you can even gain deep insights by writing automated proofs (compared to manual proofs). Ideally, practitioners should consider simple and clear explanations as their primary goal and equations and proofs as necessary supplements. Think of this as literate programming for mathematics.
Its the same misguided thinking that leads to the popular PopSci explanation of electron behavior being both wave and particle like. Why should we try to categorize the behavior a subatomic particle (an entity so distant from our experiential reality) as being analogous to one of two macroscopic entities?
I totally get the desire for intuitive understanding, and it should be encouraged, but sometimes you just have to put intuition aside and come to conclusions with pure mathematical reasoning.
It's one thing that Structure and Interpretation of Classical Mechanics (which I haven't read) got really right in concept. It's a shame, that approach hasn't been adopted more widely.
Take for example page 7 of the pdf with the heat equation. Even the description of the Laplacian is wrong. The value isn't the average of the points surrounding it. But if this simple function were written in python as a "update" function over a multi-dim array of temperature values it'd be clear exactly what's happening.
Another example of simple inconsistency in notation: superscript, does that mean squared or not? When writing two terms adjacent is it multiplication or an operator being applied? Where is the behavior of the operator defined that a student and "read the code"?
English is an extremely verbose and imprecise language. Mathematicians replaced a verbose imprecise language with a terse imprecise language. It's about time for rigorous fields to take the last step and introduce precise concepts using a precisely defined language.
The description on page 7 is split into two parts. The first part describes the value of the Laplacian correctly:
> It tells us how the temperature value at the point compares to the average value of its neighboring points.
The second part describes the long-run behavior of the differential equation
> The temperature value that this point takes is the average temperature of the points surrounding it
> The temperature value that this point takes is the average temperature of the points surrounding it
And this is not correct. Excluding a final state when all temperature values are identical, at no point is the temperature value equal to the average of the surrounding values. You aren't describing and evolution if its inaccurate for all points except a final state.
On top of that, even during the evolution the temperature isn't what's taking on the "average" of surrounding points, it the change in temperature wrt time that's being change by the average of points. And yet again, it's not an average of the surrounding temperatures is and average of the surrounding differences in temperature.
And this interaction we've had highlights exactly my point. English is an impressive language, and Mathematics is full of hand wavy explanations that come with simplicity context that isn't explicity and precisely defined. Mathematicians are used to it, and of course it's learnable like anything. My point is it's not necessary, it evolved in a different time with different constraints. Its a similar example of "the medium is the message" - math evolved when the only way to write was by hand, and duplication of definitions was manually expensive. We don't need these handwavy shortcuts now - we can make it easier for students to learn by being precise and providing definitions for them to see exactly what's happening - not expect them to learn from trying to recall every imagined context. And the fall back of "I had to learn it this way, so they should too", is a horrible excuse.
The average of the surrounding points is the attractor temperature for the system. It is an asymptote which the temperature of the point is moving towards. It's like saying an oscillator (such as a spring) wants to be at neutral, even though it never comes to rest at neutral.
I'm not engaging with your larger point, I'm just quibbling with you saying page 7 is wrong. I think that "takes" in the english description says that the temperature approaches the value over time, whereas you interpreted "takes" as referring to the temperature at every point in time.
I hear you, and I think we've probably approaching the end of the productive part of our conversation.
I do want to mention that the quote above, and interpretation of "takes" is exactly my larger point though. These definitions are all sloppy and prone to interpretation. Precise definitions would eliminate the need for all of this.
And since I can't help myself, thinking about this a little more, even your interpretation above is either faulty, or has to change the definition of "surrounding". If I have a point of average temperature, a doughnut or sphere of warmer points a small distance immediately around it and then the majority of all other points around that being colder, then the asymptote is actually towards the average of the of outer colder points, not the surrounding warmer points.
There's also no such thing as 'observation'. Fields interact by a precise and well-tested mathematical function. That function does have the effect of mutating the state, but why should there be such a thing as pure observation? There's nothing magic about conscious observation.
At a fundamental conceptual level it does not map to anything in our normal experience - attempting to force analogies etc can end up confusing things further.
I agree with your original observation btw, but it's important to recognise the limits we may be bound by in explanation and comparison.
And, or course, the vector you get from a cross product only has 3 degrees of freedom because we live in a 3D world. The cross product of two 2D vectors is a scalar, the cross product of two 4D vectors needs 6 numbers to describe it. Even in 3D, if you reflect the original vectors in a mirror, the cross product now points the opposite direction, i.e. it depends on the handednes of the coordinate system.
Look at a topographical map of a landscape and note the contour lines. As you zoom into the contour lines (if they're detailed enough), they'll start to look more and more like parallel lines, densely spaced for a steep slope, or sparsely spaced for a mild slope. These parallel lines are the gradient.
A gradient and a vector 'fit together' to give a real number. The more parallel lines the vector pierces, the bigger the number [0]. So a gradient 'eats' a vector, spitting out a real number, and vice-versa. Just like a row vector 'eats' a column vector and spits out a real number.
I'm saying 'gradient' here, but really what I mean is 'one-form'. Language deliberately imprecise for all y'all mathematicians out there.
[0] https://neutronstars.utk.edu/class/p616_s19/lect3/index.html...
Would really like to have that free time to delve and be able to suggest the alternative formulation myself.
But I think it's important to remember that mathematics is not some evil thing! There are people who train carefully so that the terse mathematical explanations are the intuition. There's nothing wrong with not being one of those people, but knocking the mathematical explanation is denying them their intuition just as their explanation denies you yours.
In a word, it takes all kinds, and I think we should all be as patient with others' ways of learning and understanding as we hope that they will be with ours.
Intuitively, divergence is how much fluid is being created (or destroyed) at a point, curl is how much a waterwheel would spin if placed at a point.
Personally, div and curl didn't quite click for me until I took fluid dynamics in my final year. I could do the homework but didn't really get why they were useful until it made sense why "div = 0 always for incompressible fluids".
source: went to intense STEM school