> 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
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.
> 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.
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'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.
> 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.