timo

The professor is preparing you for the next step where he/she says "and now assume 2 and 5 were any arbitrary value: all steps I did still hold true, so...": "When Tv1 = av1 and Tv2 = bv1: If a!=b then v1 and v2 are not colinear". Either that or for some other reason or for no reason whatsoever.

I share MigL impression that this forum is dominated by non-science topics. And not in a good sense, where people talk about other hobbies or exchange baking recipes or funny pictures of their cats. But in the sense that most activity goes into preaching/trolling (religion or culture war stuff) and people taking the bait. The problem is not only the OPs but also the people reacting to nonsense posts. Phi speaks of discussions that "appear to be a good time". And I can understand that from a moderator's perspective. But I find it a bit embarrassing. I feel this trend and the inverse relation between quality of a thread and the number of reactions it gets has been there for a long time, like 10+ years. And it is one of the reasons why my activity here has been declining to the point that ... well, my last post was in 2021. In fact, not completely coincidentally one of my last posts here contains a rant about that situation I an not sure if the "good old times" really actually existed at all. If not, then it could just be that your interests change, an everyone else feels everything is fine. It would be interesting if we could get some objective measure of this. But let's assume the trend exists: I do not think that "our interests have changed" is a likely reason - at least not on the forum level. Granted, people's interests and life situations do change. One of my drops in activity was when I started my PhD and the topics I was interested in became too narrow for a discussion forum (and also not Sci-Fi enough - not sure I saw a thread about Statistical Physics on this forum ever). Plus, much better opportunities to discuss these interests opened up. But you would expect that people dropping out are replaced by new people. I think the trend may be a symptom of something else: Forums are dead and people have moved on to other forms of communications. That's just a thought of course. I don't have data to verify it.

Joigus already mentioned is implicitly, but I think it's worth pointing it out explicitly: When it comes to the exact integral, the different methods with equal-width rectangles that approach dx->0 do approach the same limit (with some exceptions that are not relevant here). And this common limit is called "the integral". For many important functions, e.g. polynomials, we know how to compute the limit exactly. And we don't even care about the rectangle construction in these cases, and just jump to the known solution - which does not depend on the exact rectangle-method that has been used. Now: When it comes to functions for which we do not have a known solution, we often have to fall back to what is called "numerical integration". In this case, we use a single, small dx, but we do not take the limit dx->0. Then, we brute-force the approximation by summing up all the individual rectangle results (computers are very good at doing stupid, repetitive tasks very quickly). In this case, the method you propose (f(x0)/2 + f(x1)/2) is indeed considered superior over the simplest approximation (f(x0)). The calculation I showed in my previous post still holds, but N now is a fixed number that does not become arbitrarily large. In practice, the method you proposed is usually the simplest choice for numerical integration that someone with a bit of knowledge about numerical integration will use. Numerical integration routines integrated in programming languages or software libraries will often use even more complicated rules to calculate each rectangle (arguably not even a rectangle, but still dx-sized segments and a representative mean function value for each segment). Bottom line: Don't worry if you don't understand everything in this post. My point is: Your idea about improving the rule to calculate the integral is actually very good. It does not matter much for the definition of the integral (well .. it does in the sense that the definition of the integral would be broken if it gave a different result). But for numerical integration on a computer, your idea is actually very relevant.

Try to formulate your collection of mathematical symbols in English and then try to understand what it says. Then, adapt it according to the other situation. Translate back into mathematical symbols after that.

Yes, both methods approach the same limit. In this case, you can explicitly write that down: Assume you integrate from 0 to 1, and you split the range into N intervals of equal length. In the first case, your integral approximates as [math]I_{1, N} = \frac 1N \sum_{i=0}^{N-1} f(i/N) = \frac 1N \left( f(0) + f(1/N) + f(2/N) + \dots + f((N-1)/N) \right)[/math]. In the second case, your integral approximates as [math]I_{2, N} = \frac 1N \sum_{i=0}^{N-1} \frac 12 \left( f(i/N) + f((i+1)/N) \right) = \frac 1N \left( \frac 12 f(0) + f(1/N) + f(2/N) + \dots + f((N-1)/N) + \frac 12 f(1) \right) [/math]. If you compare the terms, you notice that [math]I_{1, N} - I_{2, N} = \frac 1N \frac 12 (f(0) - f(1) ) = \frac{f(0) - f(1)}{2N}.[/math] So whatever finite numbers f(0) and f(1) are, the difference between the two ways to approximate the integral becomes tiny when N becomes large enough. Btw: This editor is horrible: Preview should preview the rendered tex, not show me the raw tex I typed for different screen sizes. I want my editor from ten years ago back.

I was already chuckling when you posted this, and the thread seems to prove this part of your prediction wrong. In my experience, it is the non-scientific content that get the most attention on sfn. Probably because it is easier to respond to. I certainly put less effort into this post than into science-related posts. Possibly even less than into my one-liner that is the first reply in this thread when I thought this was a genuine question.

I helped my colleagues to move a Volkswagen E-Up for ~50 km between two cities about eight years ago - highway in one direction, smaller roads in the other. The car was used in a field test in the city I lived in, so it made sense for me to just go from/to work by car instead of by train. I drove on a cold but typical German winter day. Turning on the heating approximately halved the remaining range, and I ended up turning heating on and off periodically during the trip. I found that experience quite impressive back then, because I was not aware of such basic issues as heating before. I imagine that there is room for improvement when it comes to thermal insulation, and that newer cars perform better (the E-Up is an electric variant of a combustion engine design, so heating may not have been a big issue in the design). But that's speculation.

I agree that a lot of information is readily available. In fact, I propose that young, critical and open-minded scientists should use these information to answer the questions they might have.

Perhaps reading up on Climate Change would answer some of these questions to the young, critical and open-minded scientist.

In most cases, two days to reply to an email is not considered much (and neither would a week).

I felt the urge to quote this. Fun fact: A Greek influencer named Archimedes became famous for his obsession for lifting heavy objects with little force by using a lever.

I also think the bandwidth to transfer the satellite images that show the whole earth being flooded was very limited back then 📧. On topic, in case it wasn't clear by now: Since all the water for flooding already is on earth, and already pushes as weight on the ground (including the ice), you should expect no significant effect on the stability of the ground when it rains. Also, as studiot said, if all ice melted, the water would not cover all of the land. Here's the first Google hit I found regarding this: https://www.nationalgeographic.com/magazine/article/rising-seas-ice-melt-new-shoreline-maps (seeing the maps it is kind of funny that Australia is one of the few coal power fans in the world).

Where would the water for the rain come from?

What you talk about seems more like a method for calculation to me than an actual measurement. You can indeed calculate the volume of a body by taking a surrounding volume and then subtracting the parts of the surrounding volume that does not belong to the body (in your language: the volume filled with gap values). This may in some cases be an efficient method, e.g. for a block of stone with a cylindrical hole. In the general case, if you have a generic way to calculate the amount of gaps then you could probably use that method to calculate the volume of the body in the first place. This, as you already mentioned, is the tricky part. The most generic way to approach this is dividing a volume in tiny blocks of simply-calculated sub-volumes and use these blocks to approximate the total volume from the sum of these blocks. For example, for your stone you could use small cubes and use laser scans to determine if a cube contains rock or not. If you make these blocks finer and add a few Greek letters, then you get what in Mathematics is called "integration", which is the theoretical basis for such measurements. In many semi-geometric structures you can get good results by gluing together and substracting known geometric structures (as in the case with the block with a cylindrical hole).

I never heard that mentioned as an issue for actual projects. But yes, maybe. How, specifically, do you think that lack of water is a problem for solar power projects?