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My students are using Geogebra , Wolfram|Alpha , and Excel every week in Calculus; LaTeX in my proof-oriented classes; Mathematica in my linear algebra and Calculus 3 classes; and so on. This is mostly calculus, linear algebra, and differential equations. Instead, bring it in and teach students how to use it well.
So did that mean we were “finished” with calculus? Somewhere along the way we built out discrete calculus , asymptotic expansions and integral transforms. And in Version 14 there are significant advances around calculus. Another advance has to do with expanding the range of “pre-packaged” calculus operations.
Events are like functions, whose “arguments” are incoming tokens, and whose output is one or more outgoing tokens. Chemistry / Molecular Biology. But in thinking about molecular computing it may be crucial—and perhaps it’s also necessary for understanding molecular biology. There are many.
Events are like functions, whose “arguments” are incoming tokens, and whose output is one or more outgoing tokens. Chemistry / Molecular Biology. But in thinking about molecular computing it may be crucial—and perhaps it’s also necessary for understanding molecular biology. There are many.
The global structures of metamathematics , economics , linguistics and evolutionary biology seem likely to provide examples—and in each case we can expect that at the core is the ruliad, with its unique structure. But what about other models of computation—like cellular automata or register machines or lambda calculus?
The fall of 2021 involved really leaning into the new multicomputational paradigm , among other things giving a long list of where it might apply : metamathematics, chemistry, molecular biology, evolutionary biology, neuroscience, immunology, linguistics, economics, machine learning, distributed computing.
You can give Threaded as an argument to any listable function, not just Plus and Times : ✕. we’re adding SymmetricDifference : find elements that (in the 2-argument case) are in one list or the other, but not both. Now we can use the path function to make a “spiralling” tour video: College Calculus. In Version 13.1
But in 1798 Benjamin Thompson (Count Rumford) (1753–1814) measured the heat produced by the mechanical process of boring a cannon, and began to make the argument that, in contradiction to the caloric theory, there was actually some kind of correspondence between mechanical energy and amount of heat.
Sometimes textbooks will gloss over everything; sometimes they’ll give some kind of “common-sense-but-outside-of-physics argument”. Once one has the idea of “equilibrium”, one can then start to think of its properties as purely being functions of certain parameters—and this opens up all sorts of calculus-based mathematical opportunities.
One can view a symbolic expression such as f[g[x][y, h[z]], w] as a hierarchical or tree structure , in which at every level some particular “head” (like f ) is “applied to” one or more arguments. and zero arguments: α[ ]. ✕. One step of substitution already gives: ✕. ✕. ✕. or: ✕.
It didn’t help that his knowledge of physics was at best spotty (and, for example, I don’t think he ever really learned calculus). Richard Feynman and I would get into very fierce arguments. But suffice it say to that Ed’s old nemesis—calculus—comes in very handy. It’s just my nature.
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