Stephen Wolfram

JUNE 28, 2023

Line, Surface and Contour Integration “Find the integral of the function ” is a typical core thing one wants to do in calculus. But particularly in applications of calculus, it’s common to want to ask slightly more elaborate questions, like “What’s the integral of over the region ?”, or “What’s the integral of along the line ?”

Computer 121

Computer Calculus Construction Mathematics 121

Stephen Wolfram

JANUARY 31, 2023

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.

Energy 91

Energy Physics Argumentation Mathematics 91

Stephen Wolfram

SEPTEMBER 27, 2024

Some involve alternate functional forms; others involve introducing additional functions, or allowing multiple arguments to our function f. But it turns out that the fact that this can happen depends critically on the Ackermann function having more than one argument—so that one can construct the “diagonal” f [ m , m , m ].

Computer 112

Computer Mathematics Argumentation Science 112

National Science Foundation

SEPTEMBER 22, 2021

Similarly, reformers have focused on the timing of the course, aiming to enroll students as early as possible to open pathways to calculus and to diversify access to higher level mathematics. Yet we have not seen equal advances in achievement (National Center for Education Statistics, 2019). 2017; Stein et al.,

Algebra 76

Algebra Mathematics Flexibility Research 76

Stephen Wolfram

MARCH 7, 2022

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 at t steps gives a total number of rules equal to: &#10005. which we can read as “there exists something a for which equals a ”.

Mathematics 119

Mathematics Physics Computer Algebra 119

Stephen Wolfram

FEBRUARY 3, 2023

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.

Computer 93

Computer Energy Physics Construction 93

Stephen Wolfram

NOVEMBER 10, 2021

For integers, the obvious notion of equivalence is numerical equality. Then (by the assumed properties of equality) it follows that. But what about other models of computation—like cellular automata or register machines or lambda calculus? And we can trace the argument for this to the Principle of Computational Equivalence.

Physics 121

Physics Mathematics Computer Construction 121