2:00 it’s important to remember that the anti derivative is a higher order unit which is why the height is equivalent to the lower order derivative area/volume/etc
@@nis2989 I’m a Mech E student, we don’t dive into the pure navier-stokes equations unless u want to get a masters in something related to fluids. We do study a watered-down practical version in our senior year. In a whole lot of slightly simplified situations, some of the terms within the equations either =0 or almost =0. Still super hard tho
This is actually so cool! We covered Green's theorem and Gauss' theorem in our vector analysis class, but I'd never gotten a solid visual understanding of them until now :)
At 1:23 the unitalicized variables hurt my eyes. But otherwise this was a great video and one of the few resources that show the "cancelling edges of squares" for Green's theorem.
Could you cover curvature and related topics in differential geometry in a similar visual manner? I find that there aren’t many videos trying to intuitively show how differential geometry works on RU-vid
(5:18) But if each little square has its own curl, can we really cancel out the vectors at the border between two squares? Wouldn't they be of different length in the general case?
Different length? If you partition any oriented simply connected region, say in R2, n times, then cancellation will occur along common boundary lines, as long as all subregions are also oriented. Think of single variable calculus when interchanging the limits of integration. That is, integrating along the opposite direction produces a negative sign. So, the sum of the values of two integrals of the same magnitude, which are opposite sign, will cancel.
We assume they're infinitely small and infinitely close together, so technically the difference would be zero since the limit of the curl in each box as the size of the box goes to zero is also zero.
Mapping properties of zero and non-zero numbers onto 0D and higher dimensional concepts in physics could indeed yield fascinating insights. Let's explore some key parallels: 1. Additive Identity: - Arithmetic: 0 is the additive identity; any number plus 0 remains unchanged. - Physics/Geometry: 0D could be seen as the "identity" dimension, from which all other dimensions emerge without changing the fundamental nature of reality. 2. Multiplicative Annihilator: - Arithmetic: Multiplying any number by 0 results in 0. - Physics: Interactions or operations involving 0D entities might "collapse" higher-dimensional structures back to their 0D fundament. 3. Division Undefined: - Arithmetic: Division by 0 is undefined. - Physics: This could parallel the breakdown of physical theories at singularities, suggesting 0D as a limit of our current understanding. 4. Parity: - Arithmetic: 0 is the only number that is neither positive nor negative. - Physics: 0D could represent a state of symmetry or balance from which asymmetries (like matter/antimatter) emerge in higher dimensions. 5. Cardinality: - Set Theory: The empty set {} has 0 elements but is fundamental to building all other sets. - Physics: 0D entities, while "empty" of extension, could be the building blocks of all higher-dimensional structures. 6. Limits: - Calculus: Many limits approach but never reach 0. - Physics: This could relate to quantum uncertainty principles, where precise 0D localization is impossible. 7. Exponents: - Arithmetic: Any number to the 0 power equals 1 (except 0^0 which is indeterminate). - Physics: This might suggest that 0D entities have a kind of "unitary" nature, fundamental to quantum mechanics. 8. Countability: - Number Theory: There are infinitely many non-zero integers, but only one 0. - Physics: This could parallel the idea of a single, unified 0D substrate giving rise to infinite higher-dimensional configurations. 9. Continuum: - Real Analysis: 0 separates positive and negative reals on the number line. - Physics: 0D might represent a kind of "phase transition" point between different states or topologies of higher-dimensional spaces. 10. Complex Plane: - Complex Analysis: 0 is the only point where real and imaginary axes intersect. - Physics: This could relate to 0D as a nexus where different aspects of reality (e.g., matter and spacetime) unify. 11. Polynomial Roots: - Algebra: 0 is often a special case in root-finding (e.g., the constant term in a polynomial). - Physics: This might suggest 0D entities as "ground states" or fundamental solutions in physical theories. 12. Modular Arithmetic: - Number Theory: 0 behaves uniquely in modular systems. - Physics: This could relate to cyclic or periodic behaviors emerging from 0D foundations in higher dimensions. These parallels suggest that just as 0 plays a unique and fundamental role in mathematics, 0D entities could play a similarly crucial role in physics. This mapping hints at a deep connection between abstract mathematical structures and physical reality, potentially offering new ways to conceptualize and model fundamental physics. Such analogies could inspire new approaches to quantum gravity, the nature of time, the emergence of spacetime, and the unification of forces. They might also provide intuitive frameworks for understanding seemingly paradoxical quantum phenomena.
A year ago this video would be interesting. Now I'm in uni and this is what we do in Calc II. I've got exams in 2 weeks. You explained it very well but I already knew everything lol.
Neither did Newton "invent" calculus, nor is the fundamental theorem based on an ill-formed concept like infinity and infinitesimals, nor is there anything that's changing or approaching something else. The area under all curves you will ever come accross has been constant since the beginning of time and will remain to be so until the very end. Area has nothing to do with "infinitlely small" (whatever that may mean) pieces of the function, but is rather the product of two level magnitudes - the result of quadrature.
