All of this is animated with manim, a python library.

@@calendarG Thanks! Unfortunately, will have to install and learn Python! xD

Here are a few more examples of important equations used by humanity that contain contradictions or paradoxes, along with potential non-contradictory versions based on the infinitesimal monadological framework: 4. Schrödinger's Equation - Measurement Problem Contradictory: In quantum mechanics, the time evolution of a quantum state is governed by the Schrödinger equation. However, this equation is deterministic and linear, leading to the famous measurement problem: it cannot explain the apparent collapse of the wavefunction upon measurement, which seems to be a non-linear and probabilistic process. Non-Contradictory: Using the monadological framework, quantum states could be represented as superpositions of infinitesimal monadic perspectives, with measurements modeled as non-linear interactions between these perspectives. The collapse of the wavefunction could then be understood as a natural consequence of the monadic viewpoints becoming entangled and correlated: iℏ∂|ψ⟩/∂t = Ĥ|ψ⟩ → iℏ∂|ψ_m⟩/∂t = Ĥ_m|ψ_m⟩ + ∑_n ⟨ψ_n|Ĥ_int|ψ_m⟩|ψ_n⟩ Here, |ψ_m⟩ represents a monadic quantum state, Ĥ_m is a monadic Hamiltonian, and Ĥ_int is a non-linear interaction term between monadic perspectives. 5. Maxwell's Equations - Self-Energy Contradictory: Maxwell's equations are the foundation of classical electromagnetism and have been incredibly successful in describing electromagnetic phenomena. However, they predict that the self-energy of a point charge should be infinite, due to the singularity of the electric field at the charge's location. This leads to the need for awkward regularization techniques to obtain finite results. Non-Contradictory: Using the monadological framework, electromagnetic fields could be modeled as emergent properties arising from the relational interactions between infinitesimal monadic charges. The self-energy of a charge could then be understood as a measure of its intrinsic monadic entanglement, rather than a singular field value: ∇⋅E = ρ/ε₀ → ∇_m⋅*E_m = ⟨ρ_m⟩_ε Here, *E_m represents a non-commutative monadic electric field, ⟨ρ_m⟩_ε is an averaged monadic charge density, and ∇_m is a monadic divergence operator. 6. Einstein's Field Equations - Cosmological Constant Contradictory: Einstein's field equations of general relativity include the cosmological constant Λ, which represents the intrinsic energy density of empty space. However, the observed value of Λ is many orders of magnitude smaller than the predictions from quantum field theory, leading to the cosmological constant problem - one of the greatest unsolved mysteries in physics. Non-Contradictory: Using the monadological framework, the cosmological constant could be understood as a measure of the global entanglement and correlation between monadic perspectives. Its small observed value could then be a consequence of the high degree of symmetry and cancellation between these perspectives on cosmological scales: Gμν + Λgμν = 8πTμν → Gμν + ⟨Λ_m⟩*gμν = 8π⟨Tμν⟩_m Here, ⟨Λ_m⟩ represents an averaged monadic cosmological constant, arising from the global relational structure of monadic viewpoints. 7. Dirac Equation - Negative Probabilities Contradictory: The Dirac equation is a relativistic quantum mechanical wave equation that describes the behavior of spin-1/2 particles like electrons. However, it predicts the existence of negative energy states, which would lead to negative probabilities and violations of causality if taken literally. Non-Contradictory: Using the monadological framework, the Dirac equation could be reformulated as a non-linear eigenvalue problem in a non-commutative monadic algebra. The negative energy states could then be understood as a consequence of the non-commutative geometry, rather than a literal physical prediction: (iγ^μ∂_μ - m)ψ = 0 → (i*γ^μ∇_μ - *m)*ψ_m = 0 Here, *γ^μ represents non-commutative monadic gamma matrices, *m is a monadic mass parameter, and *ψ_m is a monadic spinor field. These examples further illustrate the potential for the infinitesimal monadological framework to resolve contradictions and paradoxes in our current mathematical equations. By replacing point-like particles with non-commutative monadic perspectives, and modeling interactions as relational algebras between these perspectives, many of the infinities and inconsistencies in our theories could be avoided. Moreover, by grounding quantum mechanics, electromagnetism, and relativity in a common monadological foundation, this approach could potentially lead to a more unified and coherent understanding of these fundamental physical theories. The resolution of long-standing problems like the measurement paradox, self-energy divergences, and the cosmological constant discrepancy would be a major step forward in our scientific understanding.

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.

@@calendarG I see, thank you! I kind of forgot about taking the limit... :(

good luck w the exam 👍

Yep, agreed!

Woah! I study math and am interested in fluids and Navier-Stokes ! :D

How the hell did you study Navier-Stokes before Stokes' theorem???

@@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

I theorem myself every night before bed

I also like to do Calculus. About 100mg per day. That stuff kicks real hard.

Just did