Learning math is learning a language, being exact and precise is the key, which requires some efforts to fully understand and practice using them. So one needs to know basis vector, basis covector, their components to achieve math fluency! d is a map (an operator)! Here is its official definition: So make it crystal clear … Continue reading To Master Mathematical Language, Exactness Is Key

Representation theory is the study of how abstract algebraic structures (like groups, algebras, Lie groups, Lie algebras) can be represented concretely as linear transformations (matrices) acting on vector spaces. Concisely, homomorphism into linear transformations! Let's talk about Lorentz representation, Spinor representation and Lie Algebra version to grasp this concept. there are three rotations Ji and … Continue reading Representation and Gauge Theory

Let's go step by step to see exactly why AμA_\mu​ must appear, and why that object is the photon.

Physics asks: how does something change as you move? Geometry asks: what does “moving” even mean if space has structure? Gauge theory answers both by saying: Forces arise because “comparison across space” is not trivial. ConceptPhysical MeaningGeometric MeaningGauge Potential ($A_\mu$)The Photon/Gluon fieldThe Connection (how to compare fibers)Field Strength ($F_{\mu\nu}$)The Electric/Magnetic ForceThe Curvature (how much the … Continue reading Force as Curvature

Math and physics are not about calculation first; they are about insight. Their purpose is to discover deep and abstract connections beneath complex phenomena, where many seemingly different things turn out to be the same at a structural level. Once such an insight is found, symbols appear—not as decoration, but as compression. Symbols allow complex … Continue reading Math and Physics: The Art of Seeing Deep Connections

On a smooth manifold, vector lives in the tangent vector space, while 1-form lives in the cotangent space. Note lot of confusion comes from the understanding of the basis vector and basis covetor. Vectors transform with Jacobian, 1-forms transform with the inverse transpose Jacobian. On a non-flat surface (a manifold), basis vectors are no longer … Continue reading Vectors vs. 1-forms: Two Different Spaces

On group level, commutation can fail, but not associativity! so x(yz) = (xy)z! How and why? From the other perspective, introducing the function to evaluate we got

The crucial tool connecting all $k$-forms is the Exterior Derivative d. This derivative is the mathematical foundation for generalizing the integral theorems of vector calculus into one single theorem: Stokes' Theorem.

Imagine the sphere as Earth, and vector filed are arrows painted on the surface.